Algebra Practice Questions for CAT with Solutions

Approximately 10-12 questions on Algebra are featuring in CAT in recent years. Most of these questions are from the following areas:

Equations
Progressions
Functions
Maxima Minima
Logarithms

To give CAT asirants a hands on experience on the variety of Algebra questions which frequently appear in CAT, we have listed around 60 questions practice on important topics from Algebra.
All these algebra questions are with detailed explanations.

Question 1:
For the given pair (x, y) of positive integers, such that 4x-17y=1 and x<1000 how many integer values of y satisfy the given conditions?
[1] 56
[2] 57
[3] 58
[4] 59

We first need to find out a solution for x & y. Once we get a solution, values of x would be in an AP with a common difference of 17 whereas values of y would be in an AP with a common difference of 4.

Valid Solutions:

x = 13, y = 3

x = 30, y = 7

x = 47, y = 11

x = 999, y = 235

No. of terms =\(\frac{{999 - 13}}{{17}} + 1 = \) = 58 + 1 = 59. Option D

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Question 2:
One year payment to the servant is Rs. 90 plus one turban. The servant leaves after 9 months and receives Rs. 65 and turban. Then find the price of the turban
[1] Rs.10
[2] Rs.15
[3] Rs.7.5
[4] Cannot be determined

Payment for 12 months = 90 + t {Assuming t as the value of a turban}

Payment for 9 months should be ¾(90 + t)

Payment for 9 months is given to us as 65 + t

Equating the two values we get

¾(90 + t) = 65 + t

270 + 3t = 260 + 4t
t = 10 Rs. Option A

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Question 3:
In CAT 2007 there were 75 questions. Each correct answer was rewarded by 4 marks and each wrong answer was penalized by 1 mark. In how many different combination of correct and wrong answer is a score of 50 possible?
[1] 14
[2] 15
[3] 16
[4] None of these

Correct (c) + Wrong (w) + Not attempted (n) = 75

4c – w + 0n= 50

Adding the two equations we get

5c + n = 125
Values of both c & n will be whole numbers in the range [0, 50]
c (max) = 25; when n = 0
c (min) = 13; when n = 60 {Smallest value of ‘c’ which will take the marks from correct questions greater than or equal to 50}
No. of valid combinations will be for all value of ‘c’ from 13 to 25 = 13. Option D

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Question 4:
How many integer solutions exist for the equation 8x – 5y = 221 such that x ´ y < 0
[1] 4
[2] 5
[3] 6
[4] 8

We first need to find out a solution for x & y. Once we get a solution, values of x would be in an AP with a common difference of 5 whereas values of y would be in an AP with a common difference of 8.

Valid Solutions:

x = 32; y = 7

x = 37; y = 15

x = 42; y = 23

But we need the solutions where one variable is negative whereas the other one is positive. so, we will move in the other direction.

x = 27; y = -1

x = 22; y = -9

x = 17; y = -17

x = 12; y = -25

x = 7; y = -33

x = 2; y = -41

So, number of integer solutions where x ´ y < 0 is 6. Option C

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Question 5:
How many integer solutions exists for the equation 11x + 15y = -1 such that both x and y are less than 100?
[1] 15
[2] 16
[3] 17
[4] 18

Valid Solutions:

x = 4; y = -3

x = 19; y = -14

x = 94; y = -69

So, there are 7 solutions of positive values of ‘x’.

x = -11; y = 8

x = -26; y = 19

x = __; y = 96

So, there are 9 solutions for positive values of ‘y’.

Total number of integer solutions = 7 + 9 = 16. Option B

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