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## Polynomials Class 10 MCQs Questions with Answers

What are you waiting for? Get all the answers to Class 10 Maths Chapter 2 MCQs! It’s time that students power up and start practicing these MCQ Questions of Polynomials Class 10 with answers. The best way of doing so would be solving them yourself in order not only to know which answer is correct but also to understand why each solution works as well.

Question 1.

If a polynomial p(y) is divided by y + 2, then which of the following can be the remainder:

(a)y + 1

(b)2y + 3

(c) 5

(d)y – 1

## Answer

Answer: (c) 5

When p(y) is divided by y + 2, then the degree of remainder < deg of (y + 2)

Question 2.

If a polynomial p(x) is divided by b – ax; the remainder is the value of p(x) at x =

(a) a

(b) \(\frac{b}{a}\)

(c) \(\frac{- b}{a}\)

(d) \(\frac{a}{b}\)

## Answer

Answer: (b) \(\frac{b}{a}\)

b – ax = 0

x = \(\frac{b}{a}\)

Question 3.

If the polynomials ax³ + 4x² + 3x – 4 and x³ – 4x + a, leave the same remainder when divided by (x – 3), then value of a is :

(a) 2b

(b) – 1

(c) 1

(d) – 2b

## Answer

Answer: (b) – 1

p(x) = ax³ + 4x² + 3x – 4

q(x) = x³ – 4x + a

p(3) = q(3)

a = – 1

Question 4.

If p(x) = 2x⁴ – ax³ + 4x² + 2x + 1 is a. multiple of 1 – 2x, then find the value of a :

(a) 25

(b) \(\frac{1}{2}\)

(c)\(\frac{- 1}{2}\)

(d) 8

## Answer

Answer: (a) 25

p(x) is a multiple of 1 – 2x.

1 – 2x is a factor of p(x)

Question 5.

If -2 is a zero of p(x) = (ax³ + bx² + x – 6) and p(x) leaves a remainder 4 when divided by (x – 2), then the values of a and b are (respectively):

(a)a = 2,b = 2

(b) a = 0,b = – 2

(c) a = 0, b = 2

(d) a = 0, b = 0

## Answer

Answer: (c) a = 0, b = 2

If – 2 is a zero =>

p(- 2) = 0

=> – 2a + b = 2

Also, p(2) = 4

2a + b = 2=>a = 0and b = 2

Question 6.

If x^{101} + 1001 is divided by x + 1, then remainder is:

(a) 0

(b) 1

(c) 1490

(d) 1000

## Answer

Answer: (d) 1000

p(x) is divided by x + 1

p(- 1) = (-1^{101}) + 1001 = 1000

Question 7.

If one zero of a polynomial p(x) = ax² + bx + c(a ≠ 0) is zero, then, which of the following is correct:

(a) b = 0

(b) c = 0

(c) other zero is also zero

(d) Nothing can be said about p(x).

## Answer

Answer: (b) c = 0

let ,α = 0

Product of the roots = αs = 0

= \(\frac{c}{a}\) = 0

Question 8.

If α, s are the zeroes of x² – lx + m, then

\(\frac{α}{s}\) + \(\frac{s}{α}\)

(a) \(\frac{l² – 2m}{m}\)

(b) \(\frac{l² + 2m}{m}\)

(c) \(\frac{l – 2m}{m}\)

(d) l² – 2m

## Answer

Answer: (a) \(\frac{l² – 2m}{m}\)

α + s = l

αs = m

Question 9.

sum of the squares of the zeroes of the polynomial p(x) = x² + 7x – k is 25, find k.

(a) 12

(b) 49

(c) – 24

(d) – 12

## Answer

Answer: (d) – 12

p(x) = x² + 7x – k

let α,s be the zeroes

α + s = – 7

αs = – k

α² + s² = 25

(α² + s) – 2αs = 25

49 + 2k = 25

k = -12

Question 10.

If one zero of 3x² – 8x + 2k + 1 is seven times the other, find k.

