REVIEW OF ALGEBRA

The Quadratic Formula

Where:

If:

Sum of the roots = Product of the roots =

Q1
Determine the value of k so that the difference of the roots of the
equation

a. 7 b. 8 c. 9 d. 10
Solution:

Let u and v be the roots (assuming u > v)

Using sum of roots,

Solving simultaneously,

Using products of the roots,

Solving for k,

Answer: (d)
Q2
If u and v are roots of the equation , find the value of .
a. 2 b. 4 c.-1 d. 1
Solution:

Squaring both sides,

Answer: (d)
 Arithmetic Progression

is an AP if

Where
d – common difference

For the nth term,

For the sum,

For the Arithmetic Mean,

 Q3
Find the sum of the integers between 2 and 100 which are divisible
by 3.

a. 1984 b. 1683 c. 1699 d. 4299

Solution:

Solving for n,

Answer (b)
Q4
The arithmetic mean of 6 numbers is 17. If two numbers are added to
the progression, the new set of numbers will have an arithmetic
mean of 19. What are the two numbers if their difference is 4?

a. 21, 25 b. 23, 27 c. 8,12 d. 16,20

Solution:

Let x and x + 4 be the two numbers

Y be the original sum (6 numbers)

Solving for x and x +4, x= 23 and x+4 = 27

Answer(b)
 Geometric Progression
is a GP if

For the Nth term,

For the Sum of the first n terms,

For the infinite sum,

For the Geometric Mean,

 Q5
There are 4 geometric means between 3 and 729. What is the sum
of all the six terms?

a. 1126 b. 1236 c. 1092 d. 1327

Solution:

GP: 3, __, __, __, __, 729

Solving for r,

Answer (c)
Q6
A rubber ball is dropped from a height of 15m. On each rebound, it
rises 2/3 of the height from which it last fell. Find the distance in
meters travelled by the ball before it comes to rest.

a. 60 b. 80 c. 85 d. 75
Solution:

Distance = 15 + 2[

Answer(d)
Q7
The geometric mean and harmonic mean of two numbers are 12
and 36/5 respectively. What are the numbers?

a. 4,36 b. 5,25 c. 6,40 d. 8,45

Solution:
Given: GP: a, 12, b

HP: a, 36/5, b
AP:

Solving simultaneously,

Answer(a)
 Binomial Theorem

Term involving

If n is even,

Middle term
 Q8
Find the middle term of the expansion of .

a. b. c. d.
Solution:

Middle term =

Answer (a)
 Q9
Find the term free of x(i.e, the constant term) in the expansion of .

a. 4210 b. 5005 c. 6030 d. 7290

Solution:
Consider the literal coefficient:

Required term: term involving .

Answer(b)
 Q10
Find the sum of the coefficients of the expansion
a. b. (

Solution:
a.

b. (1 +
 Logarithms

Where
Properties of Logarithm:
1.
2.
3.
4.
5.
6.
 Q11
If then x equals

a. 3 b. 4 c. 1 d. 2

Solution:
Let

When

Answer (d)
Q12
Solve for x in the equation

a. 1, 0.316 b. 2, 0.425 c. 3, 0.625 d. 10, 0.215

Solution:

Let

When
When
Answer (d)
Remainder Theorem

Let be a polynomial in x
If is divided by , then the remainder is

(the value of p when

 Q13
The expression when divided by leaves a remainder of and when
divided by leaves a remainder of . Find and .

a. b.
c. d.
Solution:
Let
When divided by

When divided by

Solving simultaneously,

Answer (b)
 Q14
The sum of two positive numbers is 8 and the difference of their
reciprocals is . Find the larger number.

a. 6 b. 5 c. 7 d. 3
Solution:

Let be the larger number

be the smaller number

Solving for x,

Answer (b)
Q15
If , then is

a. 1 b. c. d. -1
Solution:

Answer (c)
Q16
A man is 4 years older than his wife and 5 times as old as his son.
When the son was born, the age of the wife was six – seventh that of
her husband’s age. How old is the son?

a. 5 b. 6 c. 7 d. 8
Solution:
Let be the son’s present age
Present Past X years ago
Son x 0 (x-x)
Father 5x 4x (5x – x)
Mother 5x – 4 4x - 4 (5x - 4 – x)
In the past,

Answer (c)
 Q17
A box with an open top is to be made by taking a rectangular piece of
tin 8x10 inches and cutting a square of the same size out of each
corner and folding up the sides. If the area of the base is to be 24
square inches, what should be the length of the sides of the square?

a. 2 in b. 2.1 in c. 1.8 in d. 2.2 in

Solution:
Given: B = 24 square inches

Solving for x,

Answer (a)
 Q18
The sum of the digits of a 3 place number is 16. If the digits are
reversed and the resulting number is added to the original number,
the sum is 1049. If the resulting number is subtracted from the
original, the difference is 297. What is the number?

