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

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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, ∞)

   Explanation

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3. CAT Inequalities - Quadratic Inequalities

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

   Correct Answerx < -10 or x > 3
   

   Explanation

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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
   

   Explanation

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5. CAT Inequalities - Modulus Inequalities

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

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

   Explanation

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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, ∞)

    Explanation

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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

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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, ∞)

    Explanation

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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

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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

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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

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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}\\)

    Explanation

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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

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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

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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

    Explanation

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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.


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25. 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, ∞)

    Correct AnswerChoice B
    (-∞, -√2) ∪ (√2, ∞)

    Explanation Video Solution

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26. 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

    Correct Answer 1

    Explanation

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27. 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

    Correct Answer 29

    Explanation Video Solution

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28. 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

    Correct AnswerChoice B
    1

    Explanation Video Solution

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29. 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]

    Correct AnswerChoice D
    [-\\frac{1}{2}\\), 1]

    Explanation Video Solution

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30. 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)

    Correct AnswerChoice A
    x ∈ (\\frac{1}{3}\\), 1)

    Explanation Video Solution

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

    The set of values of x which satisfy the inequality 0.7^{2x2 - 3x + 4} < 0.343 is

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

    Correct AnswerChoice D
    (-∞, \\frac{1}{2}\\)) ∪ (1, ∞)

    Explanation Video Solution

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32. 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

    Correct AnswerChoice B
    3²⁰⁰

    Explanation Video Solution

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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)

   Correct Answer11
   

   Explanation Video Solution

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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

   Correct AnswerChoice B
   39

   Explanation Video Solution

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3. 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

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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)

   Correct Answer36
   

   Explanation Video Solution

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