(a) \(\frac{2}{3}\)

(b) \(\frac{1}{3}\)

(c) \(\frac{4}{3}\)

(d) \(\frac{5}{3}\)

## Answer

Answer: (a) \(\frac{2}{3}\)

α + 7α = 8α = \(\frac{8}{3}\)

α = \(\frac{1}{3}\)

k = \(\frac{2}{3}\)

Question 11.

Let, α, s, v be the zeroes of x³ + 4x² + x- 6 such that product of two of the zeroes is 6. Find the third zero.

(a) 6

(b) 2

(c) 4

(d) 1

## Answer

Answer: (a) 6

α s v = 6,

αs = 61

=> v = 6

Question 12.

If a, s are the zeroes of x² – 8x + λ, such

that α – s = 2, then X =

(a) 8

(b) 22

(c) 60

(d) 15

## Answer

Answer: (d) 15

α + s = 8,

αs = λ

α – s = 2

=> (α – s)2 = 4

=> α²+s²-2αs = 4

=> (α + s)² – 4as = 4

=> 64 — 4λ = 4

=> 4λ. = 60

=> X = 15

Question 13.

Find a and b so that the polynomial 6x⁴ + 8x³ – 5x² + ax + b is exactly divisible by 2x² – 5.

(a) a = 20, b = – 25

(b) a = 4, b = – 5

(c) a = 20, b = 5

(d) a = – 20, b = – 25

## Answer

Answer: (d) a = – 20, b = – 25

Divide the given polynomial by 2×2 – 5 get the remainder as (20 + a)x + (b + 25) which should be zero

Question 14.

If α, s are the zeroes of p(x) = 2x² – 5x + 7, write a polynomial with zeroes 2α+3s and 3α+2s.

(a) k(x² + \(\frac{5}{2}\)x – 41)

(b) k(x² – \(\frac{5}{2}\)x + 41)

(c) k(x² – \(\frac{5}{2}\)x – 41)

(d) k(- x² + \(\frac{5}{2}\)x + 41)

## Answer

Answer: (b) k(x² – \(\frac{5}{2}\)x + 41)

α + s = \(\frac{5}{2}\)

αs = \(\frac{7}{2}\)

k(x² – \(\frac{5}{2}\)x + 41)

Question 15.

If sum of the two zeroes of a cubic polynomial x³ – ax² + bx – c, is zero, then which of the following is true:

(a) ab = c

(b) a – b = c

(c) ab = \(\frac{c}{2}\)

(d) a = \(\frac{b}{c}\)

## Answer

Answer: (a) ab = c

Let, α, s, v be the roots =α + s + v = a

v = a

now v is a zero

ab = c

Question 16.

If a, s are the zeroes of p(x) = 2x² + 5x + k such that, α²+ s²+ αs = \(\frac{21}{4}\), then k equals,

(a) 12

(b) 4

(c) 2

(d) – 12

## Answer

Answer: (c) 2

α + s = – \(\frac{5}{2}\)

αs = \(\frac{k}{2}\)

α² + s² +αs = \(\frac{21}{4}\)

(α + s)² – αs = \(\frac{21}{4}\)

\(\frac{25}{4}\) – \(\frac{k}{2}\) = \(\frac{21}{4}\)

k = 2

Question 17.

If α, s are the zeroes of x² + px + q, then a polynomial having zeroes \(\frac{1}{α}\) and \(\frac{1}{s}\) is,

(a) x² + px + q

(b) x² + qx + p

(c) px² + qx + 1

(d) qx² + px +1

## Answer

Answer: (d) qx² + px +1

α + s = – p

αs = q

Question 18.

Find the number of zeros in the graph given:

(a) 3

(b) 2

(c) 1

(d) 0

## Answer

Answer: (b) 2

Since the graph meets X-axis at two points -2 and 1, thus it has 2 zeroes.

Question 19.

