CAT Questions | CAT Algebra Questions

A CAT Algebra question from Inequalities that appears in the Quantitative Aptitude section of the CAT Exam consists of concepts: Range of Inequalities, Modulus functions, Possible solutions and so on. The topic also involves linear and quadratic equations, finding roots, polynomials, functions and more. Inequalities is a crucial topic for CAT. Having a good foundation in this subject can help a student tackle questions in Coordinate Geometry, Functions. In CAT Exam, one can generally expect to get 2~3 questions from Inequalities. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score. If you would like to take these questions as a Quiz, head on here to take these questions in a test format, absolutely free.

1. CAT Inequalities - Integer Solutions

   How many positive integer values can x take that satisfy the inequality (x - 8) (x - 10) (x - 12).......(x - 100) < 0?

   1. 25
   2. 30
   3. 35
   4. 40

   Correct AnswerChoice B
   30

   Explanation Video Solution

   ---

2. CAT Inequalities - Cubic Inequalities

   Solve the inequality: x³ – 5x² + 8x – 4 > 0?

   1. (2, ∞)
   2. (1, 2) ∪ (2, ∞)
   3. (-∞, 1) ∪ (2, ∞)
   4. (-∞, 1)

   Correct AnswerChoice B
   (1, 2) ∪ (2, ∞)

   Video Solution Explanation

   ---

3. CAT Inequalities - Quadratic Inequalities

   Find the range of x for which (x + 2) (x + 5) > 40?

   Correct Answerx < -10 or x > 3
   

   Explanation

   ---

4. CAT Inequalities - Integer Roots, Trial and Error

   How many integer values of x satisfy the inequality x( x + 2)(x + 4)(x + 6) < 200?

   Correct AnswerThere are a total of nine values
   

   Video Solution Explanation

   ---

5. CAT Inequalities - Modulus Inequalities

   Find the range of x where ||x - 3| - 4| > 3?

   Correct Answer( -∞, -4) or (2, 4) or ( 10, ∞)
   

   Video Solution Explanation

   ---

6. CAT Inequalities - Natural Numbers

   The sum of three distinct natural numbers is 25. What is the maximum value of their product?

   Correct AnswerThe maximum product is 560.
   

   Explanation

   ---

7. CAT Inequalities - Integers, Polynomials

   If x (x + 3) (x + 5) (x + 8) < 250, how many integer values can x take?

   Correct Answer11 values
   

   Explanation

   ---

8. CAT Inequalities - Modulus, Quadratic

   (|x| - 2) (x + 5) < 0. What is the range of values x can take?

   Correct AnswerThe range is (-∞, -5) or (-2, 2)
   

   Explanation

   ---

9. CAT Inequalities and Quadratic Inequalities

   a and b are roots of the equation x² - px + 12 = 0. If the difference between the roots is at least 12, what is the range of values p can take?

   Correct Answer(-∞, -2) or (-2, 2)
   

   Explanation

   ---

10. CAT Inequalities - Integer Solutions

    If a, b, c are distinct positive integers, what is the highest value a × b × c can take if a + b + c = 31?

    1. 1080
    2. 1200
    3. 1024
    4. 1056

    Correct AnswerChoice A
    1080

    Explanation Video Solution

    ---

11. CAT Inequalities - Modulus

    a, b, c are distinct natural numbers less than 25. What is the maximum possible value of |a – b| + |b – c| – |c – a|?

    1. 44
    2. 46
    3. 23
    4. 21

    Correct AnswerChoice A
    44

    Explanation Video Solution

    ---

12. CAT Inequalities - Maximum Possible Value

    Consider integers p, q such that – 3 < p < 4, – 8 < q < 7, what is the maximum possible value of p² + pq + q²?

    1. 60
    2. 67
    3. 93
    4. 84

    Correct AnswerChoice B
    67

    Explanation Video Solution

    ---

13. CAT Inequalities - Integer Solutions

    For how many integer values does the following inequality hold good? (x + 2) (x + 4) (x + 6)........(x + 100) < 0?

    1. 25
    2. 50
    3. 49
    4. 47

    Correct AnswerChoice A
    25

    Explanation Video Solution

    ---

14. CAT Inequalities - Maximum possible value

    If a, b, c are integers such that – 50 < a, b, c < 50 and a + b + c = 30, what is the maximum possible value of abc?

    1. 1000
    2. 5000
    3. 4410
    4. 4560

    Correct AnswerChoice C
    4410

    Explanation Video Solution

    ---

15. CAT Inequalities - Properties of Inequalities

    Solve x² - |x + 3| + x > 0?

