Problems - Set 1

Conversion
1. What is the temperature degree Celsius of absolute zero?
A. -32 C. 273
B. 0 *D. -273

2. How many degrees Celsius is 100 degrees Fahrenheit?

*A. 37.8°C C. 1.334°C
B. 2.667°C D. 13.34°C

3. A comfortable room temperature is 72°F. What is this temperature, expressed in

degrees Kelvin?
A. 263 *C. 295
B. 290 D. 275

4. 225°C is equivalent to:

*A. 491°F C. 173.67°F
B. 427°F D. 109.67°F

5. At what temperature will be the °C and °F readings be equal?

A. 40° C. 32°
*B. -40° D. 0°

6. How many degrees Celsius is 80 degrees Fahrenheit?

*A. 26.67 C. 33.33
B. 86.4 D. 16.33

7. What is the absolute temperature of the freezing point of water in degree

Rankine?
A. -32 C. 428
B. 0 *D. 492

8. The angle of inclination of the road is 32°. What is the angle of inclination in mils?
A. 456.23 C. 125.36
*B. 568.89 D. 284.44

9. The angle measures x degrees. What is the measure in radians?

A. 180°x/π C. 180°π/x
*B. πx/180 D. 180πx

10. Express 45° in mils.

A. 80 mils C. 8000 mils
*B. 800 mils D. 80000 mils

11. What is the value in degrees of π radians?

A. 90° *C. 180°
B. 57.3° D. 45°

12. How many degrees is 3200 mils?

A. 360° *C. 180°
B. 270° D. 90°
13. An angular unit equivalent to 1/400 of the circumference of a circle is called:
A. Mil C. Radian
*B. Grad D. Degree

14. Carry out the following multiplication and express your answer in cubic meters
3cm x 5mm x 2m.
A. 3 x 10-3 C. 8 x 10-2
*B. 3 x 10 -4 D. 8 x 102

15. Add the following and express in meters: 3m + 2cm + 70mm.

A. 2.90 m C. 3.12 m
B. 3.14 m *D. 0.09 m

16. One nautical mile is equivalent to:

A. 5280 km C. 1.256 statute mile
B. 6280 km *D. 1.854 km

17. How many square feet is 100 square meters?

A. 328.10 *C. 1075. 84
B. 956.36 D. 1563.25

18. A tank contains 1500 gallons of water. What is the equivalent in cubic meters?
A. 4.256 C. 6
B. 5.865 *D. 5.685

19. How many cubic feet is equivalent to 100 gallons of water?

A. 74.80 *C. 13.37
B. 1.337 D. 133.7

20. How many cubic meters is 100 gallons of liquid?

A. 0.1638 cu. meter *C. 0.3785 cu. meter
B. 1.638 cu. meter D. 3.7850 cu. Meter

21. The number of board feet in a plank 3 inches thick, 1 foot wide & 20 feet long is:
A. 30 C. 120
*B. 60 D. 90

22. Which of the following is correct?

A. 1 horsepower = 748 kW
B. 1 horsepower = 0.746 watts
*C.1 horsepower = 0.746 kW
D. 1 horsepower = 546 watts

23. The acceleration due to gravity in English unit is equivalent to:

*A. 32.2 ft/sec2 C. 9.81 ft/sec2
B. 3.22 ft/sec2 D. 98.1 ft/sec2

24. The prefix nano is opposite to:

A. Mega C. Hexa
B. Tera *D. Giga

25. 10 to the 12th power is the value of the prefix:

A. Giga *C. Tera
 B. Pico D. Peta

26. Convert 630 degrees to its value in centesimal system.

A. 700 grade C. 750 grade
B. 800 grade *D. 850 grade

27. When rounded-off to four significant figures, 102.48886 becomes:

A. 102.4 C. 102.4888
B. 102.4889 *D. 102.5

28. Which of the following is a prime number?

A. 221 *C. 523
B. 63 D. 703

29. The number of significant figures in the value 500.00 is:

A. One *C. Five
B. Three D. Six

30. A line on the map was drawn at a scale of 5:100,000. If a line in the map is 290
mm long, the actual length of the line is:
A. 4.8 km C. 2.9 km
*B. 5.8 km D. 6.4 km

31. Which of following is correct?

A. 0.001 has three significant figures.
B. 100 has three significant figures.
*C. 100.00 has five significant figures.
D. 0.000249 has six significant figures.

32. The scale on the map is 1:x. a lot having an area of 64 sq. m. is represented by
an area of 25.6 cm2 on the map. What is the value of x?
*A. 500 C. 50
B. 1000 D. 10
Problems – Set 2
Exponents and Radicals

1. Solve for x: x = -(1/-270)-2/3

A. 9 *C. -9
B. 1/9 D.-1/9

2. Solve for a in the equation: a = 64x4y.

A. 4x+3y C. 256xy
B. 43xy *D. 43x+y

3. Simplify 3x - 3x-1 – 3x-2.

A. 3x-2 *C. 5 x 3x-2
B. 33x-3 D. 13 x 3x

4. Which of the following is true?

A. √−2 x √−2 =2 C. √10 = √5 + √2
B. 24 =4√6 *D. 55 + 55 + 55 + 55 + 55 = 56

5. Solve for x: x = √18 - √72 + √50

A. -2√2 C. 4
*B. 2√𝟐 D. 4√3

6. Solve for x: √𝑥 − √1 − 𝑥 = 1 - √𝑥.

A. -16/25 and 0 C. -25/16 and 0
B. 25/16 and 0 *D. 16/25 and 0

3 3 3
7. Simplify √2𝑥 4 - √16𝑥 4 + 2√54𝑥 4
3 𝟑
A. 5 √𝑥 4 *C. 5 √𝟐𝒙𝟒
3 3
B. 2 √5𝑥 4 D. 2 √𝑥 4

8. Solve for x: 3x 5x-1 = 6x+2.

A. 2.1455 C. 2.4154
B. 2.1445 *D. 2.1544

(𝑎−2 𝑏 3 )2
9. Simplify .
𝑎2 𝑏 −1
A. a-2b7 *C. a-6b7
B. a2b5 D. a-6b5

10. (10x)x is equal to:

𝟐
*A. 𝟑𝒙 C. 3xx
B. 3xxx D. 32x

11. Solve for x: 37x=1 = 6561

*A. 1 C. 3
B. 2 D. 4

12. If 3a = 7b, then 3a2/7b2 =

A. 1 *C. 7/3
B. 3/7 D. 49/9
13. Solve for 𝑈 if 𝑈 = √1 − √1 − √1 − …
A. 0.723 C. 0.852
*B. 0.618 D. 0.453

14. If 𝑥 to the ¾ power equals 8, then 𝑥 equals:

A. -9 C. 9
B. 6 *D. 16

15. If 33𝑦 = 1, what is the value of 𝑦/33?

*A. 0 C. infinity
B. 1 D. 1/33

16. Find the value of 𝑥 that will satisfy the following expression:
√𝑥 − 2 = −√𝑥 + 2
A. 𝑥 = 3/2 *C. 𝒙 = 𝟗/𝟒
B. 𝑥 = 18/ D. none of these

17. 𝑒 −3 is equal to:

A. 0.048787 *C. 0.049787
B. 0.049001 D. 0.048902

18. b to the m/nth power is equal to:

*A. 𝒏𝒕𝒉 root of 𝒃 to the 𝒎 power
B. 𝑏 to the m + 𝑛 power
C. 1/𝑛 square root of 𝑏 to the 𝑚 power
D. 𝑏 to the 𝑚 power over 𝑛

19. Find 𝑥 from the following equations:

27𝑥 = 9𝑦
𝑦 −𝑥
81 3 = 243
A. 2.5 *C. 1
B. 2 D. 1.5

20. Solve for 𝑎 from the following equation:

(𝑎𝑚 )(𝑎𝑛 ) = 100,000 𝑎𝑚𝑛 = 100,000
𝑎𝑛
= 10
𝑎𝑚
*A. 15.85 C. 12
B. 10 D. 12.56

1
−3 −
(𝑥 2 𝑦 3 𝑧 −2 ) (𝑥 −3 𝑦𝑧 3 ) 2
21. Simplify 5
(𝑥𝑦𝑧 −3 )−2
𝟏 1
*A. 𝐱𝟐 𝐲𝟕 𝐳𝟑 C. 𝑥 2 𝑦 5 𝑧 3
1 1
B. 𝑥 2 𝑦 7 𝑧 5 D. 𝑥 5 𝑦 7 𝑧 3

22. Simplify the following: 7𝑎+2- 8(7𝑎+1 ) + 5(7𝑎 ) + 49(7𝑎+2 ).

A. -5𝑎 *C. -𝟕𝒂
 B. 3 D. 7𝑎

4 3
𝑥𝑦 −1 𝑥 2 𝑦 −2
23. Simplify: (𝑥 −2 𝑦 3 ) ÷ (𝑥 −3 𝑦 3 )
A. 𝑥𝑦 3 C. 𝑥 3 𝑦
𝑦 𝟏
B. 𝑥 3 *D. 𝒙𝟑 𝒚

√5 – √3
24. Simplify the following:
√5 + √3
A. 4 + √15 C. 8 + √8
*B. 4 - √𝟏𝟓 D. 8 - √8

𝑛 𝑚
25. Which of the following is equivalent to √ √𝑎
𝑚
A. √𝑎𝑛 C. √𝑎𝑚𝑛
𝑛 𝒎𝒏
B. √𝑎𝑚 *D. √𝒂

26. Find the value of 𝑥 in (35 )(96 ) = 32𝑥 .

*A. 8.5 C. 9.5
B. 9 D. 8

𝑥𝑥
27. Solve for 𝑥: 𝑥 𝑥 = 10
*A. 1.2589 C. 1.1745
B. 2.4156 D. Cannot be solved
Problems – Set 3
Fundamentals in Algebra
1. Change 0.2272727… to a common fraction.
A. 7/44 *C. 5/22
B. 5/48 D. 9/34

2. What is the value of 7! of 7 factorial?

*A. 5040 C. 5020
B. 2540 D. 2520

3. The reciprocal of 20 is:

A. 0.50 C. 0.20
B. 20 *D. 0.05

4. If p is an odd number and q is an even number, which of the following expression

must be even?
A. p + q *C. pq
B. p – q D. p/q

5. MCMXCIV is a Roman Numeral equivalent to:

A. 2974 C. 2174
B. 3974 *D. 1994

6. What is the lowest common factor of 10 and 32?

A. 320 C. 180
*B. 2 D. 90

7. 4xy – 4x2 – y2 is equal to:

A. (2x – y)2 C. (-2x + y)2
B. (-2x + y)2 *D. –(2x – y)2

8. Factor x4 – y2 + y – x2 as completely as possible.

A. (x2 + y)(x2 + y – 1) C. (x2 – y)(x2 – y – 1)
B. (x2 + y)(x2 – y – 1) *D. (x2 – y)(x2 + y – 1)

9. Factor the expression x2 + 6x + 8 as completely as possible.

A. (x + 8)(x – 2) C. (x + 4)(x – 2)
*B. (x + 4)(x + 2) D. (x – 8)(x – 2)

10. Factor the expression x3 + 8 as completely as possible.

A. (x – 2)(x2 – 2x + 4) C. (-x + 2)(-x2 + 2x + 2)
2
B. (x + 4)(x + 2x + 2) *D. (x + 2)(x2 – 2x + 4)

11. Factor the expression (x4 – y4) as completely as possible.

A. (x + y)(x2 + 2xy + y2) *C. (x2 + y2)(x + y)(x – y)
B. (x2 + y2)(x2 – y2) D. (1 + x2)(1 + y)(1 – y2)

12. Factor the expression 3x2 + 3x2 – 18x as completely as possible.

*A. 3x(x + 2)(x + 3) C. 3x(x – 3)(x + 6)
B. 3x(x – 2)(x + 3) D. (3x2 – 6x)(x – 1)

13. Factor the expression 16 – 10x + x2.

A. (x + 8)(x – 2) C. (x – 8)(x + 2)
 *B. (x – 8)(x – 2) D. (x – 8)(x + 2)

14. Factor the expression x6 – 1 as completely as possible.

A. (x + 1)(x – 1)(x4 + x2 – 1)
B. (x + 1)(x – 1)(x4 + 2x2 +1)
C. (x + 1)(x – 1)(x4 – x2 + 1)
*D. (x + 1)(x – 1)(x4 + x2 +1)

15. The roots of the equation (x – 4)2 (x + 2) = (x + 2)2 (x – 4)

*A. 4 and -2 only C. -2 and 4 only
B. 1 only D. 1, -2, and 4 only

16. If f(x) = x2 + x + 1, then f(x) - f(x-1) =

A. 0 *C. 2x
B. x D. 3
17. Which of the following is not an identity?
A. (x – 1)2 = x2 – 2x + 1
B. (x + 3)(2x – 2) = 2(x2 + 2x – 3)
C. x2 – (x – 1)2 = 2x – 1
*D. 2(x – 1) + 3(x + 1) = 5x + 4

𝑥+3 4𝑥 2 𝑥+9
18. Solve for x: 4 + 𝑥−3 - 𝑥 2 −9 = 𝑥+3
A. -18 = -18 *C. Any value
B. 12 = 12 or -3 = -3 D. -27 = -27 or 0 = 0

19. Solve the simultaneous equations: 3x – y = 6; 9x – y = 12.

A. x = 3; y = 1 C. x = 2; y = 2
*B. x = 1; y = -3 D. x = 4; y = 2

20. Solve algebraically: 4x2 + 7y2 = 32; 11y2 – 3x2 = 41.

A. y = 4, x = ±1, and y = -4, x = ±1
*B. y = +2, x = ±1, and y = -2, x = ±1
C. x = 2, y = 3, and x = -2, y = -3
D. x = 2, y = -2, and x = 2, y = -2

21. Solve for w for the following equations:

3x – 2y + w = 11 x + 5y -2w = -9 2x + y – 3w = -6
A. 1 *C. 3
B. 2 D. 4

22. When (x + 3)(x – 4) + 4 is divided by x – k, the remainder is k. find the value of k.

A. 4 or 2 *C. 4 or -2
B. 2 or -4 D. -4 or -2

23. Find k in the equation 4x2 + kx +1 = 0 so that it will only have one real root.
A. 1 C. 3
B. 2 *D. 4

24. Find the remainder when (x12 + 2) is divided by (x - √3).

A. 652 C. 231
*B. 731 D. 851

25. If 3x3 – 4x2y + 5xy2 + 6y3 is divided by (2y + 3), the remainder is:
 *A. 0 C. 2
B. 1 D. 3

26. If (4x3 + 8y + 18y2 – 4) is divided by ( 2y + 3), the remainder is:

A. 10 C. 12
*B. 11 D. 13

27. Given: f(x) = (x + 3)(x – 4) + 4 when divided by (x – k), the remainder is k. Find k.
A. 2 *C. 4
B. 3 D. -3

28. The polynomial x3 + 4x2 – 3x + 8 is divided by x – 5. What is the remainder?

A. 281 *C. 218
B. 812 D. 182

29. Find the quotient of 3x5 – 4x3 + 2x2 + 36x + 48 divided by x3 – 2x2 + 6.
A. -3x2 – 4x + 8 C. 3x2 – 4x – 8
B. 3x2 + 4x + 8 *D. 3x2 + 6x + 8

30. If 1/x = a + b and 1/y = a – b then x – y is equal to:

A. 1/2a C. 2a/(a2 – b2)
B. 1/2b *D. 2b/(a2 – b2)

31. If x – 1/x = 1, find the value of x3 – 1/x3.

A. 1 C. 3
B. 2 *D. 4

32. If 1/x + 1/y = 3 and 2/x – 1/y = 1, then x is equal to:

A. ½ *C. ¾
B. 2/3 D. 4/3

5𝑥 𝑥+3 2𝑥+1
33. Simplify the following expression: 2𝑥 2 +7𝑥+3 - 2𝑥 2 −3𝑥−2 + 𝑥 2 +𝑥−6
A. 2/(x – 3) C. (x + 3)(x – 1)
B. (x – 3)/5 *D. 4/(x + 3)

3𝑥 2
34. If 3x = 4y then 4𝑦 2 is equal to:
A. ¾ C. 2/3
*B.4/3 D. 3/2

35. Simplify: (a + 1/a)2 – (a – 1/a)2.

A. -4 *C. 4
B. 0 D. -2/a2

36. The quotient of (x5 + 32) by (x + 2) is:

A. x4 – x3 + 8 *C. x4 – 2x3 + 4x2 – 8x + 16
B. x3 + 2x2 – 8x + 4 D. x4 + 2x3 + x2 + 16x + 8

37. Solve the simultaneous equations: y – 3x + 4 + 0, y + x2/y = 24/y

A. x = (-6 + 2√14)/5 or (-6 - 2√14)/5
y = (2 + 6√14)/5 or (-2 + 6√14)/5
B. x = (6 + 2√15)/5 or (6 - 2√15/5
y = (-2 + 6√14)/5 or (-2 - 6√15)/5
 *C. x = (6 + 2√𝟏𝟒)/5 or (6 - 2√𝟏𝟒)/5
y = (-2 + 6√𝟏𝟒)/5 or (-2 - 6√𝟏𝟒)/5
D. x = (6 + 2√14)/5 or (6 - 2√14)/5
y = (-6 - 2√14)/5 or (-6 + 2√14)/5

(𝑥 2 +4𝑥+10) 𝐴 𝐵(2𝑥+2) 𝐶
38. find the value of A in the equation: (𝑥 3 +2𝑥 2 +5𝑥) = 𝑥 + (𝑥 2 +2𝑥+5) + (𝑥 2 +2𝑥+5)
*A. 2 C. -1/2
B. -2 D. ½
𝑥+10 𝐴 𝐵
39. Find A and B such that = 𝑥−2 + 𝑥+2
𝑥 2 −4
A. A = -3; B = 2 *C. A = 3; B = -2
B. A = -3; B = -2 D. A = 3; B = 2

𝑥+2
40. Resolve 𝑥 2 −7𝑥+12 into partial fraction.
A. 6/(x – 4) – 2/(x – 3) *C. 6/(x – 4) – 5/(x – 3)
B. 6/(x – 4) + 7/(x – 3) D. 6/(x – 4) + 5/(x – 3)

41. The arithmetic mean of 80 numbers is 55. If two numbers namely 250 and 850
are removed, what is the arithmetic mean of the remaining numbers?
*A. 42.31 C. 50
B. 57.12 D. 38.62

42. The arithmetic mean of 6 numbers is 17. If two numbers are added to the
progression, the new set of number will have an arithmetic mean of 19. What are
the two numbers if their difference is 4?
A. 21, 29 C. 24, 26
*B. 23, 27 D. 2, 28

43. If 2x – 3y = x + y, then x2 : y2 =
A. 1 : 4 C. 1 : 16
B. 4 : 1 *D. 16 : 1

44. If 1/a : 1/b : 1/c = 2 : 3 : 4, then (a + b + c) : (b + c) is equal to:

*A. 13 : 7 C. 10 : 3
B. 15 : 6 D. 7 : 9

45. Find the mean proportional to 5 and 20.

A. 8 C. 12
*B. 10 D. 14

46. Find the fourth proportional of 7, 12, and 21.

*A. 36 C. 12
B. 10 D. 14

47. If (x + 3) : 10 = (3x – 2) : 8, find (2x – 1).

A. 1 *C. 3
B. 2 D. 4

48. Solve for x: -4 < 3x – 1 < 11

A. 1 < x < -4 C. 1 < x < 4
*B. -1 < x < 4 D. -1 < x < -4
49. Solve for x: x2 + 4x > 12.
*A. -6 > x > 2 C. -6 > x > -2
B. 6 > x > -2 D. 6 > x > 2

50. When the expression x4 + ax3 + 5x2 + bx + 6 is divided by (x – 2), the remainder
is 16. When it is divided by (x + 1) the remainder is 10. What is the value of the
constant a?
*A. -5 C. 7
B. -9 D. 8
Problems – Set 4
Logarithm, Binomial Theorem, Quadratic Equation
log
10 𝑥
1. If 1− log = 2, what is the value of 𝑥?
10 2

A. 1⁄4 C. 4
∗ 𝐁. 𝟐𝟓 D. 5

2. Solve for 𝑥: log 6 + 𝑥 log 4 = log 4 + log(32 + 4𝑥 )

A. 1 *C. 3
B. 2 D. 4

3. Which of the following cannot be used as a base of a system of logarithm?

A. 𝑒 C. 2
B. 10 *D. 1

4. If log 5.2 −1000 = 𝑥, what is the value of 𝑥?

*A. 4.19 C. 3.12
B. 5.23 D. 4.69

5. Find the value of 𝑎 in the equation log 𝑎 2187 = 7/2.

A. 3 *C. 9
B. 6 D. 12

6. If log 2 = 𝑥 and log 3 = 𝑦, find log 1.2.

A. 2𝑥 + 𝑦 *C. 𝟐𝒙 + 𝒚 − 𝟏
2𝑥𝑦
B. D. 𝑥𝑦 − 1
10

log𝑥 𝑦
7. is equal to:
log𝑦 𝑥
A. 𝑥 𝑦 / 𝑦 𝑥 *C. (𝒚 𝐥𝐨𝐠 𝒙)/ (𝒙 𝐥𝐨𝐠 𝒚)
B. ylog 𝑥 - 𝑥 log 𝑦 D. 1

8. If 10𝑎𝑥+𝑏 = 𝑃, what is the value of 𝑥?

*A. (1/a)(𝐥𝐨𝐠 𝑷 − 𝒃) C. (1/a)𝑃10−𝑏
B. (1/a)log( 𝑃 − 𝑏) D. (1/a)log 𝑃1𝑜

9. Find the logarithm of log(𝑎 𝑎)𝑎

A. 2𝑎log 𝑎 C. 𝑎 log 𝑎2
*B. 𝒂𝟐 𝐥𝐨𝐠 𝒂 D. (𝑎 log 𝑎)

10. Solve for 𝑥: 𝑥 = log 𝑏 𝑎 × log 𝑐 𝑑 × log 𝑑 𝑐

*A. 𝐥𝐨𝐠 𝒃 𝒂 C. log 𝑏 𝑐
B. log 𝑎 𝑐 D. log 𝑑 𝑎

11. Find the positive value of 𝑥 if log 𝑥 36= 2

A. 2 *C. 6
B. 4 D. 8

12. Find 𝑥 if log 𝑥 27+ log 𝑥 3 = 2.

 *A. 9 C. 8
B. 12 D. 7

13. Find 𝑎 if log 2 (𝑎 + 2) + log 2 (𝑎 − 2) = 5.

A. 2 *C. 6
B. 4 D. 8

14. Solve for 𝑥 if log 5 𝑥= 3.

A. 115 C. 135
*B. 125 D. 145

15. Find log 𝑃if ln 𝑃 = 8.

A. 2980.96 *C. 3.47
B. 2542.33 D. 8.57

16. If log 8 𝑥 = −𝑛, then 𝑥 is equal to:

A. 8𝑛 *C. 1/𝟖𝒏
B. 1/ 8−𝑛 D. 81/𝑛

17. If 3log10 𝑥 = −𝑛, then 𝑥 is equal to:

3
A. 𝑦 = √𝑥 *C. y = 𝒙𝟑
B. 𝑦 = √𝑥 3 D. 𝑦 = 𝑥

18. Which of the following is correct?

𝟏
A. -2log 7 = 1/49 *C. 𝐥𝐨𝐠 𝟕 (𝟒𝟗) = -2
1
B. log 7 (−2) = 1/49 D. log 7 (49) = 2

19. Log of the 𝑛th root of 𝑥 equals log of 𝑥 to the 1/𝑛 power and also equal to:
𝐥𝐨𝐠( 𝒙) log(𝑥)1/𝑛
∗ 𝐀. C.
𝒏 𝑛
𝐵. 𝑛 log(𝑥) D. (𝑛 − 1) log(𝑥)

20. What is the natural logarithm of 𝑒 to the 𝑥𝑦 power?

A. 1/𝑥𝑦 *C. 𝒙𝒚
B. 2.718/𝑥𝑦 D. 2.718𝑥𝑦

21. The logarithm of negative number is:

A. Irrational number C. imaginary number
*B. Real number D. complex number

22. What is the value of log to the base 10 of 100003.3 ?

*A. 9.9 C. 10.9
B. 99.9 D. 9.5

23. If log 𝑥 2+ log 2 𝑥=2, then the value𝑥 is:

A. 1 C. 3
*B. 2 D. 4
24. log 6 845 = ?
A. 1 C. 3
B. 2 *D. 4

25. The logarithms of the quotient and the product of two numbers are 0.352182518
and 1.556302501, respectively. Find the first number?
*A. 9 C. 11
B. 10 D. 12

26. The sum of the logarithms of two numbers is 1.748188 and the difference of their
logarithms is -0.0579919. One of the numbers is:
A. 9 *C. 8
B. 6 D. 5

𝑒𝑥
27. Solve for 𝑦: 𝑦 = ln 𝑒 𝑥−2.
*A. 2 C. -2
𝐵. 𝑥 D. 𝑥 − 2

28. What is the value of (log 5 to the base 2) + (log 5 to the base 3)?
A. 3.97 C. 9.37
B. 7.39 *D. 3.79

29. The logarithm of negative number is:

A. Irrational number C. imaginary number
B. Real number *D. complex number

30. 38. 5 to the 𝑥 power = 6.5 to the 𝑥 − 2 power, solve for 𝑥 using logarithms.
A. 2.70 *C. -2.10
B. 2.10 D. -2.02

1
31. Find the 6th term of the expansion of (2𝑎 − 3)16
22113 22113
A. − 256𝑎11 C. − 128𝑎11
𝟔𝟔𝟑𝟑𝟗 66339
∗ 𝐁. − 𝟏𝟐𝟖𝒂𝟏𝟏 D. − 256𝑎11

32. In the expansion of (𝑥 + 4𝑦)12, the numerical coefficient of the 5th term is
A. 253,440 C. 63,360
*B. 126,720 D. 506,880