Write the zero of the polynomial p(x), whose graph is given :

(a) 1

(b) 0

(c) – 1

(d) – 2

## Answer

Answer: (b) 0

Since the graph meets X-axis at x = 0

=> Zero of p(x) is ‘O’ => Correct option is (b).

Question 20.

If α, s, v are the zeros of the polynomial 2x³ – x² + 3x -1, find the value of (αsv) + (αs + sv + vα).

(a) 2

(b) \(\frac{3}{2}\)

(c) \(\frac{1}{2}\)

(d) 0

## Answer

Answer: (a) 2

p(x) = 2x³ – x² + 3x -1

αsv = – d/a = \(\frac{1}{2}\)

αs + sv + vα = c/a = \(\frac{3}{2}\)

αs + sv + vα + αsv = \(\frac{3}{2}\) + \(\frac{1}{2}\) = 2

Question 21.

If the zeros of the polynomial x³ – 3x² + x +1 are p – q,p and p + q. Find the value of q.

(a) 1

(b) 0

(c) 2

(d) ±√2

## Answer

Answer: (d) ±√2

x³ – 3x² + x +1

zeroes are p – q,p,p + q

sum of zeroes = (p – q) + p + (p + q)

= 3p

= 3

α + s + v = \(\frac{- b}{a}\)

further = αs + sv + vα = \(\frac{c}{a}\)

(p – q) p + p(p + q) + (p – q)(p + q)=1

q = ±√2

Question 22.

A quadratic polynomial has :

(a) at least 2 zeros

(b) exactly 2 zeros

(c) at most 2 zeros

(d) exactly 1 zero

## Answer

Answer: (c) at most 2 zeros

A quadratic polynomial has atmost two zeroes.

Question 23.

Which of the following Linear Graphs has no zero?

## Answer

Answer:

as it does not meet X axis.

Question 24.

If α, s are the roots of cx² – bx + a = 0 (c 0), then α + s is:

(a) \(\frac{- b}{a}\)

(b) \(\frac{b}{a}\)

(c) \(\frac{c}{a}\)

(d) \(\frac{b}{c}\)

## Answer

Answer: (d) \(\frac{b}{c}\)

sum of the roots = – \(\frac{coefficient of x}{coefficient of x²}\) = \(\frac{b}{c}\)

Question 25.

If P(x) and D(r) are any two polynomials such that D(x) ≠ 0, there exists unique polynomial Q(x) and R(x) such that, P(x) = D(x). Q(x) + R(x) where :

(a) R(x) = 0 and deg R(x) > deg Q(x)

(b) R(x) = 0 or deg R(x) > deg D(x)

(c) deg R(x) < deg Q(x)

(d) R(x) = 0 or deg R(x) < deg D(x)

## Answer

Answer: (b) R(x) = 0 or deg R(x) > deg D(x)

division algorithm

Question 26.

When we divide x³ + 5x + 7 by x⁴ – 7x² – 6 then quotient and remainder are (respectively):

(a) 0,x³ + 5x + 7

(b) x, 2x + 3

(c) 1,x⁴ – 7x²-6

(d) x², 4x – 9

## Answer

Answer: (a) 0,x³ + 5x + 7

Degree of the divisor is more than the degree of the dividend = quotient is zero and the remainder is x³ + 5x + 7

Question 27.

The value of b, for which 2x³ + 9x² – x – b is exactly divisible by 2x + 3 is:

(a) -15

(b) 15

(c) 9

(d) – 9

## Answer

Answer: (b) 15

when 2x³ + 9x² – x – b is divided by 2x + 3, remainder is – b + 15

Question 28.

If α and s are two zeros of the polynomial p(x), then which of the following is a factor of p(x):

(a) (x – α)(x – s)

(b) (x + α) (x + s)

(c) k(x – α)

(d) k(x- s)

## Answer

Answer: (a) (x – α)(x – s)

if α, s are the zeros of p(x), then (x – α)(x – s) is a factor of p(x).

Question 29.

