CAT Questions | CAT Algebra Questions

A CAT Algebra question from Polynomials that appears in the Quantitative Aptitude section of the CAT Exam consists of concepts from Number Theory and Algebra. Polynomial Remainder Theorem is an important concept in Polynomials. Sum of Squares, Sequences and Series, Finding roots of an equation all appear in Polynomials. In CAT Exam, one can generally expect to get 1~2 questions from Polynomials. 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 Polynomials - Possible pairs of solution

   How many pairs of integer (a, b) are possible such that a² – b² = 288?

   1. 6
   2. 12
   3. 24
   4. 48

   Correct AnswerChoice C
   24

   Explanation Video Solution

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2. CAT Polynomials - Polynomial Remainder Theorem

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

   1. (2,\\infty \\))
   2. (1, 2) \\cup\\) (2, \\infty\\))
   3. (-\\infty \\), 1) \\cup\\) (2, \\infty \\))
   4. (-\\infty \\), 1)

   Correct AnswerChoice B
   (1, 2) \\cup\\) (2, \\infty\\))

   Explanation

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3. CAT Polynomials - Remainder Theorem

   x⁴ – ax³ + bx² – cx + 8 = 0 divided by x – 1 leaves a remainder of 4, divided by x + 1 leaves remainder 3, find b

   1. 2.5
   2. -5.5
   3. 3.5
   4. 6.5

   Correct AnswerChoice B
   -5.5

   Explanation Video Solution

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4. CAT Polynomials - Sum of squares of nos.

   What is the sum of 1² + 3² + 5² …….31²?

   1. 9455
   2. 5456
   3. 3468
   4. 4892

   Correct AnswerChoice B
   5456

   Explanation

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5. CAT Polynomials - Polynomial Remainder Theorem

   What is the remainder when x⁴ + 5x³ – 3x² + 4x + 3 is divided by x + 2?

   1. -41
   2. -31
   3. -18
   4. 41

   Correct AnswerChoice A
   -41

   Explanation

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6. CAT Polynomials - Finding the Roots

   If x⁴ – 8x³ + ax² – bx + 16 = 0 has positive real roots, find a – b.

   1. -8
   2. 6
   3. -12
   4. -14

   Correct AnswerChoice A
   -8

   Explanation Video Solution

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7. CAT Polynomials - Remainders

   4x³ + ax² – bx + 3 divided by x – 2 leaves remainder 2, divided by x + 3 leaves remainder 3. Find remainder when it is divided by x + 2.

   1. 26.8
   2. 29.2
   3. 32.2
   4. 35.2

   Correct AnswerChoice D
   35.2

   Explanation Video Solution

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8. CAT Polynomials - Factors

   How many of the following are factors of 3²⁰⁰ – 5¹⁰⁰?
   1. 7
   2. 16
   3. 53
   4. 12
   

   1. 3
   2. 2
   3. 1
   4. All of the above

   Correct AnswerChoice A
   3

   Explanation Video Solution

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9. CAT Polynomials - Roots

   x³ – 18x² + bx – c = 0 has positive real roots, p, q and z. If geometric mean of the roots is 6, find b.

   1. 36
   2. -216
   3. 108
   4. -72

   Correct AnswerChoice C
   108

   Explanation Video Solution

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10. CAT Polynomials - Cubic Equation

    What is the value of 27x³ + 18x²y + 12xy² + y³ when x = 4, y = – 8?

    1. 64
    2. 256
    3. 512
    4. 1984

    Correct AnswerChoice D
    1984

    Explanation Video Solution

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11. CAT Polynomials - Sequence

    A sequence of numbers is defined as 2 = a_n – a_n-1. S_n is sum upto n terms in this sequence and a₃ = 5. How many values m, n exist such than S_m – S_n = 65?

    1. 4
    2. 6
    3. 2
    4. More than 6 possibilities

    Correct AnswerChoice C
    2

    Explanation Video Solution

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12. CAT Polynomials - Summation

    6 + 24 + 60 + 120 + 210 + 336 + 504 + 720…. upto 10 terms is equal to?

    1. 3680
    2. 4290
    3. 5720
    4. 6170

    Correct AnswerChoice B
    4290

    Explanation Video Solution

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13. CAT Polynomials - Factorials

    1(1!) + 2(2!) + 3(3!) + 4(4!)………….50(50!) is a multiple of prime P. P lies in the range........?

    1. 30 < P < 40
    2. 10 < P < 20
    3. 30 < P < 40
    4. P > 40

    Correct AnswerChoice D
    P > 40

    Explanation Video Solution

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14. CAT Polynomials - Fractions

    What is the sum of{ \\frac{1}{1*4}\\) + \\frac{1}{2*5}\\) + \\frac{1}{3*6}\\) + \\frac{1}{4*7}\\) +...} ?

    1. \\frac{9}{17}\\)
    2. \\frac{7}{15}\\)
    3. \\frac{11}{18}\\)
    4. \\frac{92}{173}\\)

    Correct AnswerChoice C
    \\frac{11}{18}\\)

    Explanation Video Solution

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15. CAT Polynomials - Sequences

    2 + 6 + 10 + 14 ………..upto n term is given by S_n. How many of the following statements are true?
    1. S_2m – S_2k could be a multiple of 16
    2. 18S_n is a perfect square for all n
    3. S_2n > 2S_n for all n > 1
    4. S_m+n > S_m + S_n for all m, n > 1
    

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

    Correct AnswerChoice D
    4

    Explanation

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16. CAT Polynomials - Sequences

    \\frac{(2^4 - 1)}{(2 - 1)}\\) + \\frac{(3^4 - 1)}{(3 - 1)}\\) + \\frac{(4^4 - 1)}{(4 - 1)}\\) + .. + \\frac{(10^4 - 1)}{(10 - 1)}\\) = ?