33. The middle term in the expansion of (𝑥 2 − 3)8 is:

A. -70𝑥 8 C. -5670𝑥 8
B. 70𝑥 8 *D. 5670𝒙𝟖

34. The term involving 𝑥 9 in the expansion of (𝑥 2 + 2/𝑥)12 is:

A. 25434𝑥 9 C. 25344𝑥 9
B. 52344𝑥 9 *D. 23544𝒙𝟗

1
35. The constant term in the expansion of (𝑥 + )15 is:
𝑥 3/2
 A. 3003 C. 6435
*B. 5005 D. 7365

36. Find the sum of the coefficients in the expansion of (𝑥 + 2𝑦 − 𝑧)8

*A. 256 C. 1
B.1024 D. 6

37. The sum of the coefficients in the expansion of (𝑥 + 2𝑦 + 𝑧)4 (𝑥 + 3𝑦)5 is:
A. 524,288 C. 131,072
B. 65,536 *D. 262,144

38. The sum of the coefficients in the expansion of (𝑥 + 𝑦 − 𝑧)8

*A. Less than 2 C. from 2 to 5
B. Above 10 D. from 5 to 10

39. What is the sum of the coefficients of the expansion of (2𝑥 − 1)20
A. 1 C. 215
*B. 0 D. 225

40. In the quadratic equation A𝑥 2 + B𝑥 + C = 0, the product of the roots is:

*A. C/A C. –C/A
B. –B/A D. B/A

41. If ¼ and -7/2 are the roots of the quadratic equation A𝑥 2 + B𝑥 + C = 0, what is the
value of B?
A. -28 C. -7
B. 4 *D. 26

42. In the equation 3𝑥 2 + 4𝑥 + (2ℎ - 5) = 0, find ℎ if the product of the roots is 4.

A. -7/2 *C. 17/2
B. -10/2 D. 7/2

43. If the roots of 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 are 𝑢 and𝑣, then the roots of 𝑐𝑥 2 + 𝑏𝑥 + 𝑐 = 0 are:

A. 𝑢 and 𝑣 *C. 1/𝒖 and 1/𝒗
B. – 𝑢 and 𝑣 D. -1/𝑢 and -1/𝑣

44. If the roots of the quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 are 3 and 2 and 𝑎, 𝑏 and 𝑐
are all whole numbers,
A. 12 *C. 2
B. -2 D. 6

45. The equation whose roots are the reciprocals of the roots of 2𝑥 2 − 3𝑥 − 5 = 0 is:
∗ 𝐀. 𝟓𝒙𝟐 − 𝟑𝒙 − 𝟐 = 𝟎 C. 5𝑥 2 − 2𝑥 − 3 = 0
𝐵. 3𝑥 2 − 5𝑥 − 2 = 0 D. 2𝑥 2 − 5𝑥 − 3 = 0

46. The roots of a quadratic equation are 1/3 and ¼. What is the equation?
A. 12𝑥 2 + 7𝑥 + 1 = 0 *C. 𝟏𝟐𝒙𝟐 − 𝟕𝒙 + 𝟏 = 𝟎
B. 12𝑥 2 + 7𝑥 − 1 = 0 D. 12𝑥 2 − 7𝑥 − 1 = 0
47. Find 𝑘 so the expression 𝑘𝑥 2 − 3𝑘𝑥 + 9 is a perfect square.
A. 3 C. 12
*B. 4 D. 6

48. Find 𝑘 so that 4𝑥 2 + 𝑘𝑥 + 1 = 0 will only have one real solution.

A. 1 C. 3
*B. 4 D. 2

49. The only root of the equation 𝑥 2 − 6𝑥 + 𝑘 = 0 is:

*A. 3 C. 6
B. 2 D. 1

50. Two engineering students are solving a problem leading to a quadratic equation.
One student made a mistake in the coefficient of the first-degree term, got roots
of 2 and -3. The other student made a mistake in the coefficient of the constant
term got roots of -1 and 4. What is the correct equation?
A. 𝑥 2 − 6𝑥 − 3 = 0 C. 𝑥 2 + 3𝑥 + 6 = 0
B. 𝑥 2 + 6𝑥 + 3 = 0 *D. 𝒙𝟐 − 𝟑𝒙 − 𝟔 = 𝟎
Problems – Set 5
Age, Mixture, Work, Clock, Number Problems

1. Two times the father’s age is 8 more than six times his son’s age. Ten years ago,
the sum of their ages was 44. The age of the son is;
A. 49 C. 20
*B. 15 D. 18

2. Peter’s age 13 years ago was 1/3 of his age 7 years hence. How old is Peter/
A. 15 *C. 23
B. 21 D. 27

3. A man is 41 years old and in seven years he will be four times as old as his son
is at that time. How old is his son now?
A. 9 *C. 5
B. 4 D. 8

4. A father three times as old as his son. Four years ago, he was four times as old
as his son was at that time. How old is his son?
A. 36 years C. 32 years
B. 24 years *D. 12 years

5. The ages of a mother and her daughter are 45 and 5 years, respectively. How
many years will the mother be three times as old as her daughter?
A. 5 *C. 15
B. 10 D. 20

6. Mary is 24 years old. Mary is twice as old as Ana was when Mary was as old as
Ana is now. How old is Ana?
A. 16 C. 19
*B. 18 D. 20

7. The sum of the parent’s ages is twice the sum of their children’s ages. Five years
ago, the sum of the parent’s ages is four times the sum of their children’s ages.
In fifteen years, the sum of the parent’s ages will be equal to the sum of their
children’s ages. How many children were in the family/
A. 2 C. 4
B. 3 *D. 5

8. Two thousand (2000) kg of steel containing 8% nickel is to be made by mixing

steel containing 14% nickel with steel containing 6% nickel. How much of the
steel containing 14% nickel is needed?
A. 1500kg C. 750 kg
B. 800 kg *D. 500kg

9. A 40-gram alloy containing 35% gold is to be melted with a 20-gram alloy

containing 50% gold. How much percentage of gold is the resulting alloy?

*A. 40% C. 45%

B. 30% D. 35%

10. In what ratio must a peanut costing P 240.00 per kg be mixed with a peanut
costing P 340.00 per kg so that a profit of 20% is made by selling the mixture at
P 360.00 per kg.?
 A. 1 : 2 *C. 2 : 3
B. 3 : 2 D. 3 : 5

11. A 100-kilogram salt solution originally 4% by weight. Salt in water is boiled to

reduce water content until the concentration is 5% by weight salt. How much
water is evaporated?
A. 10 *C. 20
B. 15 D. 25

12. A pound of alloy of lead and nickel weighs 14.4 ounces in water, where lead
losses 1/11 of its weight and nickel losses 1/9 of its weight. How much of each
metal is in the alloy?
A. Lead = 7.2 ounces; Nickel = 8.8 ounces
*B. Lead = 8.8 ounces; Nickel = 7.2 ounces
C. Lead = 6.5 ounces; Nickel = 5.4 ounces
D. Lead = 7.8 ounces; Nickel = 4.2 ounces

13. An alloy of silver and gold weighs 15 oz. in air and 14 oz. in water. Assuming that
silver losses 1/10 of its weight in water and gold losses 1/18 of its weight in
water, how many oz. at each metal are in the alloy/
A. Silver = 4.5 oz.; Gold = 10.5 oz.
*B. Silver = 3.75 oz.; Gold = 11.25 oz.
C. Silver = 5 oz.; Gold = 10 oz.
D. Silver = 2.75 oz.; Gold = 12.25 oz.

14. A pump can pump out a tank at 11 hours. Another pump can pump the same
tank in 20 hours. How long will it take both pumps together to pump out the tank?
A. ½ hours C. 6 hours
B. ½ hours *D. 7 hours

15. Mr. Brown can wash his car in 15 minutes, while his son John takes twice as long
to do the same job. If they work together, how many minutes can they do the
washing?
A. 6 *C. 10
B. 8 D. 12

16. One pipe can fill a tank in 5 hours and another pipe can fill the same tank in 4
hours. A drainpipe can empty the full content of the tank in 20 hours. With all the
three pipes open, how long will it take to fill the tank?
A. 2 hours C. 1.92 hours
*B. 2.5 hours D. 1.8 hours

17. A swimming pool is filled through its inlet pipe and then emptied through its outlet
pipe in a total of 8 hours. If water enter through its inlet and simultaneously
allowed to leave through its outlet, the pool is filled in 7 ½ hours. Find how long
will it take to fill the pool with the outlet closed.
A. 6 *C. 3
B. 2 D. 5

18. Three persons can do a piece of work alone in 3 hours, 4 hours, and 6 hours,
respectively. What fraction of the job can they finish in one hour working
together?
*A. ¾ C. ½
 B. 4/3 D. 2/3

19. A father and his son can dig a well if the father works 6 hours and his son works
12 hours or they can do it if the father works 9 hours and the son works 8 hours.
How long will it take for the son to dig the well alone?
A. 5 hours C. 15 hours
B. 10 hours *D. 20 hours

20. Peter and Paul can do a certain job in 3 hours. On a given day, they worked
together for 1 hour then Paul left and Peter finishes the rest of the work in 8 more
hours. How long will it take for Peter to do the job alone?
A. 10 hours *C. 12 hours
B. 11 hours D. 13 hours

21. Pedro can paint a fence 50% faster that Juan and 20% faster that Pilar and
together they can paint a given fence in 4 hours. How long will it take Pedro to
paint the same fence if he had to work alone?
*A. 10 hrs. C. 13 hrs.
B. 11 hrs. D. 15 hrs.

22. Nonoy can finish a certain job in 10 days, if Imelda will help for 6 days. The same
work can be done by Imelda In 12 days if Nonoy helps for 6 days. If they work
together, how long will it take for them to do the job?
A. 8.9 C. 9.2
*B. 8.4 D. 8

23. A pipe can fill up a tank with the drain open in three hours. If the pipe runs with
the drain open for one hour and then the drain is closed, it will take 45 more
minutes for the pipe to fill the tank. If the drain will be closed right at the start of
the filling, how long will it take for the pipe to fill the tank?
A. 1.15 hours C. 1.325 hours
*B. 1.125 hours D. 1.525 hours

24. Delia can finish a job in 8 hours. Daisy can do it in 5 hours. If Delia worked for 3
hours and then daisy was asked to help her finish it, how long will Daisy have to
work with Delia to finish the job?
A. 2/5 hour C. 28 hours
B. 25/14 hours *D. 1.923 hours

25. A job could be done by eleven workers in 15 days. Five workers started the job.
They were reinforced with four more workers at the beginning of the 6 th day. Find
the total number of days it took them to finish the job.
A. 22.36 C. 23.22
B. 21.42 *D. 20.56

26. On one job, two power shovels excavate 20,000 m 3 of earth, the larger shovel
working for 40 hours and the smaller for 35 hours. Another job, they removed
40,000 m3 with the larger shovel working 70 hours and the smaller working 90
hours. How much can the larger shovel move ion one hour?
*A. 173.91 C. 368.12
B. 347.83 D. 162.22
27. A and B can do a piece of work in 42 days, B and C in 31 days, A and C in 20
days. Working together, how many days can all of them finish the work/
*A. 18.9 C. 17.8
B. 19.4 D. 20.9
28. Eight men can dig 150 ft. of trench in 7 hours. Three men can back fill 100 ft. of
the trench in 4 hours. The time that it will take 10 men to dig and fill 200 ft. of
trench is :
*A. 9.867 hrs. C. 8.967 hrs.
B. 9.687 hrs. D. 8.687 hrs.

29. In two hours, the minute hand of the clock rotates through an angle of:
A. 45° C. 360°
B. 90° *D. 720°

30. In one day (24 hours), how many times will the hour-hand and the minute-hand
of a continuously driven clock be together?
A. 21 C. 23
*B. 22 D. 24

31. How many minutes after 3:00 P.M. will the minute hand of the clock overtakes
the hour hand?
A. 14/12 minutes *C. 16-4/11 minutes
B. 16-11/12 minutes D. 14/11 minutes

32. How many minutes after 10:00 o’clock will the hands of the clock be opposite
each other for the first time?
A. 21.41 *C. 21.81
B. 22.31 D. 22.61

33. What time between the hours of 12:00 noon and 1:00 P.M. would the hour-hand
and the minute-hand of a continuously driven clock be in straight line?
*A. 12:33 pm C. 12:37 pm
B. 12:30 pm D. 12:287 pm

34. At what time after 12:00 noon will the hour-hand and the minute-hand of a clock
first form an angle of 120°?
A. 21.818 C. 21.181
B. 12:21.181 *D. 12:21.818

35. From the time 6;15 PM to the time 7:45 PM of the same day, the minute hand of
a standard clock describe an arc of:
A. 360° *C. 540°
B. 120° D. 720°

36. It is now between 3 and 4 o’clock and in 20 minutes the minute-hand will be as
much as the hour-hand as it is now behind it. What is the time now?
*A. 3:06.36 C. 3:09.36
B. 3:07.36 D. 3:08.36

37. A man left his home at past 3:00 o’clock PM as indicated in his wall clock.
Between two to three hours after, he returned home and noticed that the hands
of the clock interchanged. At what time did he left his home?
A. 3:27.27 C. 3:22.22
*B. 3:31.47 D. 3:44.44
38. The sum of the reciprocals of two numbers is 11. Three times the reciprocal of
one of the numbers is three more than twice the reciprocal of the other number.
Find the numbers.
A. 5 and 6 *C. 1/5 and 1/6
B. 7 and 4 D. 1/7 and ¼

39. If a two-digit number has x for its unit’s digit and y for its ten’s digit, represent the
number.
A. yx C. 10x + y
*B. 10y + x D. x + y

40. One number if five less than the other number. If their sum is 135, what are the
numbers?
A. 70 and 75 *C. 65 and 70
B. 60 and 65 D. 75 and 80

41. In a two-digit number, the unit’s digit is 3 greater that the ten’s digit. Find the
number if it is four times as large as the sum of its digits.
A. 47 *C. 63
B. 58 D. 25

42. Find two consecutive even integers such that the square of the larger is 44
greater than the square of the smaller integer.
*A. 10 and 12 C. 8 and 10
B.12 and 14 D. 14 and 16

43. Twice the middle digit of a three-digit number is the sum of the other two. If the
number is divided by the sum of its digit. The answer is 56 and the remainder is
12. If the digits are reversed, the number becomes smaller by 594. Find the
number.
A. 258 *C. 852
B. 567 D. 741

44. The product of three consecutive integers is 9240. Find the third integer.
A. 20 *C. 22
B. 21 D. 23

45. The product of two numbers is 1400. If three (3) is subtracted from each number,
their product becomes 11175. Find the bigger number.
A. 28 C. 32
*B. 50 D. 40

46. The sum of the digits of a three-digit number is 14. The hundreds digit being 4
times the units digit. If 594 is subtracted from the number, the order of the digits
will be reversed. Find the number.
A. 743 C. 653
B. 563 *D. 842

47. The sum of two numbers is 21, and one number is twice the other. Find the
numbers.
*A. 7 and 14 C. 8 and 13
B. 5 and 15 D. 9 and 12
48. Ten less than four times a certain number is 14. Determine the number.
A. 4 *C. 6
B. 5 D. 7

49. The denominator of a certain fraction is three more than twice the numerator. If 7
is added to both terms of the fraction, the resulting fraction is 3/5. Find the
original fraction.
A. 8/5 C. 13/5
*B. 5/13 D. 3/5

50. Three times the first of three consecutive odd integers is three more than twice
the third. Find the third integer.
A. 9 C. 13
B. 11 *D. 15
Problems-Set 6
Motion, Variation, Percent, Miscellaneous Problems

PROBLEM 6-1 Nonoy left Pikit to drive to Davao at 6:15 PM and arrived at 11:45
PM. If he averaged 30 mph and stopped 1 hour for dinner, how far is
Davao from Pikit.
A. 128 C. 160
*B. 135 D. 256

PROBLEM 6-2 A man fires a target 420 m away and hears the bullet strike 2
seconds after he pulled the trigger. An observer 525 m away from the
target and 455 m from the man heard the bullet strike the target one
second after he heard the report of the rifle. Find the velocity of the bullet.
*A. 525 m/s C. 350 m/s
B. 360 m/s D. 336 m/s

PROBLEM 6-3 A man travels in a motorized banca at the rate of 12 kph from his
barrio to the poblacion and come back to his barrio at the rate of 10 kph. If
his total time of travel back and forth is 3 hours and 10 minutes, the
distance from the barrio to the poblacion is:
*A. 17.27 km C. 12.77 km
B. 17.72 km D. 17.32 km

PROBLEM 6-4 It takes Michael 60 seconds to run around a 440-yard track. How
long does it take Jordan to run around the track if they meet in 32 seconds
after they start together in a race around the track in opposite directions?
A. 58.76 seconds C. 65.87 seconds
*B. 68.57 seconds D. 86.57 seconds

PROBLEM 6-5 Juan can walk from his home to his office at the rate of 5 mph and
back at the rate of 2 mph. What is his average speed in mph?
*A. 2. 86 C. 4.12
B. 3.56 D. 5.89

PROBLEM 6-6 Kim and Ken traveled at the same time at the rate of 20 m/min,
from the same point on a circular track of radius 600 m. If Kim walks along
the circumference and Ken towards the center, find their distance after 10
minutes.
A. 193 m C. 241 m
B. 202 m *D. 258 m

PROBLEM 6-7 Two ferryboats ply back and forth across a river with a constant but
different speeds, turning at the riverbanks without loss of time. They leave
the opposite shores at the same instant, meet for the first time 900 meters
from one shore, and meet for the second time 500 meters from the
opposite shore. What is the width of the river?
A. 1500 m C. 2000 m
B. 1700 m *D. 2200 m

PROBLEM 6-8 A boat takes 2/3 as much time to travel downstream from C to D,
as to return, If the rate of the river’s current is 8 kph, what is the speed of
the boat in still water?
A. 38 *C. 40
 B. 39 D. 41

PROBLEM 6-9 A man rows downstream at the rate of 5 mph and upstream at the
rate of 2 mph. How far downstream should he go if he is to return in 7/4
hours after leaving?
A. 2 mi C. 3 mi
B. 3.5 mi *D. 2.5 mi

PROBLEM 6-10 A jogger starts a course at a steady rate of 8 kph. Five minutes
later, a second jogger the same course at 10 kph. How long will it take for
the second jogger to catch the first?
*A. 20 min C. 30 min
B. 25 min D. 15 min

PROBLEM 6-11 At 2:00 pm, an airplane takes off at 340 mph on an aircraft carrier.
The aircraft carrier moves due south at 25 kph in the same direction as the
plane. At 4:05 pm, the communication between the plane and the aircraft
carrier was lost. Determine the communication range in miles between the
plane and the carrier.
*A. 656 miles C. 557 miles
B. 785 mles D. 412 miles

PROBLEM 6-12 A boat going across a lake 8 km wide proceed 2 km at a certain

speed and then completes the trip at a speed ½ kph faster. By doing this,
the boat arrives 10 minutes earlier than if the original speed had been
maintained. Find the original speed of the boat.
A. 2 kph C. 9 kph
*B. 4 kph D. 5 mph

PROBLEM 6-13 Given that w varies directly as the product of x and y and inversely
as the square of z and that w=4 when x=2, y=6, and z=3. Find w when
x=1, y=4, and z=2.
A. 4 C. 1
B. 2 *D. 3

PROBLEM 6-14 If x varies directly as y and inversely as z, and x=14 when y=17
and z=2, find x when z=4, and y=16.
A. 14 *C. 16
B. 4 D. 8

PROBLEM 6-15 The electrical resistance of a cable varies directly as its length and
inversely as the square of its diameter. If a cable 600 meters long and 25
mm in diameter has a resistance of 0.1 ohm, find the length of the cable
75 mm in diameter with resistance of 1/6 ohm.
A. 600 m C. 800 m
B. 700 m *D. 900 m

PROBLEM 6-16 The electrical resistance offered by an electric wire varies directly
as the length and inversely as the square of the diameter of the wire.
Compare the electrical resistance offered by two pieces of wire of the
same material; one being 100 m long and 5 mm in diameter, and the other
is 500 m long and 3 mm in diameter.
A. R1= 0.57 R2 C. R1= 0.84 R2
 *B. R1= 0.72 R2 D. R1= 0.95 R2

PROBLEM 6-17 The time required for an elevator to lift a weight varies directly with
the weight and the distance through which it is to be lifted and inversely as
the power of the motors. If it takes 20 seconds for a 5-hp motor to lift 50
lbs. through 40 feet, what weight can an 80-hp motor lift through a
distance of 40 feet within 30 seconds?
A. 1000 lbs. C. 1175 lbs.
B. 1150 lbs. *D. 1200 lbs.

PROBLEM 6-18 The time required by an elevator to lift a weight, vary directly with
the weight and the distance through which it is to be lifted and inversely as
the power of the motor. If it takes 30 seconds for a 10-hp motor to lift 100
lbs. through 50 feet, what size of motor is required to lift 800 lbs. in 40
seconds through a distance of 40 feet?
*A. 48-hp C. 56-hp
B. 50-hp D. 58-hp

PROBLEM 6-19 In a certain department store, the monthly salary of a saleslady is

partly constant and varies as the value of her sales for the month. When
the value of her sales for the month is P10,000.00, her salary for the
month is P900.00. When her sales goes up to P12,000.00, her monthly
salary goes up to P1,000.00. What must be the value of her sales for the
month so that her salary for that month will be P2,000.00?
A. P25,000.00 *C. P32,000.00
B. P28,000.00 D. P36,000.00

PROBLEM 6-20 A man sold 100 eggs. 80 of them were sold at a profit of 30% while
the rest were sold at a loss of 40%. What is the percentage gain or loss on
the whole stock?
A. 14% *C. 16%
B. 15% D. 17%

PROBLEM 6-21 The population of the country increases 5% each year. Find the
percentage it will increase in three year.
A. 5% C. 15.15%
B. 15% *D. 15.76%

PROBLEM 6-22 Pedro bought two cars, one for P600,000.00 and the other for
P400,000.00. He sold the first at a gain of 10% and the second at a loss of
12%. What was his total percentage gain or loss?
A. 6% gain *C. 1.20% gain
B. 0% gain D. 6% loss

PROBLEM 6-23 A grocery owner raises the prices of his goods by 10%. Then he
starts his Christmas sale by offering the customers a 10% discount. How
many percent of discount does the customer actually get?
A. nothing C. 9% discount
*B. 1% discount D. they pay 1% more

PROBLEM 6-24 Kim sold a watch for P3,500.00 at a loss of 30% on the cost price.
Find the corresponding loss or gain if he sold it for P5,050.00.
A. 1% lost *C. 1% gain
 B. 10% lost D. 10% gain

PROBLEM 6-25 By selling balut at P5.00 each, a vendor gains 20%. The cost price
of egg rises by 12.5%. If he sells the balut at the same price as before,
find his new gain in percent.
A. 7.5% C. 8%
B. 5% *D. 6.25%

PROBLEM 6-26 The enrollment at college A and college B both grew up by 8% from
1980 to 1985. If the enrollment in college A grew up by 800 and the
enrollment in college B grew up by 840, the enrollment at college B was
how much greater than the enrollment in college A in 1985?
A. 650 C. 483
B. 504 *D. 540

PROBLEM 6-27 A group consists of n and n girls. If two of the boys are replaced by
two other girls, then 49% of the group members will be boys. Find the
value of n.
*A. 100 C. 50
B. 49 D. 51

PROBLEM 6-28 On his Christmas sale, a merchant marked a pair of slipper

P180.00, which is 20% off the normal retail price. If the retail price is 50%
higher than the whole sale price, what is the whole sale price of the
slipper?
A. P18.00 *C. P15.00
B. P17.00 D. P22.50

PROBLEM 6-29 A certain Xerox copier produces 13 copies every 10 seconds. If the
machine operates without interruption, how many copies will it produce in
an hour?
A. 780 C. 1,825
B. 46,800 *D. 4,680

PROBLEM 6-30 At a certain printing plant, each of the machines prints 6

newspapers every second. If all machines work together but
independently without interruption, how many minutes will it take to print
the entire run of 18,000 newspapers? (Hint: let x= no. of machines)
A. 50x *C. 50/x
B. 3000/x D. 3000

PROBLEM 6-31 A manufacturing firm maintains one product assembly line to

produce signal generators. Weekly demand for the generators in 35 units.
The line operates for 7 hours per day, 5 days per week. What is the
maximum production time per unit in hours required for the line to meet
the demand?
*A. 1 hour C. 3 hours
B. 0.75 hour D. 2.25 hours

PROBLEM 6-32 Of the 316 people watching a movie, there are 78 more children
than women and 56 more women the men. The number of men in the
movie house is:
A. 176 *C. 42
 B. 98 D. 210

PROBLEM 6-33 A certain department store has an inventory of Q units of a certain

product at time t=0. The store sells the product at a steady rate of Q/A
units per week, and exhausts the inventory in A weeks. The amount of
product in inventory at any time t is:
*A. Q - (Q/A) t C. Qt - Q/A
B. Q + (Q/A) t D. Qt - (Q/A) t

PROBLEM 6-34 A merchant has three items on sale: namely, a radio for P50, a
clock for P30, and a flashlight for P1. At the end of the day, she
has sold a total of 100 of the three items and has taken exactly P1000 on
the total sales. How many radios did he sell?
A. 80 *C. 16
B. 4 D. 20

PROBLEM 6-35 The price of 8 calculators ranges from P200 to P1000. If their
average price is P950, what is the lowest possible price of any one of the
calculators?
A. 500 *C. 600
B. 550 D. 650

PROBLEM 6-36 A deck of 52 playing cards is cut into two piles. The first pile
contains 7 times as many black cards as red cards. The second pile
contains the number of red cards that is an exact multiple as the number
of black cards. How many cards are there in the first pile?
A. 14 *C. 16
B. 15 D. 17

PROBLEM 6-37 The Population of the Philippines doubled in the last 30 years from
1967 to 1997. Assuming that the rate of population increase will remain
the same in what year will the population triple?
A. 2030 C. 2021
*B. 2027 D. 2025

PROBLEM 6-38 Determine the unit’s digit in the expansion of 3^855.