Find a cubic polynomial with the sum, sum of the product of its zeros taken two at a time and the product of its zeros as -2, +5, -3, respectively.

(a) 2x³ + 5x² + x + 3

(b) 4x³ + 5x² – 3x + 7

(c) x³ + 2x² + 5x + 3

(d) 2x³ + 5x² + 3x + 1

## Answer

Answer: (c) x³ + 2x² + 5x + 3

Let the polynomial be ax³ + bx² + cx + d

– b/a = – 2

c/a = 5

– d/a = – 3

a = 1, b = 2, c = 5 and d = 3

required polynomial is x³ + 2x² + 5x + 3

Question 30.

Write a polynomial with zeros 1, – 1 and 1.

(a) x³ + x² + x + 1

(b) x³ – x² + x + 1

(c) x³ – x² – x – 1

(d) x³ – x² – x + 1

## Answer

Answer: (d) x³ – x² – x + 1

zeros are 1, – 1 and 1.

required polynomial is

k(x – 1)(x +1)(x – 1)

= x³ – x² – x + 1

Question 31.

The graph of a polynomial is as shown, find the polynomial

(a) k(x² – x – 6)

(b) k(x³ + x² + 6x)

(c) k(x³ – x² – 6x)

(d) k(x³ – 6x)

## Answer

Answer: (c) k(x³ – x² – 6x)

zeros are – 2,0, and 3

required polynomial = k(x – 2)(x – 0)(x – 3)

= k(x³ – x² – 6x)

Question 32.

If α, s and v are the zeroes of the polynomial 2x³ – x² + 3x – 1, find the value of => (as + sv + va + asv )²

(a) \(\frac{3}{2}\)

(b) \(\frac{5}{2}\)

(c) \(\frac{1}{2}\)

(d) 4

## Answer

Answer: (d) 4

αs + sv + vα + αsv = \(\frac{3}{2}\) + \(\frac{1}{2}\) = 2

(αs + sv + vα + αsv )² = 4

Question 33.

If 2 ± √3 are the two zeros of a polynomial then the following is a factor:

(a) x² – 4x + 1

(b) x² + 4x – 1

(c) 4x² + x – 1

(d) 4x² – x + 1

## Answer

Answer: (a) x² – 4x + 1

If a, s are the zeroes => (x – α) (x – s) is a factor

=> (x – (2 + √3)) (x – (2 – √3))is a factor

=> x2 – 4x + 1 is a factor.

Question 34.

If 2 is a zero of p(x) = x² + 3x + k, find k:

(a) 10

(b) 5

(c) – 3

(d) – 10

## Answer

Answer: (d) – 10

p(x) = x² + 3x + k

p(2) = 0

=>4 + 6 + k = 0

=k = – 10

Question 35.

Given that two of the zeroes of the

polynomial, x³ + px² + rx + s are 0, then third zero

(a) 0

(b) \(\frac{p}{r}\)

(c) \(\frac{- p}{r}\)

(d) \(\frac{p}{q}\)

## Answer

Answer: (c) \(\frac{- p}{r}\)

Two zeroes are zero, let third zero = α

=> Sum of the roots = α + 0 + 0

\(\frac{Coefficient of x²}{Coefficient of x³}\)

Question 36.

Given that one of the zeroes of the

polynomial ax³ + bx² + cx + d is zero, then the product of the other two zeroes is:

(a) \(\frac{- c}{a}\)

(b) \(\frac{c}{a}\)

(c) 0

(d) \(\frac{- b}{a}\)

## Answer

Answer: (b) \(\frac{c}{a}\)

αs + sv + vα = \(\frac{c}{a}\)

now α = 0

0 + sv + 0 = \(\frac{c}{a}\)

sv = \(\frac{c}{a}\)

Question 37.

The number of polynomials having zeroes – 1 and – 5 is :

(a) 2

(b) 3

(c) 1

(d) More than 3.

## Answer

Answer: (d) More than 3.

n – number of polynomials can have zeroes -1 and -5.