Solution:

Let be the hundred’s digit

be the ten’s digit
be the unit’s digit
Solving simultaneously,

Answer:
Q19
“A” can do a job in 3 days and “B” can do the same job in 6 days.
How long will it take them if they work together?

a. 2 days b. 3 days c 4 days d. 5 days

Solution:

Let be the time for them to finish the job.

In one day,
A can finish of the job
B can finish of the job
If they work together,

Answer(a)
Q20
A’s rate of doing work is three times that of B. on a given day, A and B work
together for 4 hours, then B is called away and A finishes the rest of the job
in 2 hours. How long would it take B to do the complete job alone?

a. 20 hrs b. 21 hrs c. 22 hrs d. 23 hrs

Solution:

Let be the time for B to finish the job alone.

be the time for A to finish the job alone.
In one day,
A can finish of the job
B can finish of the job

Answer(c)
Q21
A pipe can fill a tank in 3 hrs if the drain is open. If the pipe runs with
the drain open for 1 hr and the drain is then closed, the tank will be
filled in 40 minutes more. How long does it take the pipe to fill the
tank if the drain is closed?

a. 30 mins b. 45 mins c. 1 hr d. 1.5 hrs

Solution:

Let be the time for the pipe to fill the tank with the drain closed.
be the time for the drain to empty the tank with the pipe closed.
In one hour,
The pipe can fill of the tank
The drain can remove of the tank
Solving for x,

Answer(c)
Q22
A job can be done by 20 laborers in 30 days. To finish the job earlier,
25 men were hired. However, after waiting for 20 days, 10 men quit
and were not replaced. Find the total number of days to complete the
job.

a. b. c. d.
Solution:

Let be the remaining days to finish the job

For 1 complete job, there must be

Total Days:
Answer (c)
Q23
Mary travels 2 km/hr from home to office. Should she increase her
speed by 50%, it would take her 1 hour less. How far in km is her
house from the office?

a. 1.2 b. 2 c. 4 d. 6
Solution:

Recall: Motion Problem

Note: if speed is constant, then rate = constant speed.

if speed is not constant, then rate = average speed.
Let be the time of travel at 2 km/hr in hr.

Solving for t,

Answer(d)
 Q24
A boatman rows to a place 48 miles distant and back in 14 hours but
finds that he can row 4 miles with the stream in the same time as 3
miles against the stream. Find the rate of the stream.

a. 1.5 mph b. 1 mph c. 0.8 mph d. 0.6 mph

Solution:

Let be the rate of the stream in mph

be the rate of boat in still water in mph
be the rate of boat with the stream(downstream)
be the rate of boat against the stream (upstream)
Solving for x,

Answer(b)
Q25
A car 15 ft long overtakes a truck 30 ft long which is travelling at
the rate of 45 miles per hour. How fast must the car travel to pass
the truck in 3 seconds?

a. 75 ft/s b. 77 ft/s c. 79 ft/s d. 81 ft/s

Solution:

Theory: Relative Speed

Let be the velocity of the car

Answer(d)
 Q26
Two cars A and Brace around a 1320 m circular track. The rates of A
and B are 360 m/min and 300 m/min, respectively. If they travel at
the same direction and start at the same point, when will a overtake
B for the first time?

a. 18 min b. 22 min c. 12 min d. 36 min

Solution:

Answer(b)
Q27
The gasoline tank of a car contains 50 liters of gasoline and alcohol,
the alcohol comprising 25%. How much of the mixture must be drawn
off and replaced by alcohol so that the tank contains a mixture of
which 50% is alcohol?

a. 12.67 liters b. 18.75 liters c. 16.67 liters d. 14.25 liters

Solution:
Using alcohol content,

Using gasoline content,

Answer(c)
Q28
A vessel is full of alcohol. Five liters are drawn and replaced with
water. If after drawing another five liters of the resulting mixtures,
sixteen liters of alcohol are left in the vessel, find the capacity of the
vessel in liters.

a. 15 b. 20 c. 25 d. 30
Solution:
Using Alcohol content,

Answer(c)
 Q29
If you own a store, at what price will you mark an item for sale that
cost P600 in order that you may offer 20% discount on the marked
price and still make a profit of 25% of the selling price?

a. P 1,000 b. P 1,200 c. P1,500 d. P800

Solution:

Let be the marked price

Selling price =

Answer(a)
 Q30
How soon after 2 o’clock will the hands of a clock to be together?

a. 12 minsb. 10 minsc. 13 minsd. mins

Solution:

Let be the minute spaces travelled by the minute hand

be the no. of minute spaces travelled by the hour hand
Solving for x,

Answer(b)