A. 3 *C. 7
B. 9 D. 1

PROBLEM 6-39 Find the 1987th digit in the decimal equivalent of 1785/9999 starting
from the decimal point.
A. 1 *C. 8
B. 7 D. 5

PROBLEM 6-40 Find the sum of all positive integral factors of 2048.
*A. 4095 C. 4560
B. 3065 D. 1254

PROBLEM 6-41 In how many ways can two integers be selected from the numbers
1,2,3,…50 so that their difference is exactly 5?
A. 50 *C. 45
B. 5 D. 41
PROBLEM 6-42 A box contains 8 white balls, 15 green balls, 6 black balls, 8 red
balls, and 13 yellow balls. How many balls must be drawn to ensure that
there will be three balls of the same color?
A. 8 C. 10
B. 9 *D. 11

PROBLEM 6-43 A shoe store sells 10 different sizes of shoes, each in both high-cut
and low-cut variety, each either rubber or leather, and each with white or
black color. How many different kinds of shoes does he sell?
A. 64 C. 72
*B. 80 D. 92

PROBLEM 6-44 An engineer was told that a survey had been made on a certain
rectangular field but the dimensions had been lost. An assistant
remembered that if the field had been 100 ft longer and 25 ft narrower, the
area would have been increased by 2500 sq. ft, and that if it had been 100
ft shorter and 50 ft wider, the area would have been decreased 5000 sq.
ft. What was the area of the field?
A. 25,000 ft^2 *C. 20,000 ft^2
B. 15,000 ft^2 D. 22,000 ft^2

PROBLEM 6-45 A 10-meter tape is 5 mm short. What is the correct length in

meters?
*A. 9.995 m C. 9.95 m
B. 10.05 m D. 10.005 m

PROBLEM 6-46 The distance between two points measured with a steel tape was
recorded as 916.58 ft. Later, the tape was checked and found to be only
99.9 ft long. What is the true distance between the points?
A. 935.66 ft C. 955.66 ft
B. 966.15 ft *D. 915.66 ft

PROBLEM 6-47 A certain steel tape is known to be 100,000 feet long at a

temperature of 70°F. When the tape is at a temperature of 10° F, what
tape reading corresponds to a distance of 90 ft? Coefficient of linear
expansion of the tape is 5.833ᵡ10^-6 per °F.
A. 85.931 *C. 90.031
B. 88.031 D. 93.031

PROBLEM 6-48 A line was measured with a steel tape when the temperature was
30 °C. The measured length of the line was found to be 1,256.271 feet.
The tape was afterwards tested when the temperature was 10 °C and it
was found to be 100.042 feet long. What was the true length of the line if
the coefficient of expansion of the tape was 0.000011 per °C?
A. 1,275.075 feet *C. 1,256.547 feet
B. 1,375.575 feet D. 1,249.385 feet

PROBLEM 6-49 The standard deviation of the numbers 1, 4, & 7 is:

A. 2.3567 C. 3.2256
*B. 2.4495 D. 3.8876

PROBLEM 6-50 Three cities are connected by roads forming a triangle, all of the
different lengths. It is 30 km around the circuit. One of the roads is 10 km
long and the longest is 10 km longer than the shortest. What is the length
of the longest road?
A. 5 km *C. 15 km
B. 10 km D. 20 km
Problems –Set 7
Progression, Matrix, Determinant, Venn Diagram

PROBLEM 7-1 How many terms of the sequence -9, -6, -3, … must be taken so
that the sum is 66?
A. 13 C. 4
B. 12 *D. 11

PROBLEM 7-2 The sum of the progression 5, 8, 11, 14, … is 1025. How many
terms are there?
A. 22 C. 24
B. 23 *D. 25

PROBLEM 7-3 There are seven arithmetic means between 3 and 35. Find the sum
of all the terms.
A. 169 C. 167
*B. 171 D. 173

PROBLEM 7-4 There are nine (9) arithmetic means between 11 and 51. The sum
of the progression is:
A. 279 C. 376
*B. 341 D. 254

PROBLEM 7-5 The sum of all even numbers from 0 to 420 is:
A. 43410 *C. 44310
B. 44300 D. 44130

PROBLEM 7-6 Which of the following numbers should be changed to make all the
numbers form an arithmetic progression when properly arranged?
A. 27/14 *C. 45/28
B. 33/28 D. 20/14

PROBLEM 7-7 The first term of an arithmetic progression (A. P.) is 6 and the 10th
term is 3 times the second term. What is the common difference?
A. 1 C. 3
*B. 2 D. 4

PROBLEM 7-8 The sum of five arithmetic means between 34 and 42 is:
A. 150 *C. 190
B. 160 D. 210

PROBLEM 7-9 The positive value of a so that 4x, 5x+4, 3x^2-1 will be in arithmetic
progression is:
A. 2 C. 4
*B. 3 D. 5

PROBLEM 7-10 Solve for x if x + 3x + 5x + 7x + … + 49x = 625.

A. 1/4 *C. 1
B. 1/2 D. 1 ¼

PROBLEM 7-11 The 10th term of the series a, a-b, a-2b, … is:
A. a-6b C. 2a-b
*B. a-9b D. a+9b
PROBLEM 7-12 If the sums of the first 13 terms of two arithmetic progressions are
in the ratio 7:3, find the ratio of their corresponding 7th term.
A. 3:7 *C. 7:3
B. 1:3 D. 6:7

PROBLEM 7-13 If 1/x, 1/y, 1/z are in arithmetic progression, then y is equal to:
A. x - z C. (x + z) / 2xz
B. ½ (x + 2z) *D. 2xz / (x + z)

PROBLEM 7-14 Find the 30th term of the A. P. 4, 7, 10 …

A. 88 C. 75
*B. 91 D. 90

PROBLEM 7-15 Find the 100th term of the sequence 1.01, 1.00, 0.99 …
A. 0.05 C. 0.03
B. 0.04 *D. 0.02

PROBLEM 7-16 The sum of all numbers between 0 and 10,000 which is exactly
divisible by 77 is:
A. 546,546 *C. 645,645
B. 645,568 D. 645,722

PROBLEM 7-17 What is the sum of the following finite sequence of terms? 18, 25,
32, 39, …,67.
A. 234 C. 213
B. 181 *D. 340

PROBLEM 7-18 Find x in the series: 1, 1/3, 0.2, x.

A. 1/6 *C. 1/7
B. 1/8 D. 1/9

PROBLEM 7-19 Find the fourth term of the progression ½, 0.2, 0.125, …
A. 0.102 *C. 1/11
B. 1/10 D. 0.099

PROBLEM 7-20 The 10th term of the progression 6/5, 4/3, 3/2, … is:
*A. 12 C. 12//3
B. 10/3 D. 13/3

PROBLEM 7-21 The geometric mean of 4 and 64 is:

A. 48 C. 34
*B. 16 D. 24

PROBLEM 7-22 The geometric mean of a and b is:

*A. √ab C. 1/b
B. (a + b)/2 D. ab/2

PROBLEM 7-23 Determine the sum of the infinite geometric series of 1, -1/5, +1/25,
…?
A. 4/5 C. 4/6
B. 5/7 *D. 5/6
PROBLEM 7-24 There are 6 geometric means between 4 and 8748. Find the sum of
all the terms.
*A. 13120 C. 10250
B. 15480 D. 9840

PROBLEM 7-25 Find the sum of the infinite geometric progression 6, -2, 2/3 …
A. 5/2 C. 7/2
*B. 9/2 D. 11/2

PROBLEM 7-26 Find the sum of the first 10 terms of the Geometric Progression 2,
4, 8, 16, …
A. 1023 C. 1596
*B. 2046 D. 225

PROBLEM 7-27 The 1st, 4th, and 8th terms of an A. P. are themselves geometric
progression (G. P.). What is the common ratio of the G. P.?
*A. 4/3 C. 2
B. 5/3 D. 7/3

PROBLEM 7-28 Determine x so that x, 2x + 7, 10x – 7 will form a geometric

progression.
A. -7 *C. 7
B. 6 D. -6

PROBLEM 7-29 The fourth term of a geometric progression is 189 and the sixth
term is 1701, the 8th term is:
A. 5103 C. 45927
B. 1240029 *D. 15309

PROBLEM 7-30 The sum of three numbers in arithmetical progression is 45. If 2 is

added to the first number, 3 to the second, and 7 to the third, the new
numbers will be in geometrical progression. Find the common difference in
A. P.
A. -5 C. 6
B. 10 *D. 5

PROBLEM 7-31 The geometric mean and the harmonic mean of two numbers are
12 and 36/5 respectively. What are the numbers?
*A. 36 & 4 C. 36 & 8
B. 72 & 8 D. 72 & 4

PROBLEM 7-32 If x, 4x + 8, 30x +24 are in geometric progression, find the common
ratio.
A. 2 *C. 6
B. 4 D. 8

PROBLEM 7-33 A besiege fortress is held by 5700 men who have provisions for 66
days. If the garrison loses 20 men each day, for how many days can the
provision hold out?
A. 60 *C. 76
B. 72 D. 82
PROBLEM 7-34 If one third of the air in the tank is removed by each stroke of an air
pump, what fractional part of the total air is removed in 6 strokes?
*A. 0.9122 C. 0.8211
B. 0.0877 D. 0.7145

PROBLEM 7-35 A rubber ball is dropped from a height of 15 m. On each rebound, it

rises 2/3 of the height from which it last fell. Find the distance traveled by
the ball before it comes to rest.
*A. 75 m C. 100 m
B. 96 m D. 85 m

PROBLEM 7-36 In the recent Bosnia conflict, the NATO forces captured 6400
soldiers. The provisions on hand will last for 216 meals while feeding 3
meals a day. The provisions lasted 9 more days because of daily deaths.
At an average, how many died per day?
A. 15.2 C. 18.3
*B. 17.8 D. 19.4

PROBLEM 7-37 To build a dam, 60 men must work 72 days. If all 60 men are
employed at the start but the number is decreased by 5 men at the end of
each 12-day period, how long will it take to complete the dam?
*A. 108 days C. 94 days
B. 9 days D. 60 days

PROBLEM 7-38 In a benefit show, a number of wealthy men agreed that the first
one to arrive would pay 10 centavos to enter and each later arrival would
pay twice as much as the preceding man. The total amount collected from
all of them was P104,857.50. How many wealthy men had paid?
A. 18 *C. 20
B. 19 D. 21

PROBLEM 7-39 Evaluate the following determinant: 7 8

9 4

A. 64 C. 54
B. 44 *D. -44

PROBLEM 7-40 The following equation involves two determinants:

3 x 2 -1
2 2 = x -3
The value of x is:
A. 1 *C. 4
B. 3 D. 3

PROBLEM 7-41 Evaluate the following determinant:

1 5 -2
2 1 -3
3 -2 1
A. -24 *C. -46
B. 24 D. 46

PROBLEM 7-42 Compute the value of x from the following:

4 -1 2 3
 2 0 2 1
X= 10 3 0 1
14 2 4 5

A. 27 C. 26
*B. -28 D. -29

PROBLEM 7-43 Evaluate the following determinant:

1 4 2 -1
2 -1 0 -3
D = -2 3 1 2
0 2 1 4
A. 5 C. 4
B. -4 *D. -5

PROBLEM 7-44 Given: Matrix A = 1 3

-2 1
Matrix B = -1 -2
-1 1
Find A + 2B.
A. 1 0 C. -1 3
2 1 0 1
B. 1 0 *D. -1 -1
1 3 0 3

PROBLEM 7-45 Elements of matrix B = 1 2

0 -5
Elements of matrix C = 3 6
4 1
Find the elements of the product of the two matrices, matrix BC.
*A. 11 8 C. 12 10
-20 -5 20 -4
B. 15 9 D. 15 15
-22 4 -17 -6

PROBLEM 7-46 Solve for x and y from the given relationship:

1 1 x 2
=
3 2 y 0
*A. x = -2; y = 6 C. x = -2; y = -6
B. x = 2; y = 6 D. x = 2; y = -6

PROBLEM 7-47 In a class of 40 students, 27 students like Calculus and 25 like

Geometry. How many students liked both Calculus and Geometry?
A. 10 C. 11
B. 14 *D. 12

PROBLEM 7-48 A class of 40 took examination in Algebra and Trigonometry. If 30

passed Algebra, 36 passed Trigonometry, and 2 failed in both subjects,
the number of students who passed the two subjects is:
A. 2 *C. 28
B. 8 D. 25
PROBLEM 7-49 The probability for the ECE board examinees from a certain school
to pass the Mathematics subject is 3/7 and that for the Communications
subject is 5/7. If none of the examinees failed in both subjects, and there
are 4 examinees who pass both subjects, how many examinees from the
school took the examination?
*A. 28 C. 26
B. 27 D. 32

PROBLEM 7-50 In a commercial survey involving 1000 persons on brand

preferences, 120 were found to prefer brand x only, 200 persons prefer
brand y only, 150 persons prefer brand z only, 370 prefer either brand x or
y but not z, 450 prefer brand y or z but not x, and 370 prefer either brand z
or x but not y, and none prefer all the three brands at a time. How many
persons have no brand preference with any of the three brands?
A. 120 C. 70
*B. 280 D. 320
Problems – Set 8
Permutation, Combination, Probability

Problem 8-1 How many permutations can be made out of the letters in the word
ISLAND taking four letters at a time?
*A. 360 C. 120
B. 720 D. 24
Problem 8-2 How many 4 digit numbers can be formed without repeating any
digit, from the following digits 1, 2, 3, 4, and 6.
A. 150 C. 140
*B. 120 D. 130
Problem 8-3 How many permutations can be made out of the letters of the word
ENGINEERING?
A. 39,916,800 C. 55,440
*B. 277, 200 D. 3,326,400
Problem 8-4 How many ways can 3 men and 4 women be seated on a bench if
the women are to be together?
A. 720 C. 5040
B. *B. 576 D. 1024
Problem 8-5 In how many ways can 5 people line up to pay their electric bills?
*A. 120 C. 72
B. 1 D. 24
Problem 8-6 In how many ways can 5 people line up to pay their electric bills if
two particular persons refuse to follow each other?
A. 120 C. 90
B. 72 D. 140
Problem 8-7 How many ways can 7 people be seated at a round table?
A. 5040 C. 720
B. 120 D. 840
Problem 8-8 In how many ways relative orders can we seat 7 people at a round
table with a certain 3 people side by side.
A. 144 C. 720
B. 5040 D. 1008
Problem 8-9 In how many ways can we seat 7 people in a round table with a
certain 3 people not in consecutive order?
A. 576 C. 5320
B. 3960 D. 689
Problem 8-10 The captain of a basketball team assigns himself to the 4 th place in
the batting order. In how many ways can be assign the remaining
places to his eight teammates if just three men are eligible for the
first position?
A. 2160 C. 5040
B. 40320 D. 15120
Problem 8-11 In how many ways can PICE chapter with 15 directors choose a
president, a vice-president, a secretary, a treasurer, and an auditor,
if no member can hold more than one position?
A. 630630 C. 360360
B. 3300 D. 3003
Problem 8-12 How many ways can a committee of five may be selected from an
organization with 35 members?
A. 324632 C. 125487
 B. 425632 D. 326597
Problem 8-13 How many line segments can be formed by 13 distinct point?
A. 156 C. 98
B. 36 D. 78
Problem 8-14 In how many ways can a hostess select six luncheon guests from
10 women if she is to avoid having a particular two of them together
at the luncheon?
A. 210 C. 140
B. 84 D. 168
Problem 8-15 A Semiconductor Company will hire 7 men and 4 women. In how
many ways can the company choose from 9 men and 6 women
who qualified for the position?
A. 680 C. 480
B. 840 D. 540
Problem 8-16 How many ways can you invite one or more of five friends to a
party?
A. 25 C. 31
B. 15 D. 62
Problem 8-17 A bag contains 4 red balls, 3 green balls, and 5 blue balls. The
probability of not getting a red ball in the first draw is:
A. 2 C. 1
B. 2/3 D. 1/3
Problem 8-18 Which of the following cannot be probability?
A. 1 C. 1/e
B. 0 D. 0.434343
Problem 8-19 A bag contains 3 white and 5 black balls. If two balls are drawn in
succession without replacement, what is the probability that both
balls are black?
A. 5/28 C. 5/32
B. 5/16 D. 5/14
Problem 8-20 A bag contains 3 white and 5 red balls. If two balls are drawn at
random, find the probability that both are white?
A. 3/28 C. 2/7
B. 3/8 D. 5/15
Problem 8-21 In problem 8-20, find the probability that one ball is white and the
other is red.
A. 15/56 C. 1/4
B. 15/28 D. 225/784

Problem 8-22 In problem 8-20, find the probability that all are of the same color.
A. 13/30 C. 13/38
B. 14/29 D. 15/28
Problem 8-23 The probability that both stages of a two-stage rocket to function
correctly is 0.92. The reliability of the first stage is 0.97. The
reliability of the second stage is:
A. 0.948 C. 0.968
B. 0.958 D. 0.8924
Problem 8-24 Ricky and George each throw two dice. If Ricky gets a sum of 4,
what is the probability that George will get less?
A. 1/2 C. 9/11
 B. 5/6 D. 1/12
Problem 8-25 Two fair dice thrown. What is the probability that the sum shown on
the dice is divisible by 5?
A. 7/36 C. 1/12
B. 1/9 D. 1/4
Problem 8-26 An urn contains 4 black balls and 6 white balls. What is the
probability of getting one black ball and one white ball in two
consecutive draws from the urn?
A. 0.24 C. 0.53
B. 0.27 D. 0.04
Problem 8-27 If three balls are drawn in succession from 5 white and 6 black balls
in a bag, find the probability that all are of one color, if the first ball
is replaced immediately while the second is not replaced before the
third draw.
A. 10/21 C. 28/121
B. 18/121 D. 180/14641

Problem 8-28 A first bag contains 5 white balls and 10 black balls and a second
bag contains 20 white balls and 10 black balls. The experiment
consists of selecting a bag and then drawing a ball from the
selected bag. Find the probability of drawing a white ball.
A. 1/3 C. 1/2
B. 1/6 D. 1/18
Problem 8-29 In problem 8-28, find the probability of drawing a white ball from the
first bag.
A. 5/6 C. 2/3
B. 1/6 D. 1/3
Problem 8-30 If seven coins are tossed simultaneously, find the probability that
they will just have three heads.
A. 33/128 C. 30/129
B. 35/128 D. 37/129
Problem 8-31 If seven coins are tossed simultaneously, find the probability that
there will be at least six tails.
A. 2/1 28 C. 1/6
B. 3/128 D. 2/16
Problem 8-32 A face of a coin is either head or tail. If three coins are tossed, what
is the probability of getting three tails?
A. 1/8 C. 1/4
B. 1/2 D. 1/6
Problem 8-33 The face of a coin is either head or tail. If three coins are tossed,
what is the probability of getting three tails or three heads?
A. 1/8 C. 1/4
B. 1/2 D. 1/6
Problem 8-34 Five fair coins were tossed simultaneously. What is the probability
of getting three heads and two tails?
A. 1/32 C. 1/8
B. 1/16 D. 1/4
Problem 8-35 Throw a fair coin five times. What is the probability of getting three
heads and two tails?
A. 5/32 C. 1/32
B. 5/16 D. 7/16
Problem 8-36 The probability of getting a credit in an examination is 1/3. If three
students are selected at random, what is the probability that at least
one of them got a credit?
A. 19/27 C. 2/3
B. 8/27 D. 1/3
Problem 8-37 There are three short questions in mathematics test. For each
question, one (1) mark will be awarded for a correct answer and no
mark for a wrong answer. If the probability that Mary correctly
answers a question in a test is 2/3, determine the probability that
Mary gets two marks.
A. 4/27 C. 4/9
B. 8/27 D. 2/9
Problem 8-38 A marksman hits 75% of all his targets. What is the probability that
he will hit exactly 4 of his next 10 shots?
A. 0.01622 C. 0.004055
B. 0.4055 D. 0.006122
Problem 8-39 A two-digit number is chosen randomly. What is the probability that
it is divisible by 7?
A. 7/50 C. 1/7
B. 13/90 D. 7/45
Problem 8-40 One box contains four cards numbered 1, 3, 5, and 6. Another box
contains three cards numbered 2, 4, and 7. One card is drawn from
each bag. Find the probability that the sum is even.
A. 5/12 C. 7/12
B. 3/7 D. 5/7

Problem 8-41 Two people are chosen randomly from 4 married couples. What is
the probability that they are husband and wife?
A. 1/28 C. 3/28
B. 1/14 D. 1/7
Problem 8-42 One letter is taken from each of the words PARALLEL and LEVEL
at random. What is the probability of getting the same letter?
A. 1/5 C. 3/20
B. 1/20 D. 3/4
Problem 8-43 In a shooting game, the probabilities that Botoy and Toto will hit a
target is 2/3 and 3/4 respectively. What is the probability that the
target is hit when both shoot at it once?
A. 13/5 C. 7/12
B. 5/13 D. 11/12
Problem 8-44 A standard deck of 52 playing cards is well shuffled. The probability
that the first four cards dealt from the deck will be four aces is
closest to:
A. 4x10-6 C. 3x10-6
B. 2x10-6 D. 8x10-6
Problem 8-45 A card is chosen from a pack of playing cards. What is the
probability that it is either red or a picture card?
 A. 8/13 C. 19/26
B. 10/13 D. 8/15
Problem 8-46 In a poker game consisting of 5 cards, what is the probability of
holding 2 aces and 2 queens?
A. 5!/52! C. 33/54145
B. 5/52 D. 1264/45685
Problem 8-47 Dennis Rodman sinks 50% of all his attempts. What is the
probability the he will make exactly 3 of his next 10 attempts?
A. 1/256 C. 30/128
B. 3/8 D. 15/128
Problem 8-48 There are 10 defectives per 1000 items of a product in a long run.
What is the probability that there is one and only one defective in a
random lot of 100.
A. 0.3697 C. 0.3796
B. 0.3967 D. 0.3679
Problem 8-49 The UN forces for Bosnia uses a type of missile that hits the target
with a probability of 0.3. How many missiles should be fired so that
there is at least an 80% probability of hitting the target?
A. 2 C. 5
B. 4 D. 3
Problem 8-50 In a dice game, one fair die is used. The player wins P20.00 if he
rolls either 1 or 6. He losses P10.00 if he turns up any other faces.
What is the expected winning for one roll of die?
A. P40.00 C. P20.00
B. P0.00 D. P10.00

Problems – Set 9
Complex Numbers, Vectors, Elements

Problem 9-1 In the complex number 3 + 4i; the absolute value is:
A. 10 C. 5
B. 7.211 D. 5.689
Problem 9-2 In the complex number 8 - 2i, the amplitude is:
A. 104.040 C. 345.960
B. 14.040 D. 165.960
Problem 9-3 (6 cis 1200) (4 cis 300) is equal to:
 A. 10 cis 1500 C. 10 cis 900
B. 24 cis 1500 D. 24 cis 900
Problem 9-4 30 cis 800 / 10 cis 1500 is equal to:
A. 20 cis 300 C. 3 cis 300
B. 3 cis 1300 D. 20 cis 1300
Problem 9-5 The value of x + y in the complex equation 3 + xi = y + 2i is:
A. 5 C. 2
B. 1 D. 3
Problem 9-6 Multiply (3 – 2i) (4 + 3i).
A. 12 + i C. 6 + i
B. 18 + i D. 20 + i
Problem 9-7 Divide (4 + 3i) / (2 – i).
A. (11 + 10i) / 5 C. (5 + 2i) / 5
B. (1 + 2i) D. (2 + 2i)
Problem 9-8 Find the value of i 9.

A. i C. 1
B. –i D. -1

Problem 9-9 Simplify i1997 + i1999, where i is an imaginary number.

A. 1 + i C. 1 – i
B. i D. 0
Problem 9-10 Expand (2 + √−9 ) . 3

A. 46 + 9i C. -46 – 9i
B. 46 – 9i D. -46 + 9i
Problem 9-11 Write -4 + 3i in polar form.
A. 5 36.870 C. 5 323.130
B. 5 216.870 D. 5 143.130
Problem 9-12 Simplify: i30 – 2i25 + 3i17.
A. i + 1 C. -1 + i
B. -1 – 2i D. -1 + 5i
Problem 9-13 Evaluate the value of √−10 x √−7 .
A. Imaginary C. √17
B. −√𝟕𝟎 D. √70
3
Problem 9-14 Perform the indicated operation: √−9 x √−343 .
A. 21 C. -21i
B. 21i D. -21
Problem 9-15 What is the quotient when 4 + 8i is divided by i3?
A. 8 + 4i C. 8 – 4i
B. -8 + 4i D. -8 – 4i
Problem 9-16 What is the exponential form of the complex number 4 + 3i?
A. 5ei53.13 C. 7ei53.13
B. 5ei36.87 D. 7ei36.87
Problem 9-17 What is the algebraic form of the complex number 13ei67.38?
A. 12 + 5i C. 12 – 5i
B. 5 – 12i D. 5 + 12i

Problem 9-18 Solve for x that satisfy the equation x2 + 36 = 9 – 2x2.

