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CAT Quantitative Aptitude | CAT Algebra: Inequalities 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
    Choice B
    30

  2. CAT Inequalities - Cubic Inequalities

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

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

  3. CAT Inequalities - Quadratic Inequalities

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

    x < -10 or x > 3

  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?

    There are a total of nine values

  5. CAT Inequalities - Modulus Inequalities

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

    ( -∞, -4) or (2, 4) or ( 10, ∞)

  6. CAT Inequalities - Natural Numbers

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

    The maximum product is 560.

  7. CAT Inequalities - Integers, Polynomials

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

    11 values

  8. CAT Inequalities - Modulus, Quadratic

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

    The range is (-∞, -5) or (-2, 2)

  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?

    (-∞, -2) or (-2, 2)

  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
    Choice A
    1080

  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
    Choice A
    44

  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
    Choice B
    67

  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
    Choice A
    25

  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
    Choice C
    4410

  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, ∞)
    Choice B
    x ∊ (-∞,-3] ∪ (√3, ∞)

  16. CAT Inequalities - Properties of Inequalities

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

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

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

  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
    Choice A
    98

  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
    Choice D
    61

  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
    Choice C
    3510

  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}\\)
    Choice A
    \\frac{139}{4}\\)

  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
    Choice D
    10

  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,∞)
    Choice C
    All real values of x

  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
    Choice A
    3 and 1

  25. The following questions are from IPMAT Rohtak and Indore sample papers. If you want to take these questions as a mock please click below.

    IPMAT Rohtak Sample Paper Mock
    IPMAT Indore Sample Paper Mock

    Please note that the explanation button will take you to the IPMAT solution page.


  26. IPMAT 2020 Sample Paper - IPM Rohtak Quants,Inequalities

    The set of all real numbers x for which x2 - |x + 2 |+ x > 0, is

    1. (-∞, -2) ∪ (2, ∞)
    2. (-∞, -√2) ∪ (√2, ∞)
    3. (-∞, -1) ∪ (1, ∞)
    4. (√2, ∞)
    Choice B
    (-∞, -√2) ∪ (√2, ∞)

  27. IPMAT 2019 Question Paper - IPM Indore Quants,Inequalities

    For all real values of x, \\frac{3x^{2} - 6x + 12}{x^{2} + 2x + 4}\\) lies between 1 and k, and does not take any value above k. Then k equals

    1

  28. IPMAT 2019 Question Paper - IPM Indore Quants,Inequalities

    If \|x| < 100\\) and \|y| < 100\\), then the number of integer solutions of (x, y) satisfying the equation 4x + 7y = 3 is

    29

  29. IPMAT 2019 Question Paper - IPM Indore Quants,Inequalities

    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
    Choice B
    1

  30. IPMAT 2019 Question Paper - IPM Indore Quants,Inequalities

    If a, b, c are real numbers a2 + b2 + c2 = 1, then the set of values ab + bc + ca can take is:

    1. [-1,2]
    2. [-\\frac{1}{2}\\), 2]
    3. [-1,1]
    4. [-\\frac{1}{2}\\), 1]
    Choice D
    [-\\frac{1}{2}\\), 1]

  31. IPMAT 2019 Question Paper - IPM Indore Quants,Inequalities

    The inequality \\log _{2} \frac{3x - 1}{2 - x} < 1\\) holds true for

    1. x ∈ (\\frac{1}{3}\\), 1)
    2. x ∈ (\\frac{1}{3}\\), 2)
    3. x ∈ (0, \\frac{1}{3}\\)) ∪ (1,2)
    4. x ∈ (-∞, 1)
    Choice A
    x ∈ (\\frac{1}{3}\\), 1)

  32. IPMAT 2019 Question Paper - IPM Indore Quants,Inequalities

    The set of values of x which satisfy the inequality 0.72x2 - 3x + 4 < 0.343 is

    1. (\\frac{1}{2}\\), 1)
    2. (\\frac{1}{2}\\), ∞)
    3. (-∞, \\frac{1}{2}\\))
    4. (-∞, \\frac{1}{2}\\)) ∪ (1, ∞)
    Choice D
    (-∞, \\frac{1}{2}\\)) ∪ (1, ∞)

  33. IPMAT 2019 Question Paper - IPM Indore Quants,Inequalities

    Determine the greatest number among the following four numbers

    1. 2300
    2. 3200
    3. 2100 + 3100
    4. 4100
    Choice B
    3200

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 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)

    11

  2. 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
    Choice B
    39

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

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

    8

  4. 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)

    36


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