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CAT Quantitative Aptitude | CAT Algebra: Polynomials 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.

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
Choice C
24
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)
Choice B
(1, 2) \\cup\\) (2, \\infty\\))
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
Choice B
-5.5
CAT Polynomials - Sum of squares of nos.

What is the sum of 1² + 3² + 5² …….31²?
1. 9455
2. 5456
3. 3468
4. 4892
Choice B
5456
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
Choice A
-41
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
Choice A
-8
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
Choice D
35.2
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
Choice A
3
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
Choice C
108
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
Choice D
1984
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
Choice C
2
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
Choice B
4290
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
Choice D
P > 40
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}\\)
Choice C
\\frac{11}{18}\\)
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
Choice D
4
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
Choice C
3471
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
Choice C
19522
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
Choice D
3097
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}\\)
Choice C
\\frac{99}{100}\\)
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
Choice A
5

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.

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
34
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
82
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
6

Correct: 21.52%
Incorrect: 32.13%
Unattempted: 46.35%
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
Choice A
8

Correct: 8.89%
Incorrect: 18.81%
Unattempted: 72.3%
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
Choice B
7 < x < 12

Correct: 22.35%
Incorrect: 39.73%
Unattempted: 37.92%
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
Choice C
17

Correct: 18.12%
Incorrect: 9.92%
Unattempted: 71.96%
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})
Choice C
\\frac{1}{√2})
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
Choice D
25 : 4
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)
2