Question 38.

The graph of the polynomial f(x) = 2x – 5 intersects the x – axis at

(a) (\(\frac { 5 }{ 2 }\), 0)

(b) (\(\frac { -5 }{ 2 }\), 0)

(c) (\(\frac { -5 }{ 2 }\), \(\frac { 5 }{ 2 }\))

(d) (\(\frac { 5 }{ 2 }\), \(\frac { -5 }{ 2 }\))

## Answer

Answer: (a) (\(\frac { 5 }{ 2 }\), 0)

Question 39.

If the zeroes of the quadratic polynomial Ax^{2} + Bx + C, C # 0 are equal, then

(a) A and B have the same sign

(b) A and C have the same sign

(c) B and C have the same sign

(d) A and C have opposite signs

## Answer

Answer: (b) A and C have the same sign

Question 40.

The number of polynomials having zeroes as 4 and 7 is

(a) 2

(b) 3

(c) 4

(d) more than 4

## Answer

Answer: (d) more than 4

Question 41.

If one of the zeroes of the cubic polynomial x^{3} + ax^{2} + bx + c is -1, then the product of the

other two zeroes is

(a) b – a + 1

(b) b – a – 1

(c) a – b + 1

(d) a – b – 1

## Answer

Answer: (a) b – a + 1

Question 42.

The number of zeros of a cubic polynomial is

(a) 3

(b) at least 3

(c) 2

(d) at most 3

## Answer

Answer: (d) at most 3

Question 43.

Find the quadratic polynomial whose zeros are 2 and -6

(a) x^{2} + 4x + 12

(b) x^{2} – 4x – 12

(c) x^{2} + 4x – 12

(d) x^{2} – 4x + 12

## Answer

Answer: (c) x^{2} + 4x – 12

Question 44.

If 5 is a zero of the quadratic polynomial, x^{2} – kx – 15 then the value of k is

(a) 2

(b) -2

(c) 4

(d) – 4

## Answer

Answer: (a) 2

Question 45.

The number of polynomials having zeroes as -2 and 5 is

(a) 1

(b) 2

(c) 3

(d) more than 3

## Answer

Answer: (d) more than 3

Question 46.

The zeroes of the quadratic polynomial x^{2} + 1750x + 175000 are

(a) both negative

(b) one positive and one negative

(c) both positive

(d) both equal

## Answer

Answer: (a) both negative

Question 47.

If the zeroes of the quadratic polynomial x^{2} + (a + 1) x + b are 2 and -3, then

(a) a = -7, b = -1

(b) a = 5, b = -1

(c) a = 2, b = -6

(d) a – 0, b = -6

## Answer

Answer: (d) a – 0, b = -6

Question 48.

Sum and the product of zeroes of the polynomial x^{2} +7x +10 is

(a) \(\frac { 10 }{ 7 }\) and \(\frac { -10 }{ 7 }\)

(b) \(\frac { 7 }{ 10 }\) and \(\frac { -7 }{ 10 }\)

(c) -7 and 10

(d) 7 and -10

## Answer

Answer: (c) -7 and 10

Question 49.

If x = 2 and x = 3 are zeros of the quadratic polynomial x^{2} + ax + b, the values of a and b respectively are :

(a) 5, 6

(b) – 5, – 6

(c) – 5, 6

(d) 5, – 6

## Answer

Answer: (c) – 5, 6

Question 50.

The zeroes of the quadratic polynomial 3x^{2} – 48 are

(a) both negative

(b) one positive and one negative

(c) both positive

(d) both equal

## Answer

Answer: (b) one positive and one negative

Question 14.

The zeroes of the quadratic polynomial x^{2} + kx + k, k ≠ 0,

(a) cannot both be positive

(b) cannot both be negative

(c) are always unequal

(d) are always equal

## Answer

Answer: (a) cannot both be positive

Question 51.

