Crack CAT Algebra with Confidence

Free Practice Questions to Strengthen Algebra for CAT 2025

1. CAT - Functions

If x is a positive real number then the minimum value of (x + 7) (x + 19)/(x + 3) is:

Correct Answer

36

2. CAT - Functions

If f(x) = x– 7x and g(x) = x + 3, then the minimum value of f(g(x)) – 3x is

(CAT 2021)

  1. -16
  2. -15
  3. -12
  4. -20

Correct Answer

1. -16

3. CAT - Functions

If 3x + 2 |y| + y = 7 and x + |x| + 3y = 1, then x + 2y is:
(CAT 2021)

  1. 8/3
  2. -4/3
  3. 1
  4. 0

Correct Answer

4. 0

4. CAT - Functions

Find the range of (x2 + 4x + 8)/(x2 + 4x + 5)

Correct Answer

(1,4)

5. CAT - Inequalities

Find the sum of all negative integral values of x where |x – | x – 2 | + 3 | – 4 < 3

Correct Answer

-5

6. CAT - Inequalities

Consider the function f(x) = (x + 4) (x + 6) (x + 8) ………….. (x + 98). The number of integers x for which f(x) is less than 0 are:

Correct Answer

24

7. CAT - Inequalities

How many integer values of x will satisfy this (x² – 16|x| + 60) / (x² – 14x + 49) < 0

Correct Answer

2 integer value of x

8. CAT - Inequalities

For how many positive integral values of k is (k-11)(k-15)(k − 19)… (k – 99) < 0?

Correct Answer

43

9. CAT - Inequalities

The number of integers n that satisfy the inequalities |n -60| < |n-100| < |n-20| is: (CAT 2021)

Correct Answer

9

10. CAT - Inequalities

Consider the pair of equations: x² – xy – x = 22 and y² – xy + y = 34. If x > y, then x – y equals (CAT 2021)
  1. 8
  2. 6
  3. 4
  4. 7

Correct Answer

1. 8

11. CAT - Inequalities

f(x) = (x² + 2x -15) /(x² – 7x – 18) is negative if and only if (CAT 2021)

  1. x < -5 or 3 < x < 9
  2. -5 < x < -2 or 3 < x < 9
  3. x < -5 or -2 < x < 3
  4. -2 < x < 3 or x > 9

Correct Answer

2. -5 < x < -2 or 3 < x < 9

12. CAT - Series (AP & GP)

Let k be a constant. The equations kx + y = 3 and 4x + ky = 4 have a unique solution if and only if (CAT 2020)
  1. k ≠ 2
  2. |k| = 2
  3. k = 2
  4. |k| ≠ 2

Correct Answer

4. |k| ≠ 2

13. CAT - Series (AP & GP)

Consider 2 APs 2,6,10……. and 5,12,19………….If S is a set containing the first 100 members of each progression then how many distinct elements are there in S?

Correct Answer

186

14. CAT - Series (AP & GP)

Three positive integers x, y and z are in arithmetic progression. If y − x greater than 2 and xyz = 5 (x + y + z), then z − x equals: (CAT 2021)

  1. 14
  2. 10
  3. 8
  4. 12

Correct Answer

1. 14

15. CAT - Series (AP & GP)

a + b + c + d = 4 All a,b,c,d are integers Find minimum possible value of 1/a + 1/b + 1/c + 1/d

Correct Answer

-20/7

16. CAT - Series (AP & GP)

If x, y and z are non – zero real numbers and 9x = 16y = 36z Find the value of 5[xz/(xy -yz) ]

Correct Answer

10

17. CAT - Series (AP & GP)

a + b + c + d = 4 All a,b,c,d are integers Find minimum possible value of 1/a + 1/b + 1/c + 1/d

Correct Answer

3

18. CAT - Series (AP & GP)

How many 3 term geometric progressions can be made from the series 1, 3, 32, 33, …… 348?

Correct Answer

1152

19. CAT - Logs

If loga30 = A, loga(5/3) = – B and log2 a = 1 /3, then log3 a equals: (CAT 2020)
  1. 2/(A + B) – 3
  2. 2/(A + B – 3)
  3. (A + B)/2 – 3
  4. (A + B – 3)/2

Correct Answer

2.  2 / (A + B - 3)

20. CAT - Logs

For a real number a , if (log15 a + log32 a) / [(log15 a)(log32 a)] = 4 then a must lie in range
  1. 3 < a < 4
  2. 2 < a < 3
  3. 4 < a < 5
  4. a > 5

Correct Answer

3. 4 < a < 5

21. CAT - Logs

If log2 [3 + log2 {4 + log4 (x – 1)}] – 2 = 0, then 4x equals: (CAT 2021)

Correct Answer

5

22. CAT - Logs

If 5 – log10 √(1 + x) + 4 log10 √(1 – x) = log10 (1/√(1 – x²), then 100x equals

Correct Answer

99

23. CAT - Logs

Log (x+3) (x² – x ) < 1

Correct Answer

(-3, -2) ∪ (-1, 0) ∪ (1, 3)

24. CAT - Equations

x² + 9x + |K| = 0, roots of this quadratic are integers. How many integral values of K are possible?

Correct Answer

9 integral values

25. CAT - Equations

The equation ax²+bx+c = 0 and 5x²+12x+13 = 0 have a common root where a,b,c are the side lengths of triangles ABC then find Angle C

Correct Answer

90°

26. CAT - Equations

If r a constant such that |x² – 4x – 13| = r has exactly three distinct real roots, then the value of r is (CAT 2021)
  1. 17
  2. 18
  3. 15
  4. 21

Correct Answer

1. 17

27. CAT - Equations

Suppose one of the roots of the equation ax² – bx + c = 0 is 2 + root13, where a, b and c are rational numbers and a is not equal to zero. If b = c³, then |a| equals to: (CAT 2021)

  1. 4
  2. 2
  3. 1
  4. 3

Correct Answer

2. 2

28. CAT - Equations

Let m and n be positive integers, If x² + mx + 2n = 0 and x² + 2nx + m = 0 have real roots, then the smallest possible value of m+ n is: (CAT 2020)

  1. 5
  2. 8
  3. 7
  4. 6

Correct Answer

4. 6

29. CAT - Equations

For natural numbers x, y and z, if xy + yz = 19 and yz + xz = 51, then how many such solutions are possible?

Correct Answer

2

30. CAT - Equations

Given that three roots of f(x) = x4 + ax2 + bx + c are 2, – 3 and 5, what is the value of a + b + c?

Correct Answer

79

Understanding Algebra

Algebra plays a pivotal role in the CAT exam, as it is both foundational and versatile. The questions test your ability to work with variables and equations, which are critical for solving real-world problems. Topics such as linear equations, quadratic equations, inequalities, logarithms, polynomials, and progressions are frequently covered in the exam. A typical CAT Algebra problem might involve solving systems of equations, simplifying expressions, or analyzing the behavior of functions and graphs. These problems require not just mathematical proficiency but also logical reasoning and time management skills. For instance, quadratic equations often appear with a twist, demanding innovative approaches like factorization or completing the square. Similarly, understanding arithmetic and geometric progressions can help solve pattern-based problems quickly and accurately.
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