A. ±6𝑖 C. 9i
B. ±𝟑𝒊 D. -9i
Problem 9-19 Evaluate ln (5 + 12i).
 A. 2.565 + 1.176i C. 5.625 + 2.112i
B. 2.365 – 0.256i D. 3.214 – 1.254i
Problem 9-20 Add the given vectors: (-4, 7) + (5, -9).
A. (1, 16) C. (9, 2)
B. (1, -2) D. (1, 2)
Problem 9-21 Find the length of the vector (2, 1, 4).
A. √17 C. √20
B. √𝟐𝟏 D. √19
Problem 9-22 Find the length of the vector (2, 4, 4).
A. 8.75 C. 7.00
B. 6.00 D. 5.18
Problem 9-23 What is the magnitude of the vector F = 2i + 5j +6k?
A. 6.12 C. 6.18
B. 7.04 D. 8.06
Problem 9-24 Find the value of x in the complex equation: (x + yi) (1 – 2i) = 7 – 4i.
A. 1 C. 4
B. 3 D. 2
Problem 9-25 A statement the truth of which is admitted without proof is called:
A. An axiom C. A theorem
B. A postulate D. A corollary
Problem 9-26 In a proportion of four quantities, the first and the fourth terms are
referred to as the:
A. means C. extremes
B. denominators D. numerators
Problem 9-27 Convergent series is a sequence of decreasing numbers or when
the succeeding term is _______ than the preceding term.
A. ten times more C. equal
B. greater D. lesser
Problem 9-28 It is the characteristic of a population which is measurable.
A. frequency C. sample
B. distribution D. parameter
Problem 9-29 The quartile deviation is a measure of:
A. division C. certainty
B. central tendency D. dispersion
Problem 9-30 In a complex algebra, we use a diagram to represent a complex
plane commonly called:
A. De Moivre’s Diagram C. Argand Diagram
B. Funicular Diagram D. Venn Diagram
Problem 9-31 A series of numbers which are perfect square numbers (i.e. 1, 4, 9,
16, …) is called:
A. Fourier series C. Euler’s number
B. Fermat’s number D. Fibonacci numbers
Problem 9-32 A sequence of numbers where every term is obtained by adding all
the preceding terms such as 1, 5, 14, 30, … is called:
A. Triangular number C. Tetrahedral number
B. Pyramidal number D. Euler’s number
Problem 9-33 A graphical representation of the commulative frequency
distribution in a set of statistical data is called:
A. Ogive C. Frequency polyhedron
B. Histogram D. Mass diagram
Problem 9-34 A sequence of numbers where the succeeding term is greater than
the preceding is called:
A. Dissonant series C. Isometric series
B. Convergent series D. Divergent series
Problem 9-35 The number 0.123123123… is
A. Irrational C. Rational
B. Surd D. Transcendental
Problem 9-36 An array of m x n quantities which represent a single number
system composed of elements in rows and column is known as:
A. Transpose of a matrix C. Co-factor of a matrix
B. Determinant D. Matrix
Problem 9-37 The characteristics is equal to the exponent of 10, when the
number is written in:
A. Exponential C. Logarithmic
B. Scientific notation D. Irrational
Problem 9-38 Terms that differ only in numeric coefficients are known as:
A. Unequal terms C. Like terms
B. Unlike terms D. equal terms
Problem 9-39 ______is a sequence of terms whose reciprocals form an arithmetic
progression:
A. Geometric progression C. Algebraic progression
B. Harmonic progression D. Ratio and proportion
Problem 9-40 The logarithm of a number to the base e (2.718281828…) is called:
A. Naperian logarithm C. Mantissa
B. Characteristic D. Briggsian logarithm
Problem 9-41 The ratio or product of two expressions in direct or inverse relation
with each other is called:
A. Ratio and proportion C. Means
B. Constant of variation D. Extremes
Problem 9-42 In any square matrix, when the elements of any two rows are
exactly the same the determinant is:
A. Zero C. Negative integer
B. Positive integer D. Unity

Problem 9-43 Two or more equations are equal if and only if they have the same:
A. Solution set C. Order
B. Degree D. Variable set
Problem 9-44 What is a possible outcome of an experiment called?
A. A sample space C. An event
B. A random point D. A finite set
Problem 9-45 If the roots of an equation are zero, then they are classified as:
A. Trivial solutions C. Conditional solutions
B. Extraneous roots D. Hypergolic solutions
Problem 9-46 A complex number associated with a phase-shifted sine wave in
polar form whose magnitude is in RMS and angle is equal to the
angle of the phase-shifted sine wave is known as:
A. Argand’s number C. Phasor
B. Imaginary number D. Real number
Problem 9-47 In raw data, the term, which occurs most frequently, is known as:
A. Mean C. Mode
B. Median D. Quartile
Problem 9-48 Infinity minus infinity is:
A. Infinity C. Indeterminate
B. Zero D. None of there
Problem 9-49 Any number divided by infinity is equal to:
A. 1 C. Zero
B. Infinity D. Indeterminate
Problem 9-50 The term in between any to terms of an arithmetic progression is
called:
A. Arithmetic progression C. Middle terms
B. Median D. Mean

Problem 9-51 Any equation which, because of some mathematical process, has
acquired an extra root is sometimes called a:
A. Redundant equation C. Linear equation
B. Literal equation D. Defective equation
Problem 9-52 A statement that one mathematical expression is greater than or
less than another is called:
A. Inequality C. Absolute condition
B. Non-absolute condition D. Conditional expression
Problem 9-53 A relation in which every ordered pair (x, y) has one and only one
value of y that corresponds to the values of x, is called:
A. Function C. Domain
B. Range D. Coordinates
Problem 9-54 An equation in which a variable appears under the radical sign is
called:
A. Literal equation C. Irradical equation
B. Radical equation D. Irrational equation
Problem 9-55 The number of favourable outcomes divided by the number of
possible outcomes is:
A. Permutations C. Combination
B. Probability D. Chance
Problem 9-56 Two factors are considered essentially the same if:
A. One is merely the negative of the other
B. One is exactly the same as the other
C. Both of them are negative
D. Both of them are positive
Problem 9-57 An integer is said to be prime if:
A. It is factorable by any value
B. It is an odd integer
C. It has no other integer as a factor except itself or 1
D. It is an even integer

Problem 9-58 Equations in which the members are equal for all permissible
values of unknowns are called:
A. A conditional equation C. A parametric equation
B. An identity D. A quadratic equation
Problem 9-59 Equations which satisfy only for some values of unknown are
called:
A. A conditional equation C. A parametric equation
B. An Identity D. A quadratic equation
Problem 9-60 The logarithm of 1 to any base is:
A. Indeterminate C. Infinity
B. Zero D. One

Problems - Set 10
Angles, Trigonometric Identities and Equations

Problem 10-1 Find the supplement of an angle whose compliment is 62o

A. 28o C. 152o
B. 118o D. none of these
Problem 10-2 A certain angle has a supplement five times its compliment. Find the
angle.
A. 67.5o C. 168.5o
B. 157.5o D.186o
Problem 10-3 The sum of the two interior angles of the triangle is equal to the third
angle and the difference of the two angles is equal to 2/3 of the third
angle. Find the third angle.
A. 15o C. 90o
B. 75o D. 120o
Problem 10-4 The measure of 1½ revolution counter-clockwise is:
A. 540o C. +90o
B. 520o D. -90o
Problem 10-5 The measure of 2.25 revolutions counterclockwise is:
A. -835 degrees C. 805 degrees
B. -810 degrees D. 810 degrees
Problem 10-6 Solve for θ sin θ – sec θ + csc θ – tan 2θ = -0.866
A. 40o C. 47o
B. 41o D. 43o
Problem 10-7 What are the exact values of the cosine and tangent trigonometric
functions of acute angle A, given that sin A = 3/7?
A. Cos A = 7/2 √10; tan A = 2√10 /3
B. Cos A = 2 √10 /7; tan A = 3√10 / 20
C. Cos A = 2√10 / 3; tan A = 7/2 √10
D. Cos A = 2√10 /3; tan A = 7√10 / 20
Problem 10-8 Given three angles A, B, and C whose sum is 180o. If the tan A + tan B +
tan C = x, find the value of tan A x tan B x tan C.
 A. 1 – x C. x/2
B. √x D. x

Problem 10-9 What is the sine of 820o?

A. 0.984 C. 0.866
B. -0.866 D. -0.500
o
Problem 10-10 Csc 270 = ?
A. -√3 C. √3
B. -1 D1
Problem 10-11 If coversine θ is 0.134, find the value of θ.
A. 60o C. 30o
B. 45 o D. 20o
Problem 10-12 Solve for cos 72o if the given relationship is cos 2A = 2cos2 A – 1
A. 0.309 C. 0.268
B. 0.258 D. 0.315
Problem 10-13 If sin 3A =cos 6B then:
A. A+ B = 180o C. A-2B = 30o
B. A + 2B = 30o D. A + B = 30o
Problem 10-14 Find the value of sin (arccos 15/17)
A. 8/17 C. 8/21
B. 17/9 D. 8/9
Problem 10-15 Find the value of cos [arcsin (1/3) + arctan(2/ √5)]
A. (2/9)(1+√10) C. (2/9)(√10 + 1)
B. (2/9)(√10-11) D. (2/9)(√10-1)
Problem 10-16 If sin 40 + sin 20 = sin θ, find the value of θ.
o o

A. 20o C. 120o
B. 80o D. 60o
Problem 10-17 How many different value of x from 0o to 180o for the equation
(2sin x–1)(cos x + 1) = 0?
A. 3 C. 1
B. 0 D. 2

Problem 10-18 For what value of θ (less than 2π) ill the following equation be satisfied?
Sin2θ + 4sin θ + 3 = 0
A. π C. 3π/2
B. π/4 D. π/2
Problem 10-19 Find the value of x in the equation csc x + cot x = 3
A. π/4 C. π/2
B. π/3 D. π/5
Problem 10-20 If sec A is 5/2, the quantity 1-sin2 A is equivalent to:
2

A. 2.5 C. 1.5
B. 0.6 D. 0.4
Problem 10-21 Find sin x if 2 sin x + 3 cos x – 2 = 0.
A. 1 & -5/13 C. 1 & 5/13
B. -1 & 5/13 D. -1 & -5/13
Problem 10-22 If sin A = 4/5. A in quadrant II, sin B = 7/25, B in quadrant I, find
Sin (A + B )
A. 3/5 C. 3/4
B. 2/5 D. 4/5
Problem 10-23 If sin A = 2.571x, cos A = 3.06x, and sin 2A = 3.939x, find the value of x.
A. 0.350 C. 0.100
 B. 0.250 D. 0.150
Problem 10-24 If cos θ = √3 /2, then find the value of x if x = 1 – tan2θ.
A. -2 C. 4/3
B. -1/3 D. 2/3
Problem 10-25 If sin θ – cos θ = -1/3, what is the value of sin 2θ?
A. 1/3 C. 8/9
B. 1/9 D. 4/9
Problem 10-26 If x cos θ + y sin θ = 1 and x sin θ – y cos θ = 3, what is the relationship
between x and y?
A. X2 + y2 = 20 C. x2 + y2 = 16
B. X2 – y2 = 5 D. x2 + y2 + 10

Problem 10-27 If sin x + 1 / sin x = √2, then sin2 x + 1 / sin2 x is equal to:
A. √2 C. 2
B. 1 D. 0
Problem 10-28 The equation 2 sin θ + 2 cos θ – 1 = √3 is:
A. An identity C. a conditional equation
B. a parametric equation D. a quadratic equation

Problem 10-29 If x + y= 90o, then sin x tan y / sin y tan x is equal to:
A. tan x C. cot x
B. cos x D. sin x
Problem 10-30 If cos θ = x /2, then 1-tan θ is equal to:
2

A. (2x2 + 4)/x2 C. (2x2 - 4)/x

B. (4 - 2x2)/x2 D. (2x2 - 4)/x2
Problem 10-31 Find the value in degrees of arccos (tan 24o).
A. 61.48 C. 63.56
B. 62.35 D. 60.84
Problem 10-32 arctan [2 cos (arcsin (√3 / 2)] is equal to:
A. π/3 C. π/6
B. π/4 D. π/2
Problem 10-33 Solve for x in the equation: arctan (2x) + arctan (x) = π/4
A. 0.821 C. 0.281
B. 0.218 D. 0.182
Problem 10-34 Solve for x from the given trigonometric equation:
Arctan (1-x) + arctan (1+x) = arctan 1/8
A. 4 C8
B. 6 D. 2
Problem 10-35 Solve for y if y = (1/sin x – 1/tan x)(1+cos x)
A. Sin x C. tan x
B. Cos x D. sec2 x

Problem 10-36 Solve for x: x=(tanθ + cotθ)2sin2θ-tan2θ

A. Sin θ C. 1
B. Cos θ D. 2
Problem 10-37 Solve for x: x=1-(sinθ-cosθ) 2

A. Sinθcosθ C. cos 2θ
B. -2 cos θ D. sin 2θ
Problem 10-38 Simplify cos θ – sin θ
2 2

A. 2 C. 2sin2θ + 1
B. 1 D. 2cos2θ – 1
Problem 10-39 Solve for x: x = 1-tan2a / 1+tan2
 A. Cos a C. cos 2a
B. Sin 2a D. sin a
Problem 10-40 Which of the following is different from the others?
A. 2cos2x – 1 C. cos3x – sin3x
B. Cos4x – sin4x D. 1 – 2sin2x
Problem 10-41 Find the value of y: y = (1+cos2θ) tanθ
A. Cos θ C. sin 2θ
B. Sin θ D. cos 2θ
Problem 10-42 The equation 2 sinh x cosh x is equal to:
A. ex C. sinh 2x
B. e -x D. cosh x
Problem 10-43 Simplifying the equation sin2θ (1+ cot2θ) gives:
A. 1 C. sin2θcos2θ
B. Sin2θ D. cos2θ
o
Problem 10-44 Find the value of sin (90 + A)
A. Cos A C. sin A
B. -cos A D. -sin A
Problem 10-45 Which of the following expression is equivalent to sin 2θ?
A. 2sinθcosθ C. cos2θ – sin2θ
B. Sin2θ + cos2θ D. sinθ cosθ

Problem 10-46 If tan θ = x2, which of the following is incorrect?

A. Sin θ = 1/ √1+x4 C. cos θ = 1 / √1 + x4
B. Sec θ = √1+x4 D. cscθ = √1+x4 / x2
Problem 10-47 In an isosceles right triangle, the hypotenuse is how much longer than
its sides?
A. 2 times C. 1.5 times
B. √2 times D. none of these
Problem 10-48 Find the angle in mils subtended by a line 10 yards long at a distance of
5000 yards.
A. 2.5 mils C. 4 mils
B. 2 mils D. 1 mil
Problem 10-49 The angle or inclination of ascend of a road having 8.25% grade is __
degrees.

A. 5.12 degrees C. 1.86 degrees

B. 4.72 degrees D. 4.27 degrees
Problem 10-50 The sides of a right triangle is in arithmetic progression whose common
difference if 6cm. Its area is:
A. 216cm2 C. 360cm2
B. 270cm2 D. 144cm2
Problems – Set 11
Triangles, Angle of Elevation & Depression

Problem 11-1 The hypotenuse of a triangle is 34 cm. Find the length of the shortest leg
if it is 14cm shorter than the other leg.
A. 15 cm C. 17 cm
B. 16 cm D. 18 cm
Problem 11-2 A truck travels from point M northward for 30 mi, then eastward for one
hour, then shifted N 30o W. If the constant speed is 40 kph, how far
directly from M, in km. will be it after 2 hours?
A. 43.5 C. 47.9
B. 45.2 D. 41.6
Problem 11-3 To sides of a triangle measure 6 cm, and 8 cm, and their included angle
is 40o. Find the third side.
A. 5.144 cm C. 4.256 cm
B. 5.263 cm D. 5.65 cm
o
Problem 11-4 Given a triangle: C = 100 , a = 15, b= 20. Find c:
A. 34 C. 43
B. 27 D. 35
o o
Problem 11-5 Given angle A=32 , angle B=70 , and side c=27 units. Solve for side a of
the triangle.
A. 24 units C. 14.63 units
B. 10 units D. 12 units
Problem 11-6 In a triangle, find the side c if angle C=100o, side B=20, and side a = 15.
A. 28 C. 29
B. 27 D. 26
Problem 11-7 In triangle ABC, A =45 degrees and angle C=70 degrees. The side
opposite angle C is 40 m long. What is the side opposite angle A?
A. 29.10 meters C. 30.10 meters
B. 32.25 meters C. 31.25 meters

Problem 11-8 Two sides of a triangle are 50 m. and 60 m. long. The angle included
between these sides is 30 degrees. What is the interior angle (in
degrees) opposite the longest side?
A. 92.74 C. 94.74
B. 93.74 D. 91.74

Problem 11-9 The sides of a triangle ABC are AB=15 cm, BC=18 cm. and CA=24 cm.
Determine the distance from the point of intersection of the angular
bisectors to side AB.
A. 5.21 cm C. 4.73 cm
B. 3.78 cm D. 625 cm
Problem 11-10 If AB=15 m, BC=18 m, and CA=24 m, find the point intersection of the
angular bisector from the vertex C.
A. 11.3 C. 13.4
B. 12.1 D. 14.3
Problem 11-11 In triangle ABC, angle C = 70 degrees; angle A = 45 degrees; AB = 40
m. What is the length of the median drawn from vertex A to side BC?
A. 36.8 meters C. 36.3 meters
B. 37.1 meters D. 37.4 meters
Problem 11-12 The area of the triangle whose angles are 61 o9’32”, 34o14’46”, and
84o35’42” is 680.60. The length of the longest side is:
A. 35.53 C. 52.43
B. 54.32 D. 62.54
Problem 11-13 Given a triangle ABC whose angles are A=40o, B=95o and side b=30
cm. Find the length of the bisector of angle C.
A. 21.74 C. 20.45
B. 22.35 D. 20.98
Problem 11-14 The sides of a triangular lot are 130 m, 180 m, and 190 m. The lot is to
be divided by a line bisecting the longest side and drawn from the
opposite vertex. The length of this dividing line is:
A. 100 meters C. 125 meters
B. 130 meters D. 115 meters
Problem 11-15 From a point outside of an equilateral triangle, the distance to the
vertices are 10m, 10m, and 18 m. Find the dimension of the triangle
A. 25.63 C. 19.94
B. 45.68 D. 12.25
Problem 11-16 Points A and B 1000m apart are plotted on a straight highway running
east and west From A, the bearing of a tower C is 32 degrees N of W
and from B the bearing of C is 26 degrees N of E. Approximate the
shortest distance of tower C to the highway.
A. 264 meters C. 284 meters
B. 274 meters D. 294 meters
Problem 11-17 An airplane leaves an aircraft carrier and flies South at 350 mph. The
carrier travels S 30o E at 25 mph. If the wireless communication range of
the airplane is 700miles, when will it lose contact with the carrier?
A. After 4.36 hours C. after 2.13 hours
B. After 5.57 hours D. after 4.54 hours
Problem 11-18 A statue 2 meters high stands on a column that is 3 meters high. An
observer in level with the top of the statue observed that the column and
the statue substend the same angle. How far is the observer from the
statue?
A. 5√2 meters C. 20 meters
B. 2√5 meters D. √10 meters
Problem 11-19 From the top of a building 100 m. high, the angle of depression of a
point A due east of it is 30o. From a point B due South of the building,
the angle of elevation of the top is 60o. Find the distance AB.
A. 100+3√30 C. 100√30 /3
B. 200-√30 D. 100-√3 /30
Problem 11-20 An observer found the angle of elevation of the top of the tree to be 270.
After moving 10m closer (on the same vertical and horizontal plane as
the tree), the angle of elevation become 54o. Find the height of the tree
A. 8.65 meters C. 7.02 meters
B. 7.53 meters D. 8.09 meters
Problem 11-21 From a point A at the foot of the mountain, the angle of elevation of the
top B is 60o. After ascending the mountain one (1) mile at an inclination
of 30o to the horizon, and reacting a point C, an observer finds that the
angle ACB is 135o. The height of the mountain in feet is:
A. 14386 C. 11672
B. 12493 D. 11225
Problem 11-22 A 50-meter vertical tower casts a 62.3-meter shadow when the angle of
elevation of the sun is 41.6o. The inclination of the ground is:
A. 4.72o C. 5.63o
 B. 4.33o D. 5.17o
Problem 11-23 A vertical pole is 10 m from a building. When the angle of elevation of
the sun is 45o, the pole cast a shadow on the building 1 m high. Find the
height of the pole.
A. 0 meter C. 12 meters
B. 11 meters D. 13 meters

Problem 11-24 A pole cast a shadow of 15 meters long when the angle of elevation of
the sun is 61o. If the pole has leaned 15o from the vertical directly toward
the sun, what is the length of the pole?
A. 52.43 meters C. 53.25 meters
B. 54.23 meters D. 53.24 meters
Problem 11-25 Am observer wishes to determine the height of a tower. He takes sights
at the top of the tower from A and B, which are 50ft apart, at the same
elevation on a direct line with the tower. The vertical angle at a point A is
30o and at a point B is 40o. What is the height of the tower?
A. 83.6 ft C. 110.29 ft
B. 143.97 ft D. 92.54 ft
Problem 11-26 From the top of tower A, the angle of elevation of the top of the tower B
is 46o. From the foot of tower B the angle of elevation of the top of tower
A is 28o. Both towers are on a level ground. If the height of tower B is
120m, how high is tower A?
A. 38.6 m C. 44.1 m
B. 42.3 m D. 40.7 m
Problem 11-27 Point A and B are 100 m apart and are on the same elevation as the foot
of a building. The angles of elevation of the top of the building from
points A and B are 21o and 32o respectively. How far is A from the
building?
A. 271.6 m C. 259.2 m
B. 265.4 m D. 277.9 m
Problem 11-28 A man finds the angle of elevation of the top of a tower to be 30
degrees. He walks 85 m nearer the tower and finds it angle of elevation
to be 60 degrees. What is the height of the tower?

A. 76.31 meters C. 73.31 meters

B. 73.61 meters D. 73.16 meters
Problem 11-29 The angle of elevation of a point C from a point B is 29o42’; the angle of
elevation of C from another point A 31.2 m directly below B is 59 o23’.
How high is C from the horizontal line through A?
A. 47.1 meters C. 35.1 meters
B. 52.3 meters D. 66.9 meters
Problem 11-30 A rectangular piece of land 40m x 30m is to be crossed diagonally by a
10-m wide roadway as shown If the land cost P1,500.00 per square
meter, the cost of the roadway is:
A. P401.10
B. P60,165.00
C. P601,650.00
D. P651,500.00

Problem 11-31 A man improvise a temporary

shield from the sun using a triangular
piece of wood with dimensions of 1.4m, 1.5m, and 1.3 m. With the longer
side lying horizontally on the ground, he props up the other corner of the
 triangle with a vertical pole 0.9m long. What would be the area of the
shadow on the ground when the sun is vertically overhead?
A. 0.5 m2 C. 0.84 m2
B. 0.75 m 2 D. 0.95 m2
Problem 11-32 A rectangular piece of wood 4cm x 12cm tall is tilted at an angle of 45
degree. Find the vertical distance between the lower corner and the
upper corner.
A. 4√2 cm C. 8√ 2cm
B. 2√2 cm D. 6√ 2cm
Problem 11-33 A clock has a dial face 12 inches in radius. The minute hand is 9 inches
long while the hour hand is 6 inches long. The plane of rotation of the
minute hand is 2inches above the plane of rotation of the hour hand.
Find the distance between the tips of the hands at 5:40 PM.
A. 9.17 inches C. 10.65 inches
B. 8.23 inches D. 11.25 inches
Problem 11-34 If the bearing of A from B is S 40o W, then the bearing of B from A is:
A. N 40o E C. N 50o E
B. N 40o W D. N 50o W
Problem 11-35 A plane hillside is inclined at an angle of 28 degrees with the horizontal>
A man wearing skis can climb this hillside by following a straight path
inclined at an angle of 12o to the horizontal, but one without skis must
follow a path inclined at an angle of only 5o with the horizontal. Find the
angle between the directions of the two paths.
A. 13.21o C. 15.56o
B. 18.74o D. 17.22o
Problem 11-36 Calculate the area of a spherical triangle whose radius is 5 m and whose
angles are 40o, 65o, and 110o.
A. 12.24 sq. m. C. 16.45 sq. m.
B. 14.89 sq. m. D. 15.27 sq. m.
Problem 11-37 A right spherical triangle has an angle C=90o, a=50o, and c=80o. Find
the side b.
A. 45.33o C. 74.33o
B. 78.66 o D. 75.89o
Problem 11-38 If the time is 8:00 a. GMT, hat is the time in the Philippines, which is
located at 120o East longitude?
A. 6 p.m. C. 4 p.m.
B. 4 a.m. D. 6 a.m.
Problem 11-39 An airplane flew from Manila (14o36’N, 121o05’ E) at a course of S 30o E
maintaining a certain altitude and following a great circle path. If its
groundspeed is 350knots, after how many hours will it cross the
equator?
A. 2.87 hours C. 3.17 hours
B. 2.27 hours D. 3.97 hours
Problem 11-40 Find the distance in nautical miles between Manila and San Francisco,
Manila is located at 14o 36’ N latitude and 121o 05’ E longitude. San
Francisco is situated at 37o48’N latitude and 122o24’W longitude.
A. 7856.2 nautical miles C. 6326.2 nautical miles
B. 5896.2 nautical miles D. 6046.2 nautical miles
Problem 11-41 At a certain point on the ground, the tower at the top of a 20-m high
building subtends an angle of 45o. At another point on the ground 25 m
closer the building, the tower subtends an angle of 45o. Find the height
of the tower.
A. 124.75 m C. 154.32 m
 B. 87.45 m D. 101.85 m
Problem 11-42 A wooden flagpole is embedded 3m deep at a corner A of a concrete
horizontal slab ABCD, square in form and measuring 20 ft on a side. A
storm broke the flagpole at a point one meter above the slab and inclined
toward corner C in the direction of the diagonal AC. The vertical angles
observed at the center of the slab and at corner C to the tip of the
flagpole were 65o and 35o, respectively. What is the total length of the
flagpole above the slab in yards?
A. 5.61 C. 4.32
B. 16.83 D. 12.38
Problem 11-43 From the third floor window of a building, the angle of depression of an
object on the ground is 35o58’, while from a sixth floor window, 9.75 m
above the first point of observation the angle of depression is 58 o35’.
How far is the object from the building?
A. 11.9 m C. 9.3 m
B. 10.7 m D. 15.3 m
Problem 11-44 The sides of a triangle are 18 cm, 24 cm, and 34 cm, respectively. Find
the length of the median to the 24-cm side in cm.
A. 24.4 C. 23.4
B. 21.9 D. 20.4
Problem 11-45 In the spherical triangle ABC, A=116o19’, B=55o30’, and C=80o37’. What
is the value of side a?
A. 115.57o C. 119.64o
B. 113.21o D. 115.65o
Problems – Set 12
Triangles, Quadrilaterals, and Polygons

Problem 12 – 1 The sides of a right triangle have lengths (a – b),a , and (a + b).
What is the ratio of a to b if a is greater than b and b could not be
equal to zero?
A. 1 : 4 C. 1: 4
B. 3 :1 D. 4 : 1

Problem 12 – 2 Two sides of a triangle measure 8 cm and 12 cm. Find its area if its
perimeter is 26 cm.
A. 21.33 sq. m. C. 3.306 sq. in.
B. 32.56 sq. cm. D. 32.56 sq. in.

