PAST PAPERS 2023
Category 5
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EXAM 5
1. In an airplane, the passengers are asked which languages they can speak. 41
passengers can speak Russian, 67 can speak English, 72 can speak Chinese, 15 can
speak English and Russian, 12 can speak Russian and Chinese, 35 can speak
English and Chinese, and 5 can speak English, Russian and Chinese. Find the
number of passengers.
A) 127 B) 123 C) 172 D) 140
2. Let , and . Find .
A) 17 B) 28 C) 32 D) 30
3. All the squares in the figure are congruent and have side 1 unit. Find the .
A) B) C) D)
4. Calculate. if .
A) B) C) D)
5. Calculate.
A) B) C) D)
6. ABCD is a parallelogram. Find the area of shaded region.
A) 48 B) 18 C) 54 D) 72
7. If and then calculate .
[use logarithm to evaluate]
A) 144 B) 4 C) 225 D) 15
8. Calculate. .
A) 3 B) 9 C) 1 D)
9. Simplify.
A) B) C) D)
10. If , then calculate the value of .
A) 3 B) 8 C) 4 D) 1
11.
If for a positive real number , then calculate the value of
.
A) B) 24 C) 16 D) 48
12. If , then calculate the value of .
A) 33 B) 28 C) 27 D) 29
13.
The equation of a circle in the xy-plane is given. Find the
radius of the circle.
A) 2 B) 3 C) 4 D) 9
1 1 𝑎 𝑏
14. Given that 𝑎
+ 𝑏
= 2 and 3 = 5 = 𝑥. Find the value of x.
[use logarithms to evaluate]
3
A) 𝑥 = 225 B) 𝑥 = 15 C) 𝑥 = 15 D) 𝑥 = 15
15. A 3×4×5 solid block is made up of 1×1×1 unit cubes. The outside surface of
the block is painted black. How many unit cubes have exactly one face painted
black?
A) 16 B) 18 C) 20 D) 22
16.
2
If 𝑓(𝑥) = 4 + 2𝑥 , find the integer value of k for which (2𝑘) = 18𝑘 .
A) –2 B) 0 C) 1 D) 2
𝑥+1 𝑥+2 𝑥+4 𝑥+3
17. Solve the equation 7×3 −5 =3 −5 .
A) − 2 B) − 1 C) 0 D) 1
18. Find the height of the table.
A) 54 cm B) 81 cm C) 127 cm D) 254 cm
4 3 2
19. If 𝑥 + 2 is a factor of 𝑥 + 𝑥 + 3𝑥 + 𝑘𝑥 − 10 , then find 𝑘.
A) − 5 B) − 13 C) 15 E) 5
3
[(𝑛+1)!]
20. Simplify. 3
(𝑛!)
3 3
A) 𝑛 B) 𝑛 + 1 C) 𝑛 D) (𝑛 + 1)
21. A right circular cone has a base radius of 𝑟 and a height of ℎ. If the radius is
decreased by 20 percent and the height is increased by 10 percent, which of the
following is the resulting percent change in the volume of the cone?
A) 10% decrease B) 12% decrease
C) 18.4% decrease D) 29.6% decrease
22.
2
In the 𝑥𝑦 −plane, the line 𝑦 = 2𝑥 + 𝑏 intersects the parabola 𝑦 = 𝑥 + 𝑏𝑥 + 5
at the point(3, 𝑘). If 𝑏 is a constant, what is the value of 𝑘?
A) 0 B) 1 C) 2 D) 3
23. If |4𝑥 − 4| = 8 and |5𝑦 + 10| = 15, what is the smallest possible value of
𝑥𝑦?
A) − 1 B) − 3 C) − 5 D) − 15
24. In the figure below, 𝐴𝐵 is parallel to 𝐺𝐻 and 𝐷𝐹 is parallel to 𝐵𝐶.
If |𝐷𝐸| = 1, |𝐸𝐻| = 3, |𝐸𝐺| = 2, and |𝐻𝐶| = 10, what is the length of |𝐴𝐷|
?