    1. x ∊ (-∞,-1] ∪ [√3, 3)
    2. x ∊ (-∞,-√3) ∪ (√3, ∞)
    3. x ∊ (-4,-3) ∪ (4, ∞)
    4. x ∊ (-8,-3] ∪ [2, ∞)

    Correct AnswerChoice B
    x ∊ (-∞,-√3) ∪ (√3, ∞)

    Video Solution Explanation

    ---

16. CAT Inequalities - Properties of Inequalities

    Find range of f(x) = x² – 6x + 14?

    1. (-∞, 8)
    2. (-∞, 100)
    3. (-∞, 45)
    4. (5, ∞)

    Correct AnswerChoice D
    (5, ∞)

    Explanation

    ---

17. CAT Inequalities - Properties of Inequalities

    Solve :\\frac{(x – 4) (x+3)}{(x + 4) ( x +5)}\\) > 0?

    1. x ∊ (-∞,-5) ∪ (-4 , -3) ∪ (4, ∞)
    2. x ∊ (-∞,-5) ∪ (4, ∞)
    3. x ∊ (-4,-3) ∪ (4, ∞)
    4. x ∊ (-5,-3] ∪ [4, ∞)

    Correct AnswerChoice A
    x ∊ (-∞,-5) ∪ (-4 , -3) ∪ (4, ∞)

    Video Solution Explanation

    ---

18. CAT Inequalities and Modulus

    Consider three distinct positive integers a, b, c all less than 100. If |a - b| + |b - c| = |c – a|, what is the maximum value possible for b?

    1. 98
    2. 99
    3. 50
    4. 100

    Correct AnswerChoice A
    98

    Explanation Video Solution

    ---

19. CAT Inequalities - Integers

    Consider integers m, n such that -5 < m < 4 and -3 < n < 6. What is the maximum possible value of m² - mn + n²?

    1. 65
    2. 60
    3. 50
    4. 61

    Correct AnswerChoice D
    61

    Explanation

    ---

20. CAT Inequalities - Integers

    Consider integers p, q, r such that |p| < |q| < |r| < 40. P + q + r = 20. What is the maximum possible value of pqr?

    1. 3600
    2. 3610
    3. 3510
    4. 3500

    Correct AnswerChoice C
    3510

    Explanation

    ---

21. CAT Inequalities - Minimum Value

    What is the minimum value of f(x) = x² – 5x + 41?

    1. \\frac{139}{4}\\)
    2. \\frac{149}{4}\\)
    3. \\frac{129}{4}\\)
    4. \\frac{119}{4}\\)

    Correct AnswerChoice A
    \\frac{139}{4}\\)

    Video Solution Explanation

    ---

22. CAT Inequalities - Roots

    x⁴ – 4x³ + ax² – bx = 1 = 0 has positive real roots. What is the maximum possible value of a + b?

    1. 20
    2. 12
    3. 8
    4. 10

    Correct AnswerChoice D
    10

    Explanation

    ---

23. CAT Inequalities - Range

    |x³ – 3x + 5| > -4. What range of x satisfies this?

    1. [0,∞)
    2. [-4, ∞)
    3. All real values of x
    4. [4,∞)

    Correct AnswerChoice C
    All real values of x

    Explanation

    ---

24. CAT Inequalities - Max and Min Values

    What are the maximum and minimum possible values for \\frac{|x+y|}{|x|+|y|}\\) + \\frac{|z+y|}{|z|+|y|}\\) + \\frac{|z+x|}{|x|+|z|}\\)?

    1. 3 and 1
    2. 3 and 0
    3. 4 and 0
    4. 4 and 1

    Correct AnswerChoice A
    3 and 1

    Video Solution Explanation

    ---

The Questions that follow, are from actual CAT papers. If you wish to take them separately or plan to solve actual CAT papers at a later point in time, It would be a good idea to stop here.

1. CAT 2023 Slot 2 - QA

   Any non-zero real numbers \(x, y\) such that \(y \neq 3\) and \(\frac{x}{y} \lt \frac{x+3}{y-3}\), will satisfy the condition

   1. If \(y>10\), then \(-x>y\)
   2. If \(x \lt 0\), then \(-x \lt y\)
   3. If \(y \lt 0\), then \(-x \lt y\)
   4. \(\frac{x}{y} \lt \frac{y}{x}\)

   Correct AnswerChoice C
   If \(y \lt 0\), then \(-x \lt y\)