Problem 12 – 3 If three sides of an acute triangle is 3 cm, 4 cm and “x” cm. what
are the possible value of x?
A. 1 < x < 5 C. 0 < x < 7
B. 0 < x < 5 D. 1 < x > 7

Problem 12 – 4 In triangle ABC, AB = 8m and BC = 20m. One possible dimension

CE May 1998 of CA is:
A. 13 C. 9
B. 7 D. 11

Problem 12 – 5 In a triangle BCD, BC = 25 m. and CD = 10 m. The perimeter of the

CE Nov. 1996, triangle may be.
Nov. 1996 A. 72 m. C. 69 m.
B. 70 m. D. 71 m.

Problem 12 – 6 The sides of a triangle ABC are AB = 25 cm, BC = 39 cm, and AC =

CE May 1999 40 cm. Find its area.
A. 486 sq. cm. C. 648 sq. cm.
B. 846 sq. cm. D. 468 sq. cm

Problem 12 – 7 The corresponding sides of the triangle are in the ratio of 3 : 2.

What is the ratio of their areas?
A. 3 C. 9/4
B. 2 D. 3/2

Problem 12 – 8 Find the area of the triangle whose sides are 12, 16, and 21 units.
CE Nov 1997 A. 95.45 sq. units C. 87.45 sq. units
 B. 102.36 sq. units D. 82.78 sq. units

Problem 12 – 9 The sides of a right triangle are 8, 15 and 17 units. If each side is
doubled, how many square units will be the area of the new
triangle?

A. 240 C. 320
B. 300 D. 420

Problem 12 – 10 Two triangles have equal bases. The altitude of one triangle is 3
ECE Nov. 1997 units more than its base and the altitude of the other is 3 units less
than its base. Find the altitudes if the areas of the triangle is differ
by 21 square units.
A. 5 & 11 C. 6 & 12
B. 4 & 10 D. 3 & 9

Problem 12 – 11 A triangle piece of wood having a dimension 130 cm, 180 cm and
190 cm is to be divided by a line bisecting the longest drawn from
its opposite vertex. The area of the part adjacent to the 180-cm side
is:
A. 5126 sq. cm C. 5612 sq. cm
B. 5162 sq. cm D. 5216 sq. cm

Problem 12 – 12 Find EB if the area of the inner

triangle is 1/4 of the outer triangle
as shown.
A. 32.5
B. 55.7
C. 56.2
D. 57.5

Problem 12 – 13 A piece of wire is shaped to enclose a square whose area is 169

ECE Nov. 1997 cm2. It is then reshaped to enclose a rectangle whose length is 15
cm. The area of the rectangle is:
A. 165 cm2 C. 170 cm2
B. 175 cm2 D. 156 cm2

Problem 12 – 14 The diagonal of the floor of a rectangular room is 7.50 m. The

shorter side of the room is 4.5 m. What is the area of room?
A. 36 sq. m. C. 58 sq. m.
B. 27 sq. m. D. 24 sq. m.

Problem 12 – 15 A man measuring a rectangle “x” meters by “y” meters, makes each
side 15% too small. By how many percent will his estimate for the
area be too small?
 A. 23.55% C. 27.75%
B. 25.67% D. 72.25%

Problem 12 – 16 The length of the side of a square is increased by 100%. Its

perimeter is increased by:
A. 25% C. 200%
B. 100% D. 300%

Problem 12 – 17 A Piece of wire of length 52 cm is cut into two parts. Each part is
then bent to form a square. It is found that total area of the two
squares is 97 sq. cm. The dimension of the bigger square is:
A. 4 C. 3
B. 9 D. 6

Problem 12 – 18 In the figure shown, ABCD is

a square and PDC is an
equilateral triangle, Find 0.
A. 5°
B. 15°
C. 20°
D. 25°

Problem 12 – 19 One side of a parallelogram is 10 m and its diagonals are 16 m and

24 m, respectively, Its area is:
A. 156.8 sq. cm. C. 158.7 sq. cm.
B. 185. 6 sq. m D. 142.3 sq. m.

Problem 12 – 20 If the sides of the parallelogram and an included angle are 6, 10

ECE April 1998 and 100 degrees respectively, find the length of the shorter
diagonal.
A. 10.63 C. 10.73
B. 10.37 D. 10.23

Problem 12 – 21 The area of a rhombus is 132 square cm. If its shorter diagonal is
12 cm, the length of the longer diagonal is:
A. 20 centimeter C. 22 centimeter
B. 21 centimeter D. 23 centimeter

Problem 12 – 22 The diagonals of a rhombus are 10 cm, and 8 cm., respectively, Its
area is:
A. 10 sq. cm C. 60 sq. cm
B. 50 sq. cm D. 40 sq. cm

Problem 12 – 23 Given a cyclic quadrilateral whose sides are 4 cm, 5 cm, 8 cm, and
11 cm. Its area is:
A. 40.25 sq. cm. C. 50.25 sq. cm
 B. 48.65 sq. cm. D. 60.25 sq. cm

Problem 12 – 24 A rectangle ABCD which measures 18 by 24 units is folded once,

ECE Nov. 1995 perpendicular to diagonal AC, so that the opposite vertices A and C
coincide. Find the length of the fold.
A. 2 C. 54 / 2
B. 7 / 2 D. 45 / 2

Problem 12 – 25 The sides of a quadrilateral are 10m, 8m, 16m, and 20m,
CE Nov. 1996, respectively. Two opposite interior angles have a sum of 225°. Find
CE May 1997 the area of the quadrilateral in sq. m.
A. 140.33 sq.cm. C. 150.33 sq. cm.
B. 145.33 sq.cm. D. 155.33 sq. cm.

Problem 12 – 26 A non-square rectangle is inscribed in a square so that each vertex

ECE Nov. 1996 of the rectangle is at the trisection point of the different sides of the
square. Find the ratio of the area of the rectangle to the area of the
square.
A. 5:9 C. 7:72
B. 2:7 D. 4:9

Problem 12 – 27 A trapezoid has an area of 36 m2 and altitude of 2 m. Its two bases

ECE April 1998 in meters have ratio of 4:5, the bases are:
A. 12, 1 C. 16, 20
B. 7, 11 D. 8, 10

Problem 12 – 28 Determine the area of the quadrilateral ABCD shown if OB = 80 cm,

OA = 120 cm, OD = 150
cm and 0 = 25°
A. 2272 sq. cm
B. 7222 sq. cm
C. 2572 sq. cm
D. 2722 sq. cm

Problem 12 – 29 A corner lot of hand is 35 m on one street and 25 m on the other

street. The angle between the two lines of the street being 82°. The
other to two lines of the lot are respectively perpendicular to the
lines of the streets. What is the worth of the lot if its unit price is
P2500 per square meter?
A. P1, 978,456 C. P2,234,023
B. P1, 588,045 D. P1,884,050
Problem 12 – 30 Determine the area of the quadrilateral having (8, -2), (5, 6), (4, 1),
and (-7, 4) as consecutive vertices.
A. 22 sq. units C.32 sq. units
B. 44 sq. units D. 48 sq. units

Problem 12 – 31 Find the area of the shaded

portion shown if AB is parallel
to CD.
A. 16 sq. m.
B. 18 sq. m.
C. 20 sq. m.
D. 22 sq. m.

Problem 12 – 32 The deflection angles of any polygon has a sum of:

A. 360° C. 180°(n – 3)
B. 720° D. 180°n

Problem 12 – 33 The sum of the interior angles of a dodecagon is:

A. 2160° C. 1800°
B. 1980° D. 2520°

Problem 12 – 34 Each interior angle of a regular polygon is 165°. How many sides?
A. 23 C. 25
B. 24 D. 26

Problem 12 – 35 The sum of the interior angles of a polygon is 540°, Find the
ECE March 1996 number of sides.
A. 4 C. 7
B. 6 D. 5

Problem 12 – 36 The sum of the interior angles of a polygon of n sides is 1080°. Find
CE May 1997 the value of n.
A. 5 C. 7
B. 6 D. 8

Problem 12 – 37 How many diagonals does a pentadecagon have:

A. 60 C. 80
B. 18 D. 90

Problem 12 – 38 A polygon has 170 diagonals. How many sides does it have?
A. 20 C. 25
B. 18 D. 26
Problem 12 – 39 A regular hexagon with an area of 93.53 square centimeters is
CE Nov, 1999 inscribed in a circle. The area in the circle not covered by hexagon
is:
A. 18.83 cm2 C. 19.57 cm2
B. 16.72 cm 2 D. 15.68 cm2

Problem 12 – 40 The area of a regular decagon inscribed in a circle of 15 cm

diameter is:
A. 156 sq. cm. C. 165 sq. cm.
B. 158 sq. cm D. 177 sq. cm.

Problem 12 – 41 The sum of the interior angle of a polygon is 2,520°, How many are
the sides?
A. 14 C. 16
B. 15 D. 17

Problem 12 – 42 The area of a regular hexagon inscribed in a circle of radius 1 is:

ME April 1997 A. 2,698 sq. units C. 3, 698 sq. units
B. 2,598 sq. units D. 3, 598 sq. units

Problem 12 – 43 The corners of a 2-meter square are cut off to form a regular
octagon. What is the length of the sides of the resulting octagon?
A. 0.525 C. 0.727
B. 0.626 D. 0.828

Problem 12 – 44 If a regular polygon has 27 diagonals then it is a:

ECE Nov. 1997 A. hexagon C. pentagon
B. nonagon D. heptagon

Problem 12 – 45 One side of a regular octagon is 2. Find the area of the region
ECE Nov. 1997 inside the octagon
A. 19.3 sq. units C. 21.4 sq. units
B. 13. 9 sq. units D. 31 sq. units

Problem 12 – 46 A regular octagon is inscribed in a circle of radius in 10. Find the

ECE Nov. 1997 area of the octagon.
A. 228.2 sq. units C. 282.8 sq. units
B. 288.2 sq. units D. 238.2 sq. units

Problem 12 – 47 Lot ABCDEA is a closed traverse in the form of a regular hexagon

CE May 2003 with each side equal to 100 m. The bearing of AB is N 25°E. What
is the bearing of CD?
A. S 35°E C. S 30°E
B. S 45°E D. S 40°E

Problem 12 – 48 Two sides of a parallelogram measure 68 cm and 83 cm and the

CE May 2001 shorter diagonal is 42 cm. Determine the smallest interior angle of
the parallelogram.
 A. 23. 87° C. 49.45°
B. 30.27° D. 12.65°

Problem 12 – 49 The parallel sides of a trapezoid lot measure 160 m and 240 m and
are 40 m apart. Find the length of the dividing line parallel to the
two sides that will divide the lot into two equal areas.
A. 203.96 C. 200
B. 214.25 D. 186.54

Problem 12 – 50 What do you call a polygon with 35 diagonals?

A. decagon C. dodecagon
B. dodecagon D. undecagon
Problems – Set 13
Circles, Parabola, Ellipse, &Miscellaneous Figures

PROBLEM 13-1 In the figure shown OA and OB are tangent to the circle.
If ∠AOB and 50°, find the ∠APB.
A. 45°
B. 50°
C. 60°
D. 65°

PROBLEM 13-2 In the figure shown, arc BC is half the length of arc CD.
Solve for θ.
A. 30°
B. 40°
C. 45°
D. 50°

PROBLEM 13-3 In the figure shown, MAN is a tangent to the circle of

center O. If ∠BAN = 70° find θ.
A. 220°
B. 210°
C. 140°
D. 250°

PROBLEM 13-4 In the figure shown, AB is the diameter of the circle.

Find the θ.
A. 100°
B. 105°
C. 110°
D. 210°

PROBLEM 13-5 The area of a circle 89.42 square inches. What is the
Circumference.
A. 35.33 inches
B. 32.25 inches
C. 33.52 inches
D. 35.55 inches

PROBLEM 13-6 Find the area of the circle shown.

A. 152.53 sq. units
B. 193.30 sq. units
C. 215.30 sq. units
D. 226.98 sq. units
PROBLEM 13-7 A circle whose area is 452 cm square units is into two
Segment by a chord whose distance from the center of
the circle is 6 cm. Find the area of the larger segment in
cm square.
 A. 372.5 C. 368.4
B. 363.6 D. 377.6

PROBLEM 13-8 A circle is divided into two parts by a chord, 3cm away from
the center. Find the area of the smaller part, in cm square, if
the circle has an area of 201 cm square.
A. 51.4 C. 55.2
B. 57.8 D. 53.7

PROBLEM 13-9 A quadrilateral ABCD is inscribed in a semi-circle with side AD.

coinciding with the diameter of a circle. If the sides AB, BC, and
CD are 8cm, 10cm, and 12cm long, respectively, find the area of
the circle.
A. 317 sq. cm. C. 456 sq. cm.
B. 356 sq. cm. D. 486 sq. cm.

PROBLEM 13-10 A semi-circle of radius 14 cm is formed from a piece of wire. If it

is bent into a rectangle whose length is 1 cm more than its width,
find the area of the rectangle.
A. 256.25 sq. cm. C. 386.54 sq. cm.
B. 323.57 sq. cm D. 452.24 sq. cm.

PROBLEM 13-11 Find the area of the shaded circle shown.

2
A. 3 π𝑟 2
2
B. π𝑟 2
9
4
C. π𝑟 2
3
𝟒
D. π𝒓𝟐
𝟗

PROBLEM 13-12 The angle of a sector is 30 degrees and the radius is 15 cm.
What is the area of the sector?
A. 89.5 cm2 C. 59.8 cm2
𝟐
B. 58.9 𝐜𝐦 D. 85.9 cm2

PROBLEM 13-13 A sector has a radius of 12 cm. If the length of its arc is 12 cm.
its area is:
A. 66 sq. cm. C. 144 sq. cm.
B. 82 sq. cm. D. 72 sq. cm.

PROBLEM 13-14 The perimeter of a sector is 9 cm and its radius is 3 cm. What is
the area of the sector?
11
A. 4 cm2 C. 2 cm2
𝟗 27
B. 𝐜𝐦𝟐 D. cm2
𝟐 2

PROBLEM 13-15 Find the area of the shaded portion.

A. 50π C. 20π
B. 25π D. 30π

PROBLEM 13-16 The swimming pool is to be constructed in the shape of partially

 overlapping identical circles. Each of the circles has a radius of
9m, and each passes through the center of the other. Find the
area of the swimming pool.
A. 302.33 m2 C. 398.99 m2
B. 362.55 m2 D. 409.44 𝐦𝟐

PROBLEM 13-17 Given are two concentric circles with the outer circle having radius
of 10 cm. If the area of the inner circle is half of the outer circle, find
the boarder between the two circles.
A. 2.930 cm C. 3.265 cm
B. 2.856 cm D. 2.444 cm

PROBLEM 13-18 A circle of radius 5 cm has a chord which is 6 cm long. Find the area
of the circle concentric to this circle and tangent to the given chord.
A. 14π C. 9π
B. 16π D. 4π

PROBLEM 13-19 A reversed curve on a railroad track consist of two circular arcs.
The central angle of the one side is 20° with radius 2500 feet,
and the central angle of the other is 25° with radius of 3000 feet.
Find the total lengths of the two arcs.
A. 2812 ft. C. 2821 ft.
B. 2218 ft. D. 2182 ft.

PROBLEM 13-20 Given a triangle whose sides are 24 cm, 30 cm, and 36 cm.
Find the radius of a circle which is tangent to the shortest
and the longest side of the triangle, and whose center lies
on the third side.
A. 9.111 cm. C. 12.31 cm.
B. 11.91 cm. D. 18 cm.

PROBLEM 13-21 Find the area of the largest circle that can be cut from a
Triangle whose sides are 10 cm, 18 cm, and 20 cm.
A. 11π cm2 C. 14π 𝐜𝐦𝟐
2
B. 12π cm D. 15π cm2

PROBLEM 13-22 The diameter of the circle circumscribed about a triangle

ABC with sides a, b, c is equal to:
A. 𝑎⁄sin 𝐴 C. 𝑐 ⁄sin 𝐶
B. 𝑏⁄sin 𝐵 D. all of the above

PROBLEM 13-23 The sides of a triangle are 14 cm, 15 cm, and 13 cm. Find
the area of the circumscribing circle.
A. 207.4 sq. cm. C. 215.4 sq.cm.
B. 209. 6 sq. cm. D. 220.5 sq. cm.

PROBLEM 13-24 What is the radius of the circle circumscribing an isosceles

right triangle having an area of 162 sq. cm.?
A. 13.52 C. 12.73
B. 14.18 D. 15.64

PROBLEM 13-25 If the radius of the circle is decreased by 20%, by how much
is its area decreased?
 A. 36% C. 46%
B. 26% D. 56%

PROBLEM 13-26 The distance between the center of the three circles which are
mutually tangent to each other externally are 10, 12 and 14 units
The area of the largest circle is.
A. 72π C. 64π
B. 23π D. 16π

PROBLEM 13-27 The sides of a cyclic quadrilateral measures 8cm,

9cm, 12cm, and 7cm, respectively. Find the area
of the circumscribing circle.
A. 8.65 cm2 C. 6.54 cm2
B. 186.23 cm2 D. 134.37𝐜𝐦𝟐

PROBLEM 13-28 The wheel of a car revolves n times, while the car
travels x km. The radius of the wheel in meter is:
A. 10,000 𝑥⁄(π n) C. 500,000 𝑥⁄(π n)
B. 500 𝒙⁄(𝛑 𝐧) D. 5,000 𝑥⁄(π n)

PROBLEM 13-29 If the inside wheels of car running a circular track

are going half as fast as the outside wheel, determine
the length of the track, described by outer, if the wheels
are 1.5m apart.
A. 4 π C. 6 𝛑
B. 5 π D. 8 π

PROBLEM 13-30 A goat is tied to a corner of a 30 ft by 35 ft building.

If the rope is 40 ft long and the goat can reach 1 ft
farther than the rope length, what is the maximum
area the goat can cover?
A. 5281 ft 2 C. 3961 ft 2
B. 4084 𝐟𝐭 𝟐 D. 3970 ft 2

PROBLEM 13-31 What is the area of the shaded portion shown?

A. 8 - 8π C. 8 - 4π
B. 8 - 2𝛑 D. 8 – π

PROBLEM 13-32 The interior angle of a triangle measures 2 x, x + 5,

and 2 x +15. What is the value of x?
A. 30° C. 42°
B. 66° D. 54°

PROBLEM 13-33 Two complementary angles are in the ratio of 2:1.

Find the larger angle.
A. 30° C. 75°
B. 60° D. 15°

PROBLEM 13-34 Two transmission towers 40 feet high is 200 feet apart.
 If the lowest point of the cable is 10 feet above the
ground, the vertical distance from the roadway to the
cable 50 feet from the center is:
A. 17.25 feet C. 17.75 feet
B. 17.5 feet D. 18 feet

PROBLEM 13-35 What is the area bounded by the curves 𝑦 2 = 4𝑥 and

𝑥 2 = 4𝑦 ?
A. 6.0 C. 6.666
B. 7.333 D. 5.333

PROBLEM 13-36 What is the area between y = 0, y = 3𝑥 2 , x = 0 and

x = 2?
A. 8 C. 24
B. 12 D. 6

PROBLEM 13-37 A circle having an area of 224 sq. m. is inscribed in

an octagon. Find the area of the octagon.
A. 238.6 sq. m. C. 236.6 sq. m.
B. 245. 2 sq. m. D. 246.7 sq. m.

PROBLEM 13-38 A circle is circumscribed about a hexagon. Determine

the area of the hexagon if the area outside the hexagon
but inside the circle is 15 sq. cm.
A. 73.3 sq. cm. C. 71.7 sq. cm.
B. 72.4 sq. cm. D. 74.8 sq. cm.

PROBLEM 13-39 A circle of radius 8 cm is inscribed in a sector having a

central angle of 80°. What is the area of the sector?
A. 195.63 cm2 C. 321.47 cm2
𝟐
B. 291.84 𝐜𝐦 D. 475.42 cm2

PROBLEM 13-40 A triangular piece of land has one side measuring 2 km.
The land is to be divided into two equal areas by a dividing
line parallel to the given side. What is the length of the
dividing line?
A. 6 C. 7.623
B. 8.485 D. 8

PROBLEM 13-41 A piece of wire having a total length of 72 cm was cut

into two unequal segments and bent to from two unequal
squares. If the total area of the squares is 180 sq. cm,
what is the difference in the lengths of the two segments?
A. 24 cm C. 32 cm
B. 16 cm D. 28 cm

PROBLEM 13-42 Determine the area of a regular hexagon inscribed in a circle

having an area of 170 square centimeters
A. 169.8 C. 148.2
B. 124.1 D. 140.6

PROBLEM 13-43 Three circles of radii 110, 140, and 220 are tangent to one
another. What is the area of the triangle formed by joining
 the center of the circles?
A. 39, 904 C. 32, 804
B. 25, 476 D. 47, 124

PROBLEM 13-44 A circle with area of 254.469 sq. cm. is circumscribed about
the triangle whose area is 48.23 sq. cm. If one side of the
triangle measure 18cm, determine the length of the shorter
leg of the triangle in cm.
A. 3.625 C. 8.652
B. 4.785 D. 5.643

PROBLEM 13-45 A road is tangent to a circular lake. Along the road

and 12 miles from the point of tangency, another
road opens towards the lake. From the intersection
of the two roads to the periphery of the lake, the length
of the new road is 11 miles. If the new road will be pro-
longed across the lake, find the length of the bridge to
be constructed.
A. 2.112 mi C. 2.103 mi
B. 2. 091 mi D. 2.512 mi

PROBLEM 13-46 The center of the two circles with the radii of 3 m and
5m, respectively are 4m apart. Find the area of the portion
of smaller circle outside the larger circle.
A. 11.25m2 C. 9.75m2
B. 12.15m2 D. 10.05𝐦𝟐
PROBLEM 13-47 Find the area in square centimeter of the largest square
that can be cut from a sector of a circle radius of 8cm and
central angle of 120°.
A. 21.9 C. 33.5
B. 45.2 D. 54.8

PROBLEM 13-48 A circle is inscribed in a square and circumscribed about

another. Determine the ratio of the area larger square to
the area of the smaller square.
A. 2:1 C. 1:4
B. 1:2 D. 4:1

PROBLEM 13-49 One side of a rectangle, is inscribed in a circle of diameter

17cm, is 8cm. Find the length of the other side.
A. 16 cm C. 14 cm
B. 15 cm D. 13 cm

PROBLEM 13-50 The sum of the sides of two polygons is 12 and the sum
of their diagonals is 19. The polygons are:
A. Pentagon & heptagon C. Quadrilateral & octagon
B. Both hexagon D. Triangle & nonagon
Problems – Set 14
Prisms, Pyramids, Cylinders, Cones

PROBLEM 14 – 1. If the edge of a cube is doubled, which of the following is incorrect?