A) 2.5 B) 3 C) 3.5 D) 4
25.
In the figure below, 𝐴𝐶 is a diameter of the circle. If 𝐴𝐶 = 1, which of the
following gives the area of triangle 𝐴𝐵𝐶 in terms of θ?
θ
A) 2
𝑡𝑎𝑛θ
B) 2
C) 2𝑠𝑖𝑛θ
sin𝑠𝑖𝑛 θcos𝑐𝑜𝑠 θ
D) 2
26.
The table below summarizes the distribution of living situations for residences in a
neighborhood. If a duplex in the neighborhood is to be inspected at random, what
is the probability that the residence is occupied by no more than 2 family
members?
6
A) 23
17
B) 23
15
C) 23
11
D) 23
27. Evaluate 7 5 × 4 + 1.
A) 12 B) 4 C) 16 D) 2
2
28. Find the increasing interval of function 𝑓(𝑥) = 𝑥 − 1.
A) (− ∞; − 1)∪(1; + ∞) B) (− ∞; − 1]∪[1; + ∞)
C) [1; + ∞) D) (− ∞; − 1]
3 2
29. Find the area of the figure bounded by 𝑦 = 𝑥 , 𝑥 = 𝑦 .
5 1 2
A) 1 B) 12
C) 4
D) 3
2
𝑐𝑜𝑠 𝑥
30. Solve ∫ 1−𝑠𝑖𝑛𝑥
𝑑𝑥
A) 𝑥 + 𝑐𝑜𝑠𝑥 + 𝐶 B) 𝑥 + 𝑠𝑖𝑛𝑥 + 𝐶
C) 𝑥 − 𝑐𝑜𝑠𝑥 + 𝐶 D) 1 − 𝑠𝑖𝑛𝑥 + 𝐶
31.
𝐴(2, − 1, − 3), 𝐵(− 3, 5, 2), 𝐶(− 2, 3, − 5) are the vertices of triangle
𝐴𝐵𝐶. If 𝐵𝑀 is a median of triangle 𝐴𝐵𝐶, then find the length of 𝐵𝑀.
A) 57 B) 55 C) 9 D) 61
32.
Given the arithmetic sequence 15, 20, 25, ... . Determine which term of arithmetic
sequence is equal to 265.
A) 49th B) 57th C) 51st D) 55th
33. Given that 𝑓(𝑥 − 1) = 3𝑥 + 4, find the 𝑓(2𝑥 + 1).
A) 6𝑥 + 10 B) 3𝑥 + 10
2
C) 6𝑥 + 9 D) 6𝑥 + 11𝑥 + 4
1
34. Evaluate the sum of the first six terms of the geometric sequence 2, 1, 2
, …
.
73 31 63 57
A) 22
B) 8
C) 16
D) 16
35.
The sides of the bottom and top bases of the regular frustum square pyramid are
6 𝑐𝑚 and 4 𝑐𝑚, respectively, and height 23𝑐𝑚. Find the length of the lateral edge
x of the frustum pyramid.
A) 1.5 cm
B) 4 cm
C) 3 cm
D) 5 cm
36. Find the volume of the water.
3
A) 108π 𝑐𝑚
3
B) 144π 𝑐𝑚
3
C) 72π 𝑐𝑚
3
D) 84π 𝑐𝑚
37. If , then calculate .
A) B) C) D) 4
38. Find m. or
A) -2 B) -1 C) 0 D) 1
39. A rectangular box with a square base and an open top is to be constructed using
48 square meters of material. Find the maximum volume of the box in cubic
meters.
A) 2 B) 4 C) 16 D) 32
40.
A belt snugly fit around the two wheels with radii units and units as
in the figure. Their centers are 6 units apart. If π is taken as 3, what is the length of
the belt?
A) 20 units B) 22 units C) 24 units D) 26 units