    1. 3462
    2. 3581
    3. 3471
    4. 4022

    Correct AnswerChoice C
    3471

    Explanation Video Solution

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17. CAT Polynomials - Factors

    What is the sum of all numbers less than 200 that are either prime or have more than 3 factors?

    1. 19900
    2. 19533
    3. 19522
    4. 19534

    Correct AnswerChoice C
    19522

    Explanation Video Solution

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18. CAT Polynomials - Sum of a Series

    What is the sum of { \\frac{7}{1}\\) + \\frac{26}{2}\\) + \\frac{63}{3}\\) + \\frac{124}{4}\\) + \\frac{215}{5}\\) }.... 19 terms or 7 + 13 + 21 + 31 + 43 + 57 + 73... 19 terms?

    1. 3100
    2. 3025
    3. 3044
    4. 3097

    Correct AnswerChoice D
    3097

    Explanation Video Solution

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19. CAT Polynomials - Sum of fractions

    What is the sum of { \\frac{3}{4}\\) + \\frac{5}{36}\\) + \\frac{7}{144}\\) + \\frac{9}{400}\\) + }.... + \\frac{19}{8100}\\) = ?

    1. \\frac{9}{10}\\)
    2. \\frac{11}{18}\\)
    3. \\frac{99}{100}\\)
    4. \\frac{80}{81}\\)

    Correct AnswerChoice C
    \\frac{99}{100}\\)

    Explanation Video Solution

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20. CAT Polynomials - Cubic Equation

    x³ – 4x² + mx – 2 = 0 has 3 positive roots, two of which are p and \\frac{1}{p}\\) Find m.

    1. 5
    2. -11
    3. 8
    4. -2

    Correct AnswerChoice A
    5

    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.

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1. CAT 2022 Slot 1 - QA

   For natural numbers \(x, y\), and \(z\), if \(x y+y z=19\) and \(y z+x z=51\), then the minimum possible value of \(x y z\) is

   Correct Answer34

   Video ExplanationExplanation

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2. CAT 2022 Slot 1 - QA

   Let \(A\) be the largest positive integer that divides all the numbers of the form \(3^k+4^k+5^k\), and \(B\) be the largest positive integer that divides all the numbers of the form \(4^k+3\left(4^k\right)+4^{k+2}\), where \(k\) is any positive integer. Then \((A+B)\) equals

   Correct Answer82

   Video ExplanationExplanation

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3. CAT 2021 Slot 3 - QA

   If \(n\) is a positive integer such that \((\sqrt[7]{10})(\sqrt[7]{10})^{2} \ldots(\sqrt[7]{10})^{n}>999\), then the smallest value of \(n\) is

   Correct Answer6

   Video ExplanationExplanation

   Check StatsCorrect: 21.52%
   Incorrect: 32.13%
   Unattempted: 46.35%

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4. CAT 2021 Slot 2 - QA

   Consider the pair of equations: x² - xy - x = 22 and y² - xy + y = 34. If x > y, then x - y equals

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

   Correct AnswerChoice A
   8

   Video ExplanationExplanation

   Check StatsCorrect: 8.89%
   Incorrect: 18.81%
   Unattempted: 72.3%

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5. CAT 2021 Slot 2 - QA

   For all real numbers x the condition |3x - 20| + |3x - 40| = 20 necessarily holds if

   1. 6 < x < 11
   2. 7 < x < 12
   3. 10 < x < 15
   4. 9 < x < 14

   Correct AnswerChoice B
   7 < x < 12

   Video ExplanationExplanation

   Check StatsCorrect: 22.35%
   Incorrect: 39.73%
   Unattempted: 37.92%

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6. CAT 2021 Slot 1 - QA

   If r is a constant such that |x² - 4 x - 13| = r has exactly three distinct real roots, then the value of r is

   1. 15
   2. 18
   3. 17
   4. 21

   Correct AnswerChoice C
   17

   Video ExplanationExplanation

   Check StatsCorrect: 18.12%
   Incorrect: 9.92%
   Unattempted: 71.96%

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7. CAT 2020 Question Paper Slot 2 - Polynomials

   For real x, the maximum possible value of \\frac{x}{√(1 + x^{4})}) is

   1. 1
   2. \\frac{1}{2})
   3. \\frac{1}{√2})
   4. \\frac{1}{√3})

   Correct AnswerChoice C
   \\frac{1}{√2})

   Video SolutionExplanation

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8. CAT 2017 Question Paper Slot 1 - Algebra

   If a and b are integers of opposite signs such that (a + 3)² : b² = 9 : 1 and (a - 1)² : (b - 1)² = 4 : 1, then the ratio a² : b² is:

   1. 9 : 4
   2. 81 : 4
   3. 1 : 4
   4. 25 : 4

   Correct AnswerChoice D
   25 : 4

   ExplanationVideo Solution

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9. CAT 2017 Question Paper Slot 1 - Algebra

   If a, b, c, and d are integers such that a + b + c + d = 30, then the minimum possible value of (a - b)² + (a - c)² + (a - d)² is (TITA)

   Correct Answer2
   

   ExplanationVideo Solution

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