A. the lateral area will be quadrupled
B. the volume is increased 8 times
C. the diagonal is doubled
D. the weight is doubled

PROBLEM 14 – 2. The volume of a cube is reduced by how much if all sides are halved?
A. 1/8 C. 6/8
B. 5/8 D. 7/8
PROBLEM 14 – 3. Each side of a cube is increased by 1%. By what percent is the volume
of the cube increased?
A. 23.4% C. 3%
 B. 33.1% D. 34.56%
PROBLEM 14 – 4. If the edge of a cube is increased by 30%, by how much is the surface
area increased?
A. 67 C. 63
B. 69 D. 65
PROBLEM 14 – 5. Find the approximate change in the volume of a cube of side x inches
caused by increasing its side by 1%.
A. 0.3x3 cu. in. C. 0.02x3 cu. in.
B. 0.1x3 cu. in. D. 0.03x3 cu. in.
PROBLEM 14 – 6. A rectangular bin 4 feet long, 3 feet wide, and 2 feet high is solidly
packed with bricks whose dimensions are 8 in. by 4 in. by 2 in. The
number of bricks in the bin is:
A. 68 C. 648
B. 386 D. 956
PROBLEM 14 – 7. Find the total surface area of a cube of side 6 cm.
A. 214 sq. cm. C. 226 sq. cm.
B. 216 sq. cm. D. 236 sq. cm.
PROBLEM 14 – 8. The space diagonal of a cube is 4x3 m. Find its volume.
0.5

A. 16 cubic meters C. 64 cubic meters

B. 48 cubic meters D. 86 cubic meters
PROBLEM 14 – 9. A reservoir is shaped like a square prism. If the area of its base is 225
sq. cm., how many liters of water will it hold?
A. 3.375 C. 33.75
B. 3375 D. 337.5
PROBLEM 14 – 10. Find the angle formed by the intersection of a face diagonal to the
diagonal of the cube drawn from the same vertex.
A. 35.26 C. 33.69
B. 32.56 D. 42.23
PROBLEM 14 – 11. The space diagonal of a cube (the diagonal joining two non-coplanar
vertices) is 6m. The total surface area of the cube is:
A. 60 C. 72
B. 66 D. 78
PROBLEM 14 – 12. The base edge of a regular hexagonal prism is 6 cm and its bases
are 12 cm apart. Find its volume in cu. cm.
A. 1563.45 cm3 C. 1896.37 cm3
B. 1058.45 cm 3 D. 1122.37 cm3
PROBLEM 14 – 13. The base edge of a rectangular pentagonal prism is 6 cm and its
bases are 12 cm apart. Find its volume in cu. cm.
A. 743.22 cm3 C. 587.45 cm3
B. 786.89 cm3 D. 1122.37 cm3
PROBLEM 14 – 14. The base of a right prism is a hexagon with one side 6 cm long. If
the volume of the prism is 450 cc, how fat apart are the bases?
A. 5.74 cm C. 4.11 cm
B. 3.56 cm D. 4.81 cm
PROBLEM 14 – 15. A trough has an open top 0.30 m by 6 m and closed vertical ends
which are equilateral triangles 30 cm on each side. It is filled with
water to half its depth. Find the volume of the water in cubic meters.
A. 0.058 C. 0.037
B. 0.046 D. 0.065
PROBLEM 14 – 16. Determine the volume of a right truncated prism with the following
dimensions: Let the corner of the triangular base be defined by A, B,
and C. The length AD = 10 feet, BC = 9 feet, and CA = 12 feet. The
sides at A,B, and C are perpendicular to the triangular base and have
the height of 8.6 feet, 7.1 feet, and 5.5 feet, respectively.
A. 413 ft3 C. 313 ft3
B. 311 ft 3 D. 391 ft3
PROBLEM 14 – 17. The volume of a regular tetrahedron of side 5 cm is:
 A. 13.72 cu.cm C. 15.63 cu. cm
B. 14.73 cu.cm D. 17 82 cu. cm
PROBLEM 14 – 18. A regular hexagonal pyramid whose base perimeter is 60 cm has an
altitude of 30 cm. The volume of the pyramid is:
A. 2958 cu. cm. C. 2859 cu. cm.
B. 2598 cu. cm. D. 2589 cu. cm.
PROBLEM 14 – 19. A frustum of a pyramid has an upper base 100 m by 10 m and a
lower base of 80 m by 8 m. If the altitude of the frustum is 5 m, find
its volume.
A. 4567.67 cu. m. C. 4066.67 cu m.
B. 3873.33 cu. m. D. 2345.98 cu. m.
PROBLEM 14 – 20. The altitude of the frustum of a regular rectangular pyramid is 5 m
the volume is 140 cu. m. and the upper base and the upper base is
3m by 4 m. What are the dimensions of the lower base in m?
A. 9 x 10 C. 4.5 x 6
B. 6 x 8 D. 7.50 x 10
PROBLEM 14 – 21. The frustum of a regular triangular pyramid has equilateral triangles
for its base. The lower and upper base edges are 9 m and 3 m,
respectively. If the volume is 118.2 cu.m., how far apart are the
base?
A. 9m C. 7m
B. 8m D. 10m
PROBLEM 14 – 22. A cylindrical gasoline tank, lying horizontally, 0.90m in diameter and
3m long is filled to a depth of 0.60m. How many gallons of gasoline
does it contain?
Hint: One cubic meter = 265 gallons
A. 250 C. 300
B. 360 D. 270
PROBLEM 14 – 23. A cylindrical tank is 8 feet long and 3 feet in diameter. When lying in
the horizontal position, the water is 2 feet deep. If the tank is in the
vertical position, the depth of the water in the tank is:
A. 5.67 m C. 5.82 ft
B. 5.82 m D. 5.67 ft
PROBLEM 14 – 24. A circular cylindrical is circumscribed about a right prism having a
square base one meter on an edge. The volume of cylinder is 6.283
cu. m. Find its altitude in m.
A. 5 C. 69.08
B. 4.5 D. 4
PROBLEM 14 – 25. If 23 cubic meters of water are poured into a conical vessel, it reaches
a depth of 12 cm. How much water must be added so that the depth
reaches 18 cm.?
A. 95 cubic meters C. 54.6 cubic meters
B. 100 cubic meters D. 76.4 cubic meters
PROBLEM 14 – 26. The height of a circular cone with a circular base down is h. If it
contains water to a depth of 2h/3 the ratio of the volume of water to
that of the cone is:
A. 1:27 C. 8:27
B. 2:3 D. 26:27
PROBLEM 14 – 27. A right circular cone with an altitude of 9m is divided into two
segments, one is a smaller circular cone having same vertex with an
altitude of 6m. Find the ratio of the
volume of the two cones.
A. 19:27 C. 1:3
B. 2:3 D. 8:27
PROBLEM 14 – 28. A conical vessel has a height of 24 cm. and a base diameter of 12cm.
It holds water to a depth of 18 cm. above its vertex. Find the volume
of its content in cc.
 A. 387.4 C. 383.5
B. 381.7 D. 385.2
PROBLEM 14 – 29. A right circular cone with an altitude of 8cm is divided into two
segments. One is a smaller circular cone having same vertex with
volume equal to ¼ of the original cone.
Find the altitude of the smaller cone.
A. 4.52 cm C. 5.04 cm
B. 6.74 cm D. 6.12 cm
PROBLEM 14 – 30. The slant height of the cone is 5m long. The base diameter is 6m.
What is the lateral area in sq. m?
A. 37.7 C. 44
B. 47 D. 40.8
PROBLEM 14 – 31. A right circular cone has a volume of 128 𝜋/3 cm3 and an altitude of
8 cm.
A. 16 √5 𝝅 cm2 C. 16 𝜋 cm2
B. 12 √5 𝜋 cm 2 D. 15 𝜋 cm2
PROBLEM 14 – 32. The volume of a right circular cone is 36𝜋. If its altitude is 3, find its
radius.
A. 3 C. 5
B. 4 D. 6
PROBLEM 14 – 33. A cone and hemisphere share base that is a semicircle with radius 3
and the cone is inscribed inside the hemisphere. Find the volume of
the region outside the cone and
inside the hemisphere.
A. 24.874 C. 28.274
B. 27.284 D. 28 724
PROBLEM 14 – 34. A cone was formed by rolling a thin sheet of metal in the form of a
sector of a circle 72 cm in diameter with a central angle of 210°. What
is the volume of the cone?
A. 13,602 cc C. 13,716 cc
B. 13,504 cc D. 13,318 cc

PROBLEM 14 – 35. A cone was formed by rolling a thin sheet of metal in the form of a
sector of a circle 72 cm in diameter with a central angle of 150°. Find
the volume of the cone in cc.
A. 7733 C. 7744
B. 7722 D. 7711
PROBLEM 14 – 36. A chemist’s measuring glass is conical in shape. If it is 8cm deep
and 3cm across the mouth, find the distance on the slant edge
between the markings for 1 cc and 2 cc.
A. 0.82 cm C. 0.74 cm
B. 0.79 cm D. 0.92 cm
PROBLEM 14 – 37. The base areas of a frustum of a cone are 25 sq. cm. and 16 sq. cm.,
respectively. If its altitude is 6 cm., find its volume.
A. 120 cm3 C. 129 cm3
B. 122 cm3 D. 133 cm3
PROBLEM 14 – 38. How far from the vertex is the opposite face of a tetrahedron if an
edge is 50cm long?
A. 38.618 cm C. 39.421 cm
B. 40.825 cm D. 41.214 cm
PROBLEM 14 – 39. A truncated prism has a horizontal triangular base ABC, A = 10 cm,
BC = 12 cm and CA = 8 cm. The vertical edges through A, B, and C
are 20 cm, 12 cm, and 18 cm, long respectively. Determine the
volume of the prism, in cc.
A. 661 C. 685
 B. 559 D. 574
PROBLEM 14 – 40. A lateral edge of the frustum of a regular pyramid is 1.8 m long. The
upper base is a square 1 m by 1 m and the lower base 2.4 m by 2.4
m square. Determine the volume of the frustum in cubic meters.
A. 4.6 C. 5.7
B. 3.3 D. 6.5
PROBLEM 14 – 41. A solid spherical steel ball 20 cm in diameter is placed into a tall
vertical containing water, causing the water to raise by 10 cm. What
is the radius of the cylinder?
A. 12.14 C. 10.28
B. 9.08 D. 11.55
PROBLEM 14 – 42. A conical vessel onemeter diameter at the top 60 cm high holds salt
at a depth of 36 cm from the bottom. How many cc of salt does it
contain?
A. 37,214 C. 35,896
B. 33,929 D. 31,574
PROBLEM 14 – 43. An open-top cylindrical tank is made of metal sheet having an area
of 43.82 square meter. If the diameter is 2/3 the height, what is the
height of the tank?
A. 3.24 m C. 4.23 m
B. 2.43 m D. 5.23 m
PROBLEM 14 – 44. The lateral area of a right circular cone is 386 square centimeters. If
its diameter is one-half its altitude, determine its altitude in
centimeters.
A. 24.7 C. 18.9
B. 17.4 D. 22.5
PROBLEM 14 – 45. The surface aea of a regular tetrahedron is 173.2 square centimeters.
What is its altitude?
A. 8.2 cm C. 7.2
B. 9.6 cm D. 6.5
PROBLEM 14 – 46. A cube of edge 25 cm is cut by a plane containing two diagonally
opposite edges of the cube. Find the area of the section thus formed
in sq. cm.
A. 812.3 C. 841.2
B. 912.7 D. 883.9
PROBLEM 14 – 47. A swimming pool is rectangular in shape of length 12 m and width
5.5 m. It has a sloping bottom and is 1 m deep at one end and 3.6 m
deep at the other end. The water from a full cylindrical reservoir3.6
m in diameter and 10 m deep is emptied to the pool. Find the depth
of water at the deep end.
A. 2.912 m C. 2.842 m
B. 2.695 m D. 2.754 m
PROBLEM 14 – 48. Two vertical conical tanks (both inverted) have their vertices
connected by a short horizontal pipe. One tank, initially full of water,
has an altitude of 6 m and a diameter of base 7 m. The other tank,
initially empty, has an altitude of 9 m and a diameter of base 8 m. If
the water is allowed to flow through the connecting pipe, find the level
to which the water will ultimately rise in the empty tank. (Neglect the
water in the pipe).
A. 4.254 m C. 4.687 m
B. 3.257 m D. 5.151 m
Problems - Set 15
Spheres, Prismatoid, Solids of Revolutions, Misc.

PROBLEM 15 – 1. What is the surface area of the sphere whose volume is 36 cu. m?
A. 52.7 m2 C. 46.6 m2
B. 48.7 m 2 D. 54.6 m2
PROBLEM 15 – 2. If the surface area of the sphere is increased by 21%, its volume is
increased by:
A. 13.31% C. 21%
B. 33.1% D. 30%
PROBLEM 15 – 3. The surface area of the sphere is 4𝜋r . Find the percentage increase
2

in its diameter
when the surface area increases by 21%.
A. 5% C. 15%
B. 10% D. 20%
PROBLEM 15 – 4. Find the percentage increase in volume of a sphere if its surface area
is increased by
21%.
A. 30.2% C. 34.5%
B. 33.1% D. 30.9%
PROBLEM 15 – 5. The volume of a sphere is increased by how much if its radius is
increased by 20%?
A. 32.6% C. 44%
B. 33% D. 72.8%
PROBLEM 15 – 6. Given two spheres whose combined volume is known to be 819 cu.
m. If their radii are in the ratio 3:4, what is the volume of the smaller
sphere?
A. 576 cu. m. C. 343 cu. m.
B. 243 cu. m. D. 476 cu. m.
PROBLEM 15 – 7. How much will the surface area of the sphere be increased if its radius
is increased by 5%.
A. 25% C. 12.5%
B. 15.5% D. 10.25%
PROBLEM 15 – 8. The volume of a sphere is 904.78 cu. m. Find the volume of the
spherical segment of height 4m.
A. 234.57 cu. m. C. 145.69 cu. m.
B. 256.58 cu. m. D. 124.58 cu. m.
PROBLEM 15 – 9. A sphere of radius r just fits into a cylindrical container of radius r and
altitude 2r. Find the empty space in the cylinder.
A. (8/9) 𝜋 r3 C. (4/5) 𝜋 r3
B. (20/27) 𝜋 r 3 D. (2/3) 𝝅 r3
PROBLEM 15 – 10. If a solid steel ball is immersed in an 8 cm diameter cylinder, it
displaces water to a depth of 2.25 cm. The radius of the ball is:
A. 3 cm C. 9 cm
B. 6 cm D. 12 cm
PROBLEM 15 – 11. The diameters of two spheres are in the ratio 2:3. If the sum of their
volumes is 1260 cu. m., the volume of the larger sphere is:
A. 972 cu. m. C. 856 cu. m.
B. 927 cu. m. D. 865 cu. m.
PROBLEM 15 – 12. A hemispherical bowl of radius 10 cm is filled with water to such a
depth that the water surface area is equal to 75𝜋 sq. cm. The volume
of water is:
A. 625/3 cm3 C. 625π /2 cm3
B. 625 π /3 cm 3 D. 625π cm3

PROBLEM 15 – 13. A water tank is in the form of a spherical segment whose base radii
are 4m and 3m and whose altitude is 6m. The capacity of the tank in
gallons is:
A. 91,011 C. 95,011
B. 92,011 D. 348.72
PROBLEM 15 – 14. Find the volume of a spherical sector of altitude 3 cm. and radius 5
cm.
A. 75π cu. cm. C. 50π cu. cm.
B. 100π cu. cm. D. 25π cu. cm.
PROBLEM 15 – 15. A water tank is in the form of a spherical segment whose base radii
are 4m and 3m and whose altitude is 6m. The capacity of the tank in
gallons is:
A. 91,011 C. 95,011
B. 92,011 D. 348.72
PROBLEM 15 – 16. How far from the center of the sphere of a radius 10 cm should a
plane be passed so the ratio of the areas of two zones is 3:7.
A. 3 cm C. 5 cm
B. 4 cm D. 6 cm
PROBLEM 15 – 17. A 2-m diameter spherical tank contains 1396 liters of water. How
many liters of water be added for the water to reach a depth of
1.75m?
 A. 2613 C. 2542
B. 2723 D. 2472
PROBLEM 15 – 18. Find the volume of a spherical segment of radius 10m and altitude
5m.
A. 654.5 cu. m. C. 675.2 cu. m.
B. 659.8 cu. m. D. 680.5 cu. m.
PROBLEM 15 – 19. Find the volume of a spherical wedge of radius 10 cm. and altitude
12 in.
A. 425.66 sq. m. C. 581.78 sq. m.
B. 431.25 sq. m. D. 444.56 sq. m.
PROBLEM 15 – 20. Determine the area of the zone of the sphere of radius 8 in. and an
altitude 12 in.
A. 192π sq. in. C. 185π sq. in.
B. 198π sq. in. D. 195π sq. in.
PROBLEM 15 – 21. The corners of the cubical block touch the closed spherical shell that
encloses it. The volume of the box is 2744 cc. What volume in cc,
inside the shell is not occupied by block?
A. 1356 cu. cm. C. 3423 cu. cm.
B. 4721 cu. cm. D. 7623 cu. cm.
PROBLEM 15 – 22. A cubical container that measures 2 inches on each side is tightly
packed with 8 marbles and is filled with water. All 8 marbles are in
contact with the walls of the and the adjacent marbles. All of the
marbles are of the same size. What is the volume of the water in the
container?
A. 0.38 cu. in. C. 3.8 cu. in.
B. 2.5 cu. in. D. 4.2 cu. in.
PROBLEM 15 – 23. The volume of the water in the spherical tank is 1470.265 cm 3.
Determine the depth of the water if the tank has a diameter of 30 cm.
A. 8 C. 4
B. 6 D. 10
PROBLEM 15 – 24. The volume of the water in the spherical tank having a diameter of
4m is 5.236 m3. Determine the depth of the water in the tank.
A. 1.0 C. 1.2
B. 1.4 D. 1.6

PROBLEM 15 – 25. A mixture compound from equal parts of two liquids, one white and
the other black, was placed in a hemispherical bowl. The total depth
of the two liquids is 6”. After for a long time the mixture separated the
white liquid settling below the black liquid is 2”, find the radius of the
bowl in inches.
A. 7.53 C. 7.73
B. 7.33 D. 7.93
PROBLEM 15 – 26. 20.5 cubic meters of water is inside a spherical tank whose radius
is 2m. Find the height of the water surface above the bottom of the
tank, in m.
A. 2.7 C. 2.3
B. 2.5 D. 2.1
PROBLEM 15 – 27. The volume of sphere is 36𝜋 cu. m. The surface area of this sphere
in sq. m. is:
A. 36 π C. 18 π
B. 24π D. 12 π
PROBLEM 15 – 28. Spherical balls 1.5 cm in diameter area packed in a box measuring6
cm by 3 cm by 3. If as many balls as possible are packed in the box,
how much free space remains in the box?
A. 28.41 cc C. 29.87 cc
B. 20.47 cc D. 25.73 cc
PROBLEM 15 – 29. A solid has a circular base of radius r. Find the volume of the solid if
every plane perpendicular to a given diameter is a square.
A. 16 r3/3 C. 6 r3
B. 5 r3 D. 19r3/3
PROBLEM 15 – 30. A solid has circular base of diameter 20 cm. Find the volume of the
solid if every cutting plane perpendicular to the base along a given
diameter is an equilateral triangle.
A. 2514 cc C. 2309 cc
B. 2107 cc D. 2847 cc
PROBLEM 15 – 31. The base of a certain solid is a triangle of base b and altitude h. If all
sections perpendicular to the altitude of the triangle are regular
hexagons, find the volume of the solid.
A. ½ √3 b2 h C. √3b2 h/3
B. 2 √3 b h2 D. √3b2 h
PROBLEM 15 – 32. The volume generated by the circle x2+y24x-6y-12 = 0 revolved about
the line 2x-3y-= 0 is:
A. 3242 cubic units C. 3452 cubic units
B. 3342 cubic units D. 3422 cubic units
PROBLEM 15 – 33. The volume generated by rotating the curve 9x2+ 4y2= 36 about the
line 4x+3y = 20 is:
A. 48 π C. 42 π
B. 58 π 2 D. 48 π 2
PROBLEM 15 – 34. Find the volume generated by revolving the area bounded by the
ellipse y/9 = x2/4 =1 the line x = 3.
A. 347.23 cu. units C. 378.43 cu. units
B. 355.31 cu. units D. 389.51 cu. units
PROBLEM 15 – 35. The area of the second quadrant of the circle x2 + y2 = 36 is revolved
about the line y + 10 = 0
A. 2218.6 C. 2233.4
B. 2228.8 D. 2208.5
PROBLEM 15 – 36. A square area of edge “a” revolves about a line through one vertex,
making an angle 𝜃 with an edge and not crossing the square. Find
the volume generated.
A. 3𝜋 a3 (sinθ + cosθ) C. 2𝜋 a3 (sinθ + cosθ)
B. π a3 (sinθ + cosθ)/2 D. 𝛑 a3 (sinθ + cosθ)
PROBLEM 15 – 37. Given an ellipse whose semi-major axis is 6 cm. and semi-major axis
is 3 cm. What is the volume generated if it is revolved about the minor
axis?
A. 36π cu. cm. C. 96π cu. cm.
B. 72π cu. cm. D. 144π cu. cm.
PROBLEM 15 – 38. A square hole 2” x 2” is cut through a 6-inch diameter log along its
diameter and perpendicular to its axis. Find the volume of wood that
was removed.
A. 27.32 cu. in. C. 21.78 cu. in.
B. 23.54 cu. in. D. 34.62 cu. in.
PROBLEM 15 – 39. Find the of the spherical wedge whose volume is 12 cu. m. with a
central angle of 1.8 radians.
A. 2.36 m C. 2.52 m
B. 2.73 m D. 2.15 m
PROBLEM 15 – 40. By using Pappus theorem, determine the volume generated by
revolving the area in the first and second quadrants bounded by the
ellipses 4x2 + 25y2 = 100 and the x- axis, about the x-axis.
A. 85.63 C. 95.35
B. 93.41 D. 83.78
PROBLEM 15 – 41. Determine the volume of a spherical wedge of radius 2 m and a
central angle of 1.25 radians.
A. 6.67 m3 C. 9.85 m3
 B. 8.64 m3 D.5.74 m3
PROBLEM 15 – 42. Find the volume generated by revolving the triangle whose vertices
are (2,2),(4,8), and (6,2) about the line 3x – 4y – 24 = 0.
A. 365.45 C. 543.65
B. 498.12 D. 422.23
PROBLEM 15 – 43. A light bulb is placed at a certain distance from the surface of a
spherical globe of radius 20 cm. If it illuminates one-third of the total
surface of the globe, how far is it from the surface?
A. 30 cm C. 60 cm
B. 35 cm D. 40 cm
PROBLEM 15 – 44. A conoid has a circular base of radius 25 cm and an altitude of 30
cm. Find its volume
in cc.
A. 32,457 C. 24,486
B 29,452 D. 18,453
PROBLEM 15 – 45. A sphere having a diameter of 30 cm is cut into 2 segments. The
altitude of the first segment is 8 cm. What is the ratio of the second
segment to that of the first?
A. 3.2 C. 5.8
B. 4.7 D. 2.5

Problems-Set 16
Points, Lines, Circles

1. State the quadrant in which the coordinate (15, -2) lies.

a. I c. II
b. IV d. III
2. Of what quadrant is A, if sec A is positive and csc A is negative?
a. III c. IV
b. I d. II
3. The segment from (-1, 4) to (2, -2) is extended three times its own length. The
terminal point is
a. (11, -18) c. (11, -20)
b. (11, -24) d. (-11, -20)
4. The midpoint of the line segment between P1(x,y) and P2(-2,4) is Pm(2,-1). Find
the coordinate of P1+
a. (6, -5) c. (6, -6)
b. (5, -6) d. (-6, 6)
5. Find the coordinate of the point P(2, 4) with respect to the translated axis with
origin at (1, 3).
 a. (1, -1) c. (-1, -1)
b. (1, 1) d. (-1, 1)
6. Find The median through (-2, -5) of the triangle whose vertices are (-6, 2), (2, -2),
and (-2, -5).
a. 3 c. 5
b. 4 d. 6
7. Find the centroid of a triangle whose vertices are (2, 3), (-4, 6), and (2, -6).
a. (0, 1) c. (-1, -1)
b. (0, -1) d. (-1, 1)
8. Find the area of triangle whose vertices are A (-3, -1), B (5, 3), and C (2, -8)
a. 34 c. 38
b. 36 d. 32
9. Find the distance between the points (4, -2) and (-5, 1)
a. 4.897 c. 7.149
b. 8.947 d. 9.487
10. Find the distance between A (4, -3) and B (-2, 5).
a. 11 c. 9
b. 8 d. 10
11. If the distance between the points (8, 7) and (3, y) is 13, what is the value of y?
a. 5 c. 10 or -5
b. -19 d. 5 or -19
12. The distance between the points (sin x, cos x) and (cos x, -sin x) is:
a. 1 c. 2 sin x cos x
b. √𝟐 d. 4 sin x cos x
13. Find the distance from the point (2, 3) to the line 3x + 4y + 9 = 0.
a. 5 c. 5.8
b. 5.4 d. 6.2

14. Find the distance from the point (5, -3) to the line 7x – 4y - 28 = 0.
a. 2.62 c. 2.48
b. 2.36 d. 2.54
15. How far is the line 3x – 4y + 15 = 0 from the origin?
a. 1 c. 3
b. 2 d. 4
16. Determine the distance from (5, 10) to the line x – y = 0
a. 3.86 c. 3.68
b. 3.54 d. 3.72
17. The two points on the lines 2x + 3y + 4 = 0 which are at distance 2 from the line
3x + 4y – 6 = 0 are:
a. (-8, 8) and (-16, -16) c. (-5.5, 1) and (-5, 2)
b. (-44, 64) and (-5, 2) d. (64, -44) and (4, -4)
18. The intercept form for algebraic straight-line equation is
a. a/x + y/b = 1 c. Ax + By + C = 0
b. y = mx + b d. x/a + y/b = 1
19. Find the slope of the line defined by y – x = 5
a. 1 c. ¼
b. -1/2 d. 5 + x
20. The slope of the line 3x + 2y + 5 = 0 is:
a. -2/3 c. 3/2
b. -3/2 d. 2/3
21. Find the slope of the line whose parametric equation is y = 5 – 3t and x = 2 + t.
a. 3 c. 2
b. -3 d. -2
22. Find The slope of the curve whose parametric equations are x = -1 + t and y = 2t.
a. 2 c. 1
b. -3 d. 4
23. Find the angle that the line 2y – 9x – 18 = 0 makes with the x-axis
 a. 74.77o c. 47.77o
b. 4.5o d. 77.47o
24. Which of the following is perpendicular to the line x/3 + y/4 = 1?
a. x – 4y – 8 = 0 c. 3x – 4y – 5 = 0
b. 4x – 3y – 6 = 0 d. 4x + 3y – 11 = 0
25. Find the equation of the bisector of the obtuse angle between the lines 2x + y = 4
and 4x – 2y = 7
a. 4y = 1 c. 2y = 3
b. 8x = 15 d. 8x + 4y = 6
26. The equation of the line through (1, 2) and parallel to the line 3x – 2y + 4 = 0 is:
a. 3x – 2y + 1 = 0 c. 3x + 2y + 1 = 0
b. 3x – 2y + 1 = 0 d. 3x + 2y – 1 = 0
27. If the points (-3, -5), (x, y) and (3, 4) lie on a straight line, which of the following is
correct?
a. 3x + 2y – 1 = 0 c. 2x + 3y – 1 = 0
b. 2x + 3y + 1 = 0 d. 3x – 2y – 1 = 0
28. One line passes through the points (1, 9) and (2, 6), another line passes through
(3, 3) and (-1, 5). The acute angle between the two lines is:
a. 30o c. 60o
b. 45o d. 135o
29. The two straight lines 4x – y + 3 = 0 and 8x – 2y + 6 = 0
a. intersects at the origin c. are parallel
b. are coincident d. are perpendicular
30. A line which passes through (5, 6) and (-3, -4) has an equation of
a. 5x + 4y + 1 = 0 c. 5x – 4y + 1 = 0
b. 5x – 4y – 1 = 0 d. 5x + y – 4 = 0