   Video ExplanationExplanation

   ---

2. CAT 2023 Slot 2 - QA

   If a certain amount of money is divided equally among \(n\) persons, each one receives Rs 352 . However, if two persons receive Rs 506 each and the remaining amount is divided equally among the other persons, each of them receive less than or equal to Rs 330 . Then, the maximum possible value of \(n\) is

   Correct Answer16

   Video ExplanationExplanation

   ---

3. CAT 2021 Slot 3 - QA

   The number of distinct pairs of integers (m, n) satisfying |1+mn| < |m+n| < 5 is

   Correct Answer12

   Video ExplanationExplanation

   Check StatsCorrect: 1.15%
   Incorrect: 40.77%
   Unattempted: 58.08%

   ---

4. CAT 2021 Slot 2 - QA

   For all possible integers n satisfying 2.25 ≤ 2 + 2^{n + 2} ≤ 202, the number of integer values of 3 + 3^{n + 1} is

   Correct Answer7

   Video ExplanationExplanation

   Check StatsCorrect: 10.73%
   Incorrect: 33.35%
   Unattempted: 55.92%

   ---

5. CAT 2021 Slot 1 - QA

   The number of integers n that satisfy the inequalities |n - 60| < |n - 100| < |n - 20| is

   1. 21
   2. 18
   3. 20
   4. 19

   Correct AnswerChoice D
   19

   Video ExplanationExplanation

   Check StatsCorrect: 12.02%
   Incorrect: 10.03%
   Unattempted: 77.95%

   ---

6. CAT 2020 Question Paper Slot 2 - Inequalities

   If x and y are non-negative integers such that x + 9 = z, y + 1 = z and x + y < z + 5, then the maximum possible value of 2x + y equals

   Correct Answer: 23

   Correct Answer Video Solution Explanation

   ---

7. CAT 2018 Question Paper Slot 2 - Number Theory | Inequalities

   The smallest integer n for which 4ⁿ ＞ 17¹⁹ holds, is closest to

   1. 33
   2. 39
   3. 37
   4. 35

   Correct AnswerChoice B
   39

   Explanation Video Solution

   ---

8. CAT 2018 Question Paper Slot 2 - Polynomials | Inequalities

   The smallest integer n such that n³ - 11n² + 32n - 28 ＞ 0 is (TITA)

   Correct Answer8
   

   Explanation Video Solution

   ---

9. CAT 2018 Question Paper Slot 2 - Inequalities

   If a and b are integers such that 2x² − ax + 2 ＞ 0 and x² − bx + 8 ≥ 0 for all real numbers x, then the largest possible value of 2a − 6b is (TITA)

   Correct Answer36
   

   Explanation Video Solution

   ---

10. CAT 2017 Question Paper Slot 1 - Inequalities

    For how many integers n, will the inequality (n – 5) (n – 10) – 3(n – 2) ≤ 0 be satisfied? (TITA)

    Correct Answer11
    

    Explanation Video Solution

    ---

The Questions that follow, are from actual XAT papers. If you wish to take them separately or plan to solve actual XAT papers at a later point in time, It would be a good idea to stop here.

1. XAT 2018 Question Paper - QADI

   If 2 ≤ |x – 1|×|y + 3| ≤ 5 and both x and y are negative integers, find the number of possible combinations of x and y.

   1. 4
   2. 5
   3. 6
   4. 8
   5. 10

   Correct AnswerChoice E
   10

   Video SolutionExplanation

   ---

The Questions that follow, are from actual IPMAT papers. If you wish to take them separately or plan to solve actual IPMAT papers at a later point in time, It would be a good idea to stop here.

1. IPMAT 2020 Question Paper - IPM Indore Quants

   Consider the following statements:
   (i) When 0 < x < 1, then \\frac{1}{1+x}) < 1 - x + x²
   (ii) When 0 < x < 1, then \\frac{1}{1+x}) > 1 - x + x²
   (iii) When -1 < x < 0, then \\frac{1}{1+x}) < 1 - x + x²
   (iv) When -1 < x < 0, then \\frac{1}{1+x}) > 1 - x + x²
   Then the correct statements are

   1. (i) and (ii)
   2. (ii) and (iv)
   3. (i) and (iv)
   4. (ii) and (iii)

   Correct AnswerChoice C
   (i) and (iv)

   Explanation

   ---

2. IPMAT 2019 Question Paper - IPM Indore Quants

   If x ∈ (a, b) satisfies the inequality \\frac{x - 3}{x^{2} + 3x + 2} \geq 1,\\) then the largest possible value of b - a is

   1. 3
   2. 1
   3. 2
   4. No real values of x satisfies the inequality

   Correct AnswerChoice B
   1

   ExplanationVideo Solution

   ---