The sum and product of the zeroes of the polynomial x^{2}-6x+8 are respectively

(a) \(\frac { -3 }{ 2 }\) and – 1

(b) 6 and 8

(c) \(\frac { -3 }{ 2 }\) and 1

(d) \(\frac { 3 }{ 2 }\) and 1

## Answer

Answer: (b) 6 and 8

Question 52.

If the point (5,0), (0-2) and (3,6) lie on the graph of a polynomial. Then which of the following is a zero of the polynomial?

(a) 5

(b) 6

(c) not defined

(d) -2

## Answer

Answer: (a) 5

Question 53.

If a and ß are the zeroes of the polynomial 5x^{2} – 7x + 2, then sum of their reciprocals is:

(a) \(\frac { 14 }{ 25 }\)

(b) \(\frac { 7 }{ 5 }\)

(c) \(\frac { 2 }{ 5 }\)

(d) \(\frac { 7 }{ 2 }\)

## Answer

Answer: (d) \(\frac { 7 }{ 2 }\)

Question 54.

If one zero of the quadratic polynomial x^{2} + 3x + k is 2, then the value of k is

(a) 10

(b) -10

(c) 5

(d) -5

## Answer

Answer: (b) -10

Question 55.

The zeroes of the quadratic polynomial x^{2} + px + p, p ≠ 0 are

(a) both equal

(b) both cannot be positive

(c) both unequal

(d) both cannot be negative

## Answer

Answer: (b) both cannot be positive

Question 56.

The zeroes of the quadratic polynomial x^{2} + 99x + 127 are

(a) both positive

(b) both negative

(c) one positive and one negative

(d) both equal

## Answer

Answer: (b) both negative

Fill in the blanks:

1. A quadratic equation can have ________ two roots, (exactly/atleast/atmost)

## Answer

Answer: atmost

2. If a is a zero of p(x), then ________ is a factor of p(x).

## Answer

Answer: (x – α)

3. The sum of the zeroes of a cubic polynomial is ______

## Answer

Answer: – \(\frac{coefficient of x²}{coefficient of x³}\)

4. Division Algorithm for polynomials states that, Dividend = _________ x _________ + Remainder.

## Answer

Answer: Divisor × coefficient

5. If a polynomial p(x) does not touch _______ axis, then it has no zeroes.

## Answer

Answer: X – axis

Match the following:

Question 1.

Linear polynomial (one zero) | Touches x axis at one point only -2 | |

Quadratic Polynomial (2 zeros) | intersects X-axis at x = l. | |

Quadratic Polynomial (no zero) | Does not meet X-axis. | |

Linear Polynomial (One zero) | Passes through origin. | |

Quadratic Polynomial (One zero) | Meets X- axis at 2 points x = 1 and x = -l |

## Answer

Answer:

Quadratic Polynomial (2 zeros) | Meets X- axis at 2 points x = 1 and x = -l | |

Linear Polynomial (One zero) | intersects X-axis at x = l. | |

Linear Polynomial (One zero) | Passes through origin. | |

Quadratic Polynomial (one zero) | Meets X- axis at -2 | |

Quadratic Polynomial (no zero) | Does not meet X-axis. |

Question 2.

(a) p(x) = ax + b | No. of Zeroes = 3 | 3 Zeroes | αsv= – \(\frac{d}{a}\) |

(b) q(x) = ax^{2} + bx +c (a ≠ 0) |
Cubic
Polynomial |
2 Zeroes | Sum of the zeroes = 0 |

(c) r(x) = ax^{3} + bx^{2 }+ cx + d(a≠ 0) |
Linear
Polynomial |
Meets X-axis at 3 | α + s = –\(\frac{b}{a}\) |

Quadratic
Polynomial |
One zero | –\(\frac{b}{a}\) |

## Answer

Answer:

(a) p(x) = ax + b | Linear
Polynomial |
One zero | –\(\frac{b}{a}\) |

(b) q(x) = ax^{2} + bx +c (a ≠ 0) |
Quadratic
Polynomial |
2 Zeroes | α + s = –