31. Find the equation of the line with slope of 2 and y-intercept of -3.
a. y = -3x + 2 c. y = 2/3 x + 1
b. y = 2x – 3 d. y = 3x – 2
32. What is the equation of the line that passes through (4, 0) and is parallel to the
line x – y – 2 = 0?
a. y + x + 4 = 0 c. y – x – 4 = 0
b. y – x + 4 = 0 d. y + x – 4 = 0
33. Determine B such that 3x + 2y – 7 = 0 is perpendicular to 2x – By + 2 = 0.
a. 2 c. 4
b. 3 d. 5
34. The equation of the line that intercepts the x-axis at x = 4 ant the y-axis at y = -6
is:
a. 2x – 3y = 12 c. 3x – 2y = 12
b. 3x + 2y = 12 d. 2x – 37 = 12
35. How far from the y-axis is the center of the curve 2x2 + 2y2 + 10x – 6y – 55 = 0?
a. -3.0 c. -3.25
b. 2.75 d. 2.5
36. Find the area of the circle whose center is at (2, -5) and tangent to the line 4x +
3y – 8 = 0.
a. 6π c. 3π
b. 9π d. 12π
37. Determine the area enclosed by the curve x2 – 10x + 4y + y2 = 196
a. 15π c. 12π
b. 225π d. 144π
38. Find the shortest distance from the point (1, 2) to a point on the circumference of
the circle defined by the equation x2 + y2 + 10x + 6y + 30 = 0
a. 5.61 c. 5.81
b. 5.71 d. 5.91
39. Determine the length of the chord common to the circles x2 + y2 = 64 and x2 + y2
– 16x = 0.
 a. 13.86 c. 13.25
b. 12.82 d. 12.28
40. If (3, -2) lies on a circle with center (-1, 1), then the area of the circle is:
a. 5π c. 4π
b. 25π d. 3π
41. The Radius of the circle 2x + 2y – 3x + 4y - 1 = 0 is:
2 2

a. √𝟑𝟑/𝟒 c. √33/3
b. 33 / 16 d. 17
42. What is the radius of a circle with the following equation? x2 – 6x + y2 – 4y – 12
=0
a. 3.46 c. 7
b. 5 d. 6
43. The diameter of a circle described by 9x2 + 9y2 = 16 is:
a. 16/9 c. 4
b. 4/3 d. 8/3
44. Find the center of the circle x + y – 6x + 4y – 23 = 0.
2 2

a. (3, -2) c. (-3, 2)

b. (3, 2) d. (-3, -2)

45. Determine the equation of the circle whose center is at (4, 5) and tangent to the
circle whose equation is x2 + y2 + 4x + 6y – 23 = 0
a. x2 + y2 – 8x + 10y – 25 = 0
b. x2 + y2 + 8x - 10y + 25 = 0
c. x2 + y2 - 8x - 10y + 25 = 0
d. x2 + y2 - 8x - 10y - 25 = 0
46. the equation of the circle with center at (-2, 3) and which is tangent to the line 20x
-21y – 42 = 0 is:
a. x2 + y2 + 4x - 6y - 12 = 0
b. x2 + y2 + 4x - 6y + 12 = 0
c. x2 + y2 + 4x + 6y - 12 = 0
d. x2 + y2 - 4x - 6y - 12 = 0
47. A circle has a diameter whose ends are at (-3, 2) and (12, -6). Its equation is:
a. 4x2 + 4y2 – 36x + 16y + 192 = 0
b. 4x2 + 4y2 – 36x + 16y - 192 = 0
c. 4x2 + 4y2 – 36x - 16y - 192 = 0
d. 4x2 + 4y2 + 36x + 16y - 192 = 0
48. Find the equation of the circle with center on x + y = 4 and 5x + 2y + 1 = 0 and
having a radius of 3.
a. x2 + y2 + 6x - 16y + 64 = 0
b. x2 + y2 + 8x - 14y + 25 = 0
c. x2 + y2 + 6x - 14y + 49 = 0
d. x2 + y2 + 6x - 16y + 36 = 0
49. If (3, -2) lies on the circle with center (-1, 1) then the equation of the circle is:
a. x2 + y2 + 2x - 2y – 23 = 0
b. x2 + y2 + 4x - 2y - 21 = 0
c. x2 + y2 + 2x - y - 33 = 0
d. x2 + y2 + 4x - 2y - 27 = 0
50. Find the equation of k for which the equation x2 + y2 + 4x - 2y -k = 0 represents a
point circle.
a. 5 c. 6
b. -5 d. -6
Problems-Set 17
Parabola, Ellipse, Hyperbola, Polar, Space

1. The vertex of the parabola y2 - 2x + 6y + 3 = 0 is at:

a. (-3, 3) c. (-3, 3)
b. (3, 3) d. (-3, -3)

2. The length of the latus rectum of the parabola y2 = 4px is:

a. 4p c. p
b. 2p d. -4p
3. Given the equation of the parabola: y2 – 8x – 4y – 20 = 0. The length of its latus
rectum is:
a. 2 c. 6
b. 4 d. 8
4. What is the length of the latus rectum of the curve x2 = -12y?
a. 12 c. 3
b. -3 d. -12
5. Find the equation of the directrix of the parabola y2 = 16x.
a. x = 8 c. x = -8
b. x = 4 d. x = -4
2
6. The curve y = x + x + 1 opens:
 a. upward c. to the right
b. to the left d. downward
7. The parabola y = -x2 + x + 1 opens:
a. to the right c. upward
b. to the left d. downward
8. Find the equation of the axis of symmetry of the function y = 2x2 – 7x + 5.
a. 4x + 7 = 0 c. 4x – 7 = 0
b. x - 2 = 0 d. 7x + 4 = 0
9. Find the equation of the locus of the center of the circle which moves so that it is
tangent to the y-axis and to the circle of radius one (1) with center at (2, 0).
a. x2 + y2 - 6x + 3 = 0 c. 2x2 + y2 - 6x + 3 = 0
2
b. x - 6x + 3 = 0 d. y2 - 6x + 3 = 0
10. Find the equation of the parabola with vertex at (4, 3) and focus at (4, -1).
a. y2 - 8x + 16y - 32 = 0 c. x2 + 8x - 16y + 32 = 0
b. y2 + 8x - 16y - 32 = 0 d. x2 - 8x + 16y - 32 = 0
11. Find the area bounded by the curves x2 + 8y + 16 = 0, x – 4 = 0, the x-axis, and
the y-axis.
a. 10.67 sq. units c. 9.67 sq. units
b. 10.33 sq. units d. 8 sq. units
12. Find the area (in aq. Units) bounded by the parabolas x2 - 2y = 0 and x2 + 2y – 8
= 0.
a. 11.7 c. 9.7
b. 10.7 d. 4.7

13. The length of the latus rectum of the curve (x – 2)2 / 4 + (y + 4)2 / 25 = 1 is:
a. 1.6 c. 0.80
b. 2.3 d. 1.52
14. Find the length of the latus rectum of the following ellipse:
25x2 + 9y2 -300x – 144y + 1251 = 0
a. 3.4 c. 3.6
b. 3.2 d. 3.0
15. If the length of the major and minor axes of an ellipse is 10 cm and 8 cm,
respectively, what is the eccentricity of the ellipse?
a. 0.50 c. 0.70
b. 0.60 d. 0.80
16. The eccentricity of the ellipse x2 / 4 + y2 / 16 = 1 is:
a. 0.725 c. 0.689
b. 0.256 d. 0.866
17. An ellipse has the equation 16x + 9y2 + 32x – 128 = 0. Its eccentricity is:
2

a. 0.531 c. 0.824
b. 0.66 d. 0.93
18. The center of the ellipse 4x2 + y2 - 16x – 6y - 43 = 0 is at:
a. (2, 3) c. (1, 9)
b. (4, -6) d. (-2, -5)
19. Find the ratio of the major axis to the minor axis of the ellipse:
9x2 + 4y2 – 24y - 72x - 144 = 0
a. 0.67 c. 1.5
b. 1.8 d. 0.75
20. The area of the ellipse 9x2 + 25y2 - 36x – 189 = 0 is equal to:
a. 15π sq. units c. 25π sq. units
b. 20π sq. units d. 30π sq. units
21. The area of the ellipse is given as A = 3.1416 ab, Find the area of the ellipse
25x2 + 16y2 - 100x + 32y = 284.
a. 86.2 square units c. 68.2 square units
b. 62.8 square units d. 82.6 square units
22. The semi-major axis of an ellipse is 4 and its semi-minor axis is 3. The distance
from the center to the directrix is:
a. 6.532 c. 0.6614
b. 6.047 d. 6.222
23. Given an ellipse x2/36 + y2/32 = 1. Determine the distance between foci.
a. 2 c. 4
b. 3 d. 8
24. How far apart are the directrices of the curve 25x2 + 9y2 - 300x – 144y + 1251=0?
a. 12.5 c. 13.2
b. 14.2 d. 15.2
25. The major axis of the elliptical path in which the earth moves around the sun is
approximately 186,000,000 miles and the eccentricity of the ellipse is 1/60.
Determine the apogee of the earth.
a. 94,550,000 miles c. 91,450,000 miles
b. 94,335,100 miles d. 93,000,000 miles
26. Find the equation of the ellipse whose center is at (-3, -1), vertex at (2, -1), and
focus at (1, -1).
a. 9x2 + 36y2 - 54x + 50y – 116 = 0
b. 4x2 + 25y2 + 54x - 50y – 122 = 0
c. 9x2 + 25y2 + 50x + 50y + 109 = 0
d. 9x2 + 25y2 + 54x + 50y – 119 = 0
27. Point P(x, y) moves with a distance from point (0, 1). One-half of its distance from
line y = 4, the equation of its locus is
a. 4x2 + 3y2 = 12 c. x2 + 2y2 = 4
2 2
b. 2x + 4y = 5 d. 2x2 + 5y3 = 3
28. The cords of the ellipse 64x2 + 25y2 = 1600 having equal slopes of 1/5 are
bisected by its diameter. Determine the equation of the diameter of the ellipse.
a. 5x – 64y = 0 c. 5x + 64y = 0
b. 64x – 5y = 0 d. 64x + 5y = 0
29. Find the equation of the upward asymptote of the hyperbola whose equation is
(x – 2)2 / 9 – (y + 4)2 / 16.
a. 3x + 4y – 20 = 0 c. 4x + 3y – 20 = 0
b. 4x – 3y – 20 = 0 d. 3x – 4y – 20 = 0
30. The semi-conjugate axis of the hyperbola x2 / 9 - y2 / 4 = 1 is:
a. 2 c. 3
b. -2 d. -3
31. What is the equation of the asymptote of the hyperbola
𝑥2 𝑦2
− 4 = 1?
9
a. 2x – 3y = 0 c. 2x – y = 0
b. 3x – 2y = 0 d. 2x + y = 0
32. The graph y = (x – 1) / (x + 2) is not defined at:
a. 0 c. -2
b. 2 d. 1
33. The equation x2 + Bx + y2 + Cy + D = 0 is:
a. hyperbola c. ellipse
b. parabola d. circle
34. The general second degree equation has the form Ax2 + Bxy + Cy2 + Dx + Ey + F
= 0 and describes an ellipse if:
a. B2 – 4AC = 0 c. B2 – 4AC = 1
b. B2 – 4AC > 0 d. B2 – 4AC <0
35. Find the equation of the tangent to the circle x2 + y2 – 34 = 0 through point (3, 5).
a. 3x + 5y – 34 = 0 c. 3x + 5y + 34 = 0
b. 3x – 5y – 34 = 0 d. 3x – 5y + 34 = 0
36. Find the equation of the tangent to the curve x2 + y2 + 4x + 16y – 32 = 0, through
(4, 0).
a. 3x – 4y + 12 = 0 c. 3x + 4y + 12 = 0
 b. 3x – 4y – 12 = 0 d. 3x + 4y – 12 = 0
37. Find the equation of the normal to the curve y2 + 2x + 3y = 0 through point (-5, 2)
a. 7x + 2y + 39 =0 c. 2x – 7y – 39 = 0
b. 7x – 2y + 39 = 0 d. 2x + 7y – 39 = 0
38. Determine the equation of the line tangent to the graph y = 2x2 + 1, at the point
(1, 3).
a. y = 4x + 1 c. y = 2x - 1
b. y = 4x – 1 d. y = 2x + 1
39. Find the equation of the tangent to the curve x2 + y2 = 41 through (5, 4)
a. 5x + 4y = 41 c. 4x + 5y = 41
b. 4x – 5y = 41 d. 5x – 4y = 41
40. Find the equation of a aline normal to the curve x2 = 16y at (4, 1),
a. 2x – y – 9 = 0 c. 2x + y – 9 = 0
b. 2x – y + 9 = 0 d. 2x + y + 9 = 0
41. What is the equation of the tangent to the curve 9x2 + 25y2 – 225 = 0 at (0, 3)?
a. y + 3 = 0 c. x – 3 = 0
b. x + 3 = 0 d. y – 3 = 0
42. What is the equation of the normal to the curve x2 + y2 = 25 at (4,3)?
a. 3x – 4y = 0 c. 5x – 3y = 0
b. 5x + 3y = 0 d. 3x + 4y = 0
43. The polar form of the equation 3x + 4y – 2 = 0 is:
a. 3r sin 𝜃 + 4r cos 𝜃 = 2 c. 3r 𝐜𝐨𝐬 𝜽 + 4r 𝐬𝐢𝐧 𝜽 = 2
b. 3r sin 𝜃 + 4r cos 𝜃 = -2 d. 3r sin 𝜃 + 4r tan 𝜃 = -2
44. The polar form of the equation 3x + 2y2 = 8 is:
2

a. r2 = 8 c. r = 8
2
b. r = 8 / cos 𝜃 + 2 d. r2 = 8 / cos2 𝜽 + 2
45. The distance between points (5, 30o) and (-8, -50o) is:
a. 9.84 c. 6.13
b. 10.14 d. 12.14
46. Convert 𝜃 = 𝜋/3to Cartesian equation.
a. x = √3x c. 3y = √3x
b. y = x d. y = √𝟑x
47. The point of intersection of the planes x + 5y – 2z = 9, 3x – 2y + z = 3, and x + y
+ z = 2 is:
a. (2, 1, -1) c. (-1, 1, -1)
b. (2, 0, -1) d. (-1, 2, 1)
48. A warehouse roof needs a rectangular skylight with vertices (3, 0, 0), (0, 0, 4),
and (0, 0, 4). If the units are in meter, the area of the skylight is:
a. 12 sq. m. c. 15 sq. m.
b. 20 sq. m. d. 9 sq. m.
49. The distance between points in space whose coordinates are (3,4,5) and (4,6,7)
is:
a. 1 c. 3
b. 2 d. 4
50. What is the radius of the sphere with center at origin and which passes through
the point (8, 1, 6)?
a. 10 c. √𝟏𝟎𝟏
b. 9 d. 10.5
51. Points C (5, 7, z) and D (4, 1, 6) are 7.28 cm apart. Find the value of z.
a. 3 cm c. 2 cm
b. 4cm d. 1 cm
52. What is the total length of the curve r = 4 sin ∅?
a. 8𝜋 c. 2𝜋
b. 𝜋 d. 𝟒𝝅
53. A triangle have vertices at A(-3, -2), B(2, 6), and C(4, 2). What is the abscissa of
the centroid of the triangle?
 a. ¾ c. 3/2
b. 5/4 d. 1
54. What is the distance between the vertices of the following ellipse: 64x2 + 25y2 +
16x – 16y – 648 = 0?
a. 6.324 c. 10.21
b. 12.54 d. 5.105
55. Determine the equation of the curve such that the sum of the distances of any
point of the curve from two points whose coordinates are (-3, 0) and (3,0) is
always equal to 8.
a. 4x2 + 49y2 – 343 = 0 c. 7x2 + 16y2 – 112 = 0
b. 7x2 + 16y2 – 122 = 0 d. 7x2 + 16y2 – 112 = 0
56. Find the volume of the tetrahedron bounded by the coordinate planes and the
plane 8x + 12y + 4z – 24 = 0.
a. 5 c. 6
b. 9 d. 12
57. The distances from the focus to the vertices of the ellipse are 4 and 6 units.
Determine the ellipse flatness.
a. 0.0202 c. 0.0312
b. 0.206 d. 0.0187
58. If the length of the latus rectum of an ellipse is three-fourth of the length of the
minor axis, determine its eccentricity.
a. 0.775 c. 0.661
b. 0.332 d. 0.553
59. Transform r = 3 / 3+2cos 𝜃 into Cartesian coordinates.
a. 5x2 - 9y2 + 12x + 9 = 0 c. 5x2 + 9y2 + 12x + 9 = 0
b. 5x2 + 9y2 - 12x - 9 = 0 d. 5x2 + 9y2 + 12x - 9 = 0
60. A warehouse roof needs a rectangular skylight with vertices (3, 0, 0), (3, 3, 0), (0,
3, 4) and (0, 0, 4). If the units are in meter, the area of the skylight is:
a. 12 sq. m. c. 15 sq. m.
b. 20 sq. m. d. 9 sq. m.

Problems-Set 18
Recent Board Exams

PROBLEM 18-1. Solve the system.

xy = 4
y - 2x + 2 = 0
A. {(-3,4)} C. {(4,6),(-3,-8)}
B. {4,-8),(-3,6)} D. {(-3,4),(-8,6)}
PROBLEM 18-2. Solve the equation: √x² - √3x = 2x - 6
A. 12 C. 3
B. 4 D. 5
PROBLEM 18-3. √8 + 3 √18 - 7√2 =
A. 3 √2 C. 4 √2
B. 5 - 3 √2 D. 2 √2
PROBLEM 18-4. If -9 < x < -4 and -12 < y <-6 , then
A. 108 < xy < 24 C. 108 < xy < 120
B. 108 > xy >24 D. -21 < x + y < -10
PROBLE 18-5. What is the value of E in the following equation?
(2x⁴ + 3xᶟ + 7x² + 10x +10) / (x - 1)(x² + 3)² = A / (x -1) +
 (Bx + C) / (x² + 3) + (Dx + E) / ( x² + 3)²
A. 2 C. 0
B. 3 D. -1

PROBLEM 18-6. If 3ˣ = 54 , then 3ˣ⁻² is equal to :

A. 6 C. 8
B. 5 D. 9
PROBLEM 18-7. Which o the following is the value of x² + 1/x² when
x + 1 / x = 5?
A. 28 C. 23
B. 24 D. 25
PROBLEM 18-8. Evaluate the following logarithm: log ₂ √2 / 8.
A. 2.5 C. -2.5
B. 5.2 D. -5.2
PROBLEM 18-9. Given log ₁₀ 2 = 0.3010, find log ₁₀ 32.
A. 1.863 C. 1.365
B. 0.256 D. 1.505
PROBLEM 18-10. Given log (2x – 3) = ½. What is the value of x if the base of
the logarithm is 9?
A. 1 C. 3
B. 2 D. 4
PROBLEM 18-11. If log N = 2/3, then N is equal to:
A. 4 C. 3
B. 2 D. 5
PROBLEM 18-12. What is the value of x if log (base x) 1296 = 4?
A. 5 C. 3
B. 4 D. 6
PROBLEM 18-13. Solve for x if 2 log x – log (30 – 2x) = 1.
A. 12 C. 30
B. 10 D. 15

PROBLEM 18-14. If log ₓ 2 = 0.69 and log ₓ 3 = 1.10, find logₓ 6?

A. 1.7836 C. 1.1236
B. 1.5698 D. 2.1475
PROBLEM 18-15. If log ₓ 6 = 1.12925, what is the value of log ₓ 11?
A. 1.65 C. 1.42
B. 1.73 D. 1.51
PROBLEM 18-16. What is the product (AB) of the following complex numbers?
Express your answer in polar form.
A. 50 < 90⁰ C. 50 < 45⁰
B. 50 < 30⁰ D. 50 < 60⁰
PROBLEM 18-17. Add the following complex numbers: 3 + 4i and 2 – 5i.
A. 12 + 2i C. 6 – i
B. 24 – 5i D. 5 – i
PROBLEM 18-18. The function is defined as f(x) = 1/(1 + x) . For what value of
x is f(x) undefined?
A. { -1,-1/2} C. {-1,0}
B. {-1,-2} D. {0}
PROBLEM 18-19. If f(x) = 2x² + 2, find the value of f(x + 4).
 A. 2x² + 16x +24 C. 2x² + 6x + 4
B. 2x² + 16x D. 2x² + 16x + 34
PROBLEM 18-20. If f(x) = (3√x 4)², then how much does f(x) increase as x goes
from 2 to 3.
A. 1.372 C. 1.273
B. 1.732 D. 1.723
PROBLEM 18-21. If f(x) = x³ - x - 1, what is the set of all c if f(c) = f(-c)?
A. All real umbers C. {0}
B. {-1,0,1} D. {0,1}
PROBLEM 18-22. If f(x) = 3x – 1, then f⁻¹(x) is equal to:
A. x – ⅓ C. ⅓ x+ ⅓
B. 3x + 1 D. ⅓ x – 2
PROBLEM 18-23. If f(x) = x - 2x² + 2, what is f(a – 2)?
A. 9a – 2a² – 8 C. 2a – 9a² – 8
B. 9a – 2a² + 8 D. 2a + 9a² – 9
PROBLEM 18-24. If f(x) = 3x + 2 and g(x) = x, then g(x) is equal to:
A. x – 2/3 C. ⅓ (x – 2)
B. 4x + 9 D. 3x – 2

PROBLEM 18-25. Solve for x in the following equation:

x + 4x +7x + 10x + ... + 64x = 1430
A. 3 C. 5
B. 4 D. 2
PROBLEM 18-26. Four positive integers form an arithmetic progression. If the
product of the first and the last term is 70 and the second
and third term is 88, what is the first term?
A. 3 C. 5
B. 14 D. 8
PROBLEM 18-27. What is the value of x if 1, ¼, 1/x, 1/10 ... form a harmonic
progression?
A. 6 C. 8
B. 7 D. 5
PROBLEM 18-28. Find the sum o the infinite progression: 2⁻¹, 2⁻³, 2⁻⁵,...
A. 3/4 C. 2/3
B. 1/2 D. 1/3
PROBLEM 18-29. The sides of a right triangle are in the arithmetic progression
whose common difference is 6. Find the hypotenuse.
A. 6 C. 18
B. 30 D. 24
PROBLEM 18-30. If 3, 5, 8.333, and 13.889 are the first four terms of a
sequence then, which of the following cold define the
sequence?
A. A₀ = 3; Aₓ = Aₓ₋₁ + 40/9
B. A₀ = 3; Aₓ+₁= Aₓ+₂
C. A₀ = 3; Aₓ = 5/3 Aₓ₋₁
D. A₀ = 3; Aₓ+₁= 2Aₓ+₂
PROBLEM 18-31. Two numbers differ by 40 and their arithmetic mean exceeds
their geometric mean by 2. What is the smaller number?
A. 81 C. 64
B. 96 D. 45
PROBLEM 18-32. If kx² - (k + 3)x² + 13 is divided by x – 4 the remainder is 157.
What is the value of k?
A. 5 C. 3
B. 4 D. 7
PROBLEM 18-33. The average rate of production of PCB is one (1) unit for
every 2 hours work by two workers. How many PCB’s can
be produced in one month by 80 workers working 200 hours
during the month?
A. 2000 C. 3000
B. 5000 D. 4000
PROBLEM 18-34. A can d a job 50% faster than B ad 20% faster than C.
Working altogether, they can finish the job in 4 days. How
many days will it take A to finish the job if he work alone?
A. 16 C. 10
B. 18 D. 12
PROBLEM 18-35. If John can paint a room in 30 minutes and Tom can paint it
in 1 hour, how many minutes will it take them to paint the
room if they work together?
A. 15 C. 12
B. 20 D. 10
PROBLEMS 18-36. A salesman started walking from office A at 9:30 am at the
rate of 2.5 kph. He arrived office B 12 seconds late. Had he
started at A at 9:00 am and walked at 1.5 kph, he would
have arrived at B one minute before the required time. At
what time was he supposed to be at B?
A. 10:13 am C. 10:22 am
B. 10:16 am D. 10:18 am
PROBLEM 18-37. A man walks a certain distance at the rate of 5 kph and
returns at rate of 4 kph, if the total time that it takes hi is 3
hour and 6 minutes, what is the total distance that he
walked?
A. 8 C. 18
B. 9 D. 16
PROBLEM 18-38. John can shovel a driveway in 50 minutes. If Mary can
shovel the driveway in 20 minutes, how long will it take them,
to the nearest minute, to shove the driveway if they work
together?
A. 12 C. 14
B. 13 D. 16
PROBLEM 18-39. Carpenter Pedro can make a bookshelf in 5 working days
alone. Carpenter Juan can do the same bookshelf in 10 days
working alone. How long will it take them to extend in the
opposite direction for the first time?
A. 4 – 1/3 days C. 3 – 1/3 days
B. 3 – 2/3 days D. 2– 2/3 days
PROBLEMS 18-40. At exactly what time after 2 o’clock will the hour hand and
the minute hand of a continuously driven clock extend in the
opposite direction for the first time?
A. 2:43:6.8 C. 2:43:12.2
B. 2:43:58.1 D. 2:43:38.2
PROBLEM 18-41. A speed boat can make a trip of 100 km in one hour and 30
minutes if it travels upstream. If it travels downstream, it will
 take one hour and 15 minutes to travel the same distance.
What is the speed of the boat in calm water?
A. 80 kph C. 66.67 kph
B. 123.33 kph D. 73.33 kph
PROBLEM 18-42. A train can pass an average of 3 stations every 10 minutes,
at this rate, how many stations will t pass in one hour?
A. 30 C. 18
B. 26 D. 15
PROBLEM 18-43. Going against the wind, a domestic plane can travel 5/8 of
the distance it can travel in one hour if it is going with the
wind. If the plane can fly 300 kilometers an hour in calm air,
what is the velocity of the wind?
A. 69.23 kph C. 63.92 kph
B. 62.93 kph D. 96.23 kph
PROBLEMS 18-44. John is four times as old as Harry. In six years, John will be
twice as old as Harry. What is the age of Harry now?
A. 2 C. 4
B. 3 D. 5
PROBLEMS 18-45. What is the value of the following determinants?
1 4 6 1
2 3 8 2
1 4 6 1
2 1 5 2
A. 1 C. 4
B. 3 D. 0
PROBLEM 18-46. Find the number of ways two balls, four dolls, and six toy
guns can be given to 12 children, if each child gets a toy.
A. 12,606 C. 13,860
B. 10,620 D. 1,640
PROBLEM 18-47. If three coins are tossed, how many possible ways are there
for at least one coin showing tails?
A. 5 C. 8
B. 6 D. 7
PROBLEM 18-48. What is the indicated root ³√-125?
A. -25 C. -5
B. 5 D. 25

PROBLEM 18-49. Given the w varies directly as the product of x and y and
inversely as the square of z and that w = 4 when x = 2, y = 6,
and z = 3. What is the value of w when x = 1, y = 4, and z =
2?
A. 3 C. 5
B. 4 D. 2

PROBLEM 18-50. If 16 are four more than 3x, then x² + 5 is equal to:
A. 18 C. 8
B. 21 D. 10
PROBLEM 18-51. A professional organization is composed of x ECE’s and 2x
CE’s. If 6 ECE’s are placed by 6 CE’s, 1/6 of the members
will be ECE’s. What is the value of x?
A. 24 C. 12
B. 36 D. 18
PROBLEM 18-52. An hour – long test has 60 problems. If a student completes
30 problems in 20 minutes, how many seconds does he
have o average for completing each of the remaining
problems?
A. 60 C. 120
B. 90 D. 80
PROBLEM 18-53. The sum of two numbers is 10 and the sum of the square of
the numbers is 52. Find the product of the two numbers.
A. 26 C. 32
B. 24 D. 16
PROBLEM 18-54. After 8:00 pm, a ride in a cab cost P25.00 plus P3.00 for
every fifth of a kilometer travelled. If a passenger travel x
kilometres, what is the cost of the trip in pesos, as a function
of x?
A. 28x C. 25 + 15x
B. 25 + 3x D. 25 + 0.6x
PROBLEM 18-55. On a scaled map, a distance of 10 cm represents 5 km. If a
street is 750 meters long, what is the length on the map, in
centimetres?
A. 15 C. 150
B. 1.5 D. 0.15
PROBLEM 18-56. Between 1950 and 1960, the population of Singapore
increased by 3.5 million. If the amount of increase between
1960 and 1970 was 1.75 million more than the increase from
1950 to 1960, compute for the total amount of increase in
the population of Singapore between 1950 and 1970.
A. 5.75 million C. 5.25 million
B. 4.25 million D. 8.75 million
PROBLEM 18-57. In a typical month ½ of the UFO sightings in the California
State are attributable to airplanes and ⅓ of the remaining
sightings are attributable to weather balloons. If there were
108 sightings during one typical month, how many would be
attributable to weather balloons?
A. 36 C. 24
B. 54 D. 18
PROBLEM 18-58. If x is an integer, which of the following must be an odd
integer?
A. 4(x – 1) C. 3x² – 2
B. 4x – 1 D. x² – 3
PROBLEM 18-59. Ernie’s average in 6 subjects is 83. If his lowest grade is
disregarded, the average o his remaining subject is 84. What
is his lowest grade?
A. 76 C. 79
B. 77 D. 78
PROBLEM 18-60. A product has a current selling price of P325. If it’s selling
price is expected to decline at the rate of 10 % per annum
because of adolescence, what will be its selling price four
years hence?
A. P302.75 C. P213.23
B. P202.75 D. P156.00
PROBLEM 18-61. Instead of multiplying the number by 17, Karla divided it by
17. I the answer she obtained was1, what should have been
the correct answer?
A. 285 C. 289
B. 276 D. 295

PROBLEM 18-62. Kim was sent to the store to get 11 boxes of sardines. Kim
could carry only two boxes at a time. How many trips would
Kim have to make?
A. 4 C. 7
B. 6 D. 5
PROBLEM 18-63. Simplify cos (30⁰ + A) as a function of angle A only.
A. sin A C. cos A
B. tan A D. sec A
PROBLEM 18-64. In triangle ABC, sinA/sinB = 7/10 and sinB/sinC = 5/2. If
angle A,B, and C are opposite side a, b, and c, respectively,
and the triangle had a perimeter of 16, what is the value of
a?
A. 12.33 C. 4.67
B. 8 D. 5.33
PROBLEM 18-65. At one side of a road is a pole 25 ft high fixed on top of a
wall 5 ft high. On the other side of the road, at a point on the
ground directly opposite, the flag staff and the wall subtend
equal angles. Find the width of the road.
A. 30 feet C. 45 feet
B. 20 feet D. 50 feet
PROBLEM 18-66. If sec 2A = 1/ sin13A, determine the value of A.
A. 3⁰ C. 7⁰
B. 6⁰ D. 5⁰

PROBLEM 18-67. The sides of a triangle are 8, 15, and 17 units. If each side is
doubled, by how many square units will the area of the
triangle be increased?
A. 120 C. 180
B. 60 D. 240
PROBLEM 18-68. Find the area of the spherical triangle ABC having the
following parts:
Angle A = 140⁰ Angle C = 86⁰
Angle B = 75⁰ radius of the sphere = 4m
A. 32.78 m² C. 33.79 m²
B. 41.41 m² D. 4.56 m²
PROBLEM 18-69. What is the area of a rhombus whose diagonals are 12 and
24?
A. 144 C. 108
B. 164 D. 132
PROBLEM 18-70. A wire with length of 52 cm is cut into two unequal lengths.
Each pair is bent to form a square. If the sum of the area of
the two square is 97 cm², what is the area of the smaller
square?
A. 16 C. 64
B. 81 D. 49
PROBLEM 18-71. The perimeter of a circular sector, whose central angle is 60⁰
is 14 feet. Find the radius of the circle.
A. 3.68 feet C. 6.32 feet
B. 4.59 feet D. 8.74 feet
PROBLEM 18-72. An equilateral triangle has a altitude of 5√3 cm long. Find the
area of the triangle.
A. 25√3 C. 15√3
B. 3√5 D. 10√3

PROBLEM 18-73. A circle with radius r has a circumference equal to the

perimeter of a square whose side is S. Which of the
following is correct?
A. r = S/2π C. r = 2π/S
B. r = S D. r = 2S/π

PROBLEM 18-74. How long is each side of a regular hexagon with perimeter
108 centimeters?
A. 18 C. 32
B. 24 D. 12

PROBLEM 18-75. How many sides have a polygon if the sum of the measures
of the angles is 900⁰?
A. 9 C. 8
B. 6 D. 7
PROBLEM 18-76. A rectangular room is to contain 125 square meters of
flooring. If its length is to be five times its width, what should
its dimension be?
A. 5m x 75m C. 3m x 15m
B. 25m x 125m D. 5m x 25m
PROBLEM 18-77. If the total number of diagonals o an N-gon is 77, then what
is the value of N?
A. 14 C. 12
B. 13 D. 15
PROBLE 18-78. A circle has a circumference that is numerically equal t its
area If a certain square has the same area as the circle,
what would be the length of the side?
A. 3√π C. π√2
B. π√3 D. 2√π

PROBLEM 18-79. A rectangular solid has dimensions 3,4, and 5. What is its
diagonal?
A. 7.071 C. 6.325
B. 7.253 D. 9.125
PROBLEM 18-80. Find the volume of the pyramid formed in the firs octant by
the plane 6x+ 10y + 5z – 30 = 0.
A. 15 C. 12
B. 13 D. 14
PRBLEM 18-81. The upper and lower bases of the frustum of a rectangular
pyramid are 3m y 4m and 6m by 8m, respectively. If the
volume of the solid is 140m³, how far apart are the bases?
A. 6.5m C. 5m
 B. 4.5m D. 3.5m

PROBLEM 18-82. By how many percent will the volume of the cube increase if
its edge is increased by 20%?
A. 44 C. 1.728
B. 72.8 D. 7.28
PROLEM 18-83. A closed conical vessel wit base radius of 1 m and altitude
2.5 m has its axis vertical. In upright position (vertex
uppermost), the depth f water in it is 50 cm. If the vessels
was inverted (vertex lowermost), how deep is the water in it?
A. 196.8cm C. 204.6cm
B. 187.5cm D. 174.8cm
PROBLEM 18-84. When an irregular-shaped object is placed in a cylindrical
vessel of radius 8 cm, containing water, the water surface
rises 6 cm. What is the volume of the object if it is completely
submerged in water?
A. 384π cc C. 632π cc
B. 525π cc D. 245π cc

PROBLEM 18-85 A right prism has a base in the shape of regular octagon
inscribed in a square 10 cm by 10 cm. If its altitude of 15 cm,
find its volume in cc.

A. 1,242.6 C. 1,359.7
B. 1,163.4 D. 1,421.6
PROBLEM 18-86 If two points in number line have coordinates 1 and 7, find
the coordinate of a point on the line which is twice as far
from 1 as from 7.
A. 3 C. 6
B. 4 D. 5

PROBLEM 18-87 Find the equation of the line with slope of 2/3 and which
passes through point of intersection of the lines 4x–2y +1 = 0
and x – 2y + 4 = 0.

A. 4x – 6y + 9 = 0 C. 6x – 4y + 11 = 0
B. 6x – 4y + 9 = 0 D. 4x – 6y + 11 = 0

PROBLEM 18-88. Find x if the point (x,2) is equidistant from (3,2) and (7,2).

A. 5 C. 8
B. 4 D. 5

PROBLEM 18-89. A rectangle with sides parallel to the axes has vertices at
(-3,2), (2,5), and (-3,5). What is the coordinate of the fourth
vertex?

A. (-5,2) C. (5,-2)
B. (2,2) D. (-3,-2)

PROBLEM 18-90. The triangle defined by the points A(6,1), B(2,4) and
C(-2,1) is what?
 A. right C. isosceles
B. obtuse D. scalene

PROB LEM 18-91. Given the curve 36x² + 9y² - 36 = 0. What is the length of its
latus rectum?

A. 0.6 C. 1.5
B. 0.75 D. 1
PROBLEM 18-92. Which of the following points (1,0), (-1,0), (4,4), and (9,7)
belong to the graph of the equation y = x² - x ?
A. (-1,0) C. (4,4) and (1,0)
B. (9,7) D. (1,0)
PROBLEM 18-93. A circle is described by the equation x² + y² - 16x = 0. What
is the length of the chord that is 4 units from the center of the
circle?
A. 12.563 C. 8.523
B. 13.856 D. 9.632
PROBLEM 18-94. What is the equation of the line that passes through (-3,5)
and is parallel to the line 4x – 2y + 2 = 0?
A.4y – 2x + 22 = 0 C. 4x – 2y + 11 = 0
B. 2x – y + 10 = 0 D. 2x – y + 11 = 0
PROBLEM 18-95. A circle with its center in the first quadrant is tangent to both
x and y-axes. If the radius is 4, what is the equation of the
circle?

A. (x + 4)² + (y + 4)² = 16 C. (x – 4)² + (y + 4)² = 16

B. (x – 4)² + (y – 4)² = 16 D. (x + 2)² + (y – 2)² = 16
PROBLEM 18-96. What s the distance between the line x + 2y + 8 = 0 and the point
(5,-2)?
A. 4.025 C. 4.205
B. 4.502 D. 4.052

PROBLEM 18-97. Find the area enclosed by the following curve:

x² + y² - 10x – 10y + 25 = 0

A. 85.47 square units C. 78.55 square units

B. 95.61 square units D. 68.53 square units

PROBLEM 18-98. What s the diameter of a circle with the following equation:
x² + y² - 6x + 4y – 12 = 0
A. 10 C.16
B. 5 D. 25
PROBLEM 18-99. What conic section is described by the equation
x² + y² - 4x + 2y – 20 = 0
A. circle C. hyperbola
B. parabola D. ellipse

PROBLEM 18-100. The directrix of the parabola is y – 5 = 0 and its focus is at

(4,-3).Find the length of its latus rectum.
 A. 16 C. 2
B. 12 D. 4
PROBLEM 18-101. Find the area bounded by the parabola x² = 16(y – 1) and its
latus rectum.
A. 56.33 C. 36.67
B. 46.67 D. 42.67
PROBLEM 18-102. 4x² - y² = 16 is the equation of a:
A. circle C. hyperbola
B. parabola D. ellipse
PROBLEM 18-103. Find e area bounded by the parabola 4y = x²- 2x + 1 and its
latus rectum.
A. 8/3 C. 2/3
B. 4/3 D. 10/3
PROBLEM 18-104 Point (3,4) is the center of a circle that is tangent to the X-
axis. What is the point of tangency?
A. (4,0) C. (3,0)
B. (0,4) D. (0,3)
PROBLEM 18-105. An arch 18 m high has a form of parabola with a vertical
axis. The length of a horizontal beam placed across the arch
8 m from the top is 64 m . Find the width at the bottom.
A. 81 C. 64
B. 96 D. 74
PROBLEM 18-106. Find k such that the line y = 4x + 3 is tangent to the curve
y = x² + k.
A. 7 C. 5
B. 6 D.4
PROBLEM 18-107. A triangle have vertices at (0,0), (6, 30⁰), and (9, 70⁰). What
is the perimeter of the triangle?
A. 29.45 units C. 20.85 units
B. 25.74 units D. 23.74 units
PROBLEM 18-108. The square root of a variance is referred to as _________.
A. dispersion C. median
B. central tendency D. standard deviation
PROBLEM 18-109. To find the angles of a triangle, given only the length of the
sides, one would use:
A. law of sines C. orthogonal functions
B. law of tangents D. law of cosines
PROBLEMS 18-110. A curve generated by a point which moves in uniform
circular motion about an axis while travelling with a constant
speed parallel to the axis.
A. epicycloid C. spiral of Archimedes
B. cycloid D. helix
Problem 19-1 Which of the following mathematically represent to digit number
with 𝑥 as the units digit and 𝑦 as it’s ten’s digit .

A. 10𝑦 + 10𝑥 C. 𝟏𝟎𝒚 + 𝒙

B. 10𝑦 − 𝑥 D. 10𝑥 + 𝑦

Problem 19-2 What is the equation form of the statement; the amount by which
100 exceeds four times a given number?

A. 4𝑥(100) C. 100 − 4𝑥
B. 𝟏𝟎𝟎 + 𝟒𝒙 D. 4𝑥 − 100

Problem 19-3 Solve for 𝑥 if 8𝑥 = 2𝑦+2 and 163𝑥−𝑦 = 4𝑦 .

A. 4 C. 3
B. 2 D. 1

Problem 19-4 From the equations 7𝑥 2 + (2𝑘 − 1)𝑥 − 3𝑘 + 2 = 0, determine the

value of 𝑘 so that the sum and product of the roots are equal.

A. 4 C. 2
B. 3 D. 1

Problem 19-5 The arithmetic mean six numbers is 17. If two numbers are added
to the progression, the new arithmetic mean is 21. What are the two
if their difference is 4?

A. 28 and 32 C. 26 and 30
B. 30 and 34 D. 34 and 38

Problem 19-6 What is the sum of all odd integers between 10 and 500?

A. 65,955 C. 87,950
B. 62,475 D. 124,950

Problem 19-7 How many terms of the progression 3,5,7,…should there to be so

that their sum will be 2600?

A. 54 C. 50
B. 52 D. 55

Problem 19-8 If the first term of geometric progression is 27 and the fourth term is
-1, the third term is?

A. 3 C. -2
B. 2 D. -3
Situation(9-11) The fourth term of a geometric progression is 6 and the 10 th term is
384.

Problem 19-9 What is the common ratio of the G.P.?

A. 1.5 C. 2.5
B. 3 D. 2

Problem 19-10 What is the first term?

A. 0.75 C. 3
B. 1.5 D. 0.5

Problem 19-11 What is the seventh term?

A. 24 C. 48
B. 32 D. 96

Problem 19-12 Candle 𝐴 and candle 𝐵 of equal length are to be lighted at the
same time and burning until candle 𝐴 is twice as long as candle 𝐵.
Candle 𝐴 is designed to fully burn 8 hours while candle 𝐵 for 4
hours. How long will they be lighted?

A. 2 hrs and 40 min C. 3 hrs and 40 min

B. 3 hrs and 30 min D. 2 hrs and 30 min

Problem 19-13 At approximately what time between 6 and 7 o’clock will the minute
and hour hands coincide?

A. 27 min and 41 sec after 6 o’clock

B. 32 min and 72 sec after 6 o’clock
C. 25 min and 38 sec after 6 o’clock
D. 32 min and 43 sec after 6 o’clock

Problem 19-14 In how many minutes after 2 p.m. will the hands of the clock extend
in the opposite directions for the first time?

A. 40.522 C. 45.575
B. 43.636 D. 41.725

Problem 19-15 How many ounce of pure nickel must added to 150 ounce of alloy
70% pure to make an alloy 85% pure?

A. 150 C. 125
B. 225 D. 175

Problem 19-16 If there are nine distinct items taken three at time, how many
permutations will be there?

A. 252 C. 504
B. 720 D. 336

Problem 19-17 A bag contains 3 white and 5 red balls. If two balls are drawn in
succession without returning the first ball drawn, what is the
probability that the balls drawn are both red?
 A. 0.299 C. 0.357
B. 0.237 D. 0.107

Problem 19-18 A janitor with a bunch of nine keys is to open a door but only one
key can open. What is the probability that he will succeed in three
trials?

A. 0.375 C. 0.333
B. 0.425 D. 0.255

Problem 19-19 Evaluate the expression: (1 + 𝑖 2 )10 , where 𝑖 is an imaginary

number.

A. 0 C. 𝑖
B. -1 D. 1

Situation(20-22) Given the following matrix:

2 5 1 1 0 0
𝐴 = {3 1 0} And 𝐵 = {0 1 0}
6 7 4 0 0 1

Problem 19-20 Determine the minor corresponding to the 5 entry in matrix 𝐴.

A. 15 C. 12
B. 8 D. 4

Problem 19-21 Determine the cofactor corresponding to the 3 entry in matrix 𝐴.

A. 13 C. 4
B. -13 D. -4

Problem 19-22 What is the product 𝐴 x 𝐵?

1 2 5 5 2 1
A. {0 3 1} C. {1 3 0}
4 6 7 7 6 4
2 1 5 𝟐 𝟓 𝟏
B. {3 0 1} D. {𝟑 𝟏 𝟎}
6 4 7 𝟔 𝟕 𝟒

Problem 19-23 If sin 𝐴 = 4/5 and sin 𝐵 = 7/25, what is sin(𝐴 + 𝐵) if 𝐴 is in the 3rd
quadrant and 𝐵 is in the 2nd quadrant?

A. 3/5 C. -3/5
B. 2/5 D. 4/5

Problem 19-24 If the side of right triangle are 3,4,5 m, the area of inscribe circle is
___________?

A. 2𝜋 square m C. 3/4𝜋 square m

B. 3𝜋/2 square m D. 𝝅 square m

Situation(25-27) A flagpole is stands on top of a pedestal. From a point O at same

elevation as the base of the pedestal, the vertical angle of the top of
the flagpole is 60° and the vertical angle of the bottom of the
flagpole is 52°. Point O is 14.8 meters away from the base of
pedestal.
Problem 19-25 How high is the pedestal in meters?

A. 18.94 C. 15.24
B. 25.36 D. 14.28

Problem 19-26 How long is the flagpole in meters?

A. 3.68 C. 8.12
B. 5.88 D. 6.69

Problem 19-27 What is the distance from O to the of the flagpole?

A. 33.2 m C. 15.4 m
B. 29.6 m D. 22.8 m

Situation(28-30) One of the diagonals of a rhombus measure 12 inches. If the area

of the rhombus is 132 square inches, determine the following:

Problem 19-28 The length of the other diagonal in inches.

A. 24 C. 22
B. 8 D. 14

Problem 19-29 The measure of the side of the rhombus in inches.

A. 13.85 C. 8.52
B. 10.96 D. 12.53

Problem 19-30 The measure of acute angle between the side of the rhombus in
degrees.

A. 28.61 C. 57.22
B. 45.26 D. 32.78

Situation(31-33) A triangular lot 𝐴𝐵𝐶 have side 𝐴𝐵 = 400 and angle 𝐵 = 50°. The lot
is to be segregated by a dividing line 𝐷𝐸 parallel to 𝐵𝐶 and 150 m
long. The area of segment 𝐵𝐶𝐷𝐸 is 50,977.4 𝑚2 .

Problem 19-31 Calculate the area of lot 𝐴𝐵𝐶, in square meter.

A. 62,365 C. 57,254
B. 59,319 D. 76,325
Problem 19-32 Calculate the area of lot 𝐴𝐷𝐸, in square meter.

A. 8,342 C. 6,569
B. 14,475 D. 11,546

Problem 19-33 Calculate the value of angle 𝐶, in degrees.

A. 57 C. 63
B. 42 D. 68
Problem 19-34 A railroad curve is to be laid in a circular path. What should be the
radius if the track is to change direction by 30 degrees at a distance
of 300 m?

A. 352 m C. 287 m
B. 573 m D. 452 m

Situation(35-37) A swimming pool is shaped from two intersecting circles 9 m in

radius with their centers 9 m apart.

Problem 19-35 What is the area common to the two circles, in square meter?

A. 85.2 C. 128.7
B. 63.7 D. 99.5

Problem 19-36 What is the total water surface area, in square meter?

A. 409.4 C. 387.3
B. 524.3 D. 427.5

Problem 19-37 What is the perimeter of the pool, in meters?

A. 63.5 C. 82.4
B. 75.4 D. 96.3

Problem 19-38 Spheres of the same size are piled in the form of pyramid with an
equilateral triangle as its base. Compute the total number of
spheres in the pile if each side contains 4 spheres.

A. 20 C. 16
B. 64 D. 28

Problem 19-39 What is the volume (in 𝑐𝑚3 ) of a right hexagonal prism 15 cm high
and width one of its sides equal to 6 cm?

A. 985 C. 818
B. 929 D. 1185

Problem 19-40 Eight balls are tightly packed in cubical container that measure 8
cm on each side. The balls are arranged with 4 balls per layer and
in contact with the walls of the container and the adjacent balls. If
the container is filled with water, what is the volume of the water (in
cubic cm)?

A. 355 C. 268
B. 335 D. 244

Situation(41-43) A closed conical vessel has a base radius of 2 m and is 6 m high.

When upright position, the depth of water in the vessel is 3 m.

Problem 19-41 What is the volume of water in cubic meter?

A. 22 C. 28
 B. 25 D. 32

Problem 19-42 If the vessel is held in inverted position, how deep is the water, in
meters?

A. 8.56 C. 4.12
B. 5.74 D. 6.87

Problem 19-43 What is the weight of water in quintals. Unit weight of water is
9800N/𝑚3 ?

A. 263.4 C. 219.7
B. 195.4 D. 247.2

Problem 19-44 The total volume of two spheres is 100𝜋 cubic cm. The ratio of
areas is 4:9. What is the volume of the smaller sphere, in cubic
cm?

A. 75.85 C. 71.79
B. 314.16 D. 242.36

Problem 19-45 What is the distance between the points (3,7) and (-4,-7)?

A. 18.65 C. 5.25
B. 6.54 D. 15.65

Problem 19-46 What is the area bounded by the curved defined by the equation
𝑥 2 − 8𝑦 = 0 and its latus rectum?

A. 5.33 C. 10.67
B. 7.33 D. 3.66

Situation(47-49) Given the ellipse 9𝑥 2 + 16𝑦 2 − 144 = 0.

Problem 19-47 Determine the length of the arc in the first quadrant.

A. 5.55 C. 7.58
B. 6.32 D. 9.97

Problem 19-48 Determine the equation of its diameter bisecting all chords having
equal slope of 3.

A. 4 + 25𝑦 C. 𝟔𝒙 + 𝟑𝟐𝒚 = 𝟎
B. 9𝑥 + 16𝑦 = 0 D. 9𝑥 + 32𝑦 = 0

Problem 19-49 What is the volume generated if the area on the first and second
quadrants is resolved about the 𝑋-axis?

A. 201.1 C. 175.4
B. 150.8 D. 165.7

Situation(50-52) Given the equilateral hyperbola 𝑥𝑦 = 8

Problem 19-50 What is the length of the conjugate axis of the hyperbola?
 A. 12 C. 8
B. 16 D. 4

Problem 19-51 How far apart are the vertices of the hyperbola?

A. 4 C. 16
B. 8 D. 12

Problem 19-52 Determine the eccentricity of the hyperbola.

A. 1.414 C. 1.368
B. 1.732 D. 1.521

Situation(53-55) Given the ellipse 16𝑥 2 + 25𝑦 2 = 400.

Problem 19-53 Compute is perimeter.

A. 32.653 C. 24.785
B. 28.448 D. 36.896

Problem 19-54 Determine its second eccentricity.

A. 0.75 C. 0.67
B. 0.82 D. 0.6

Problem 19-55 What is the equations of its diameter bisecting the chords having
equal slope of 1/5?

A. 5𝑥 − 16𝑦 = 0 C. 𝟏𝟔𝒙 + 𝟓𝒚 = 𝟎
B. 5𝑥 + 16𝑦 = 0 D. 16𝑥 − 5𝑦 = 0
Christian Abell Balona
Nelmar Laguna
Francis Maranan
Jenelito Dioneda
Don Paul Frivaldo
Ruben John Dometita
Carmela Latuga
Dyssa Marie Golloso
Joseph Ortiz
Rafael Baleza
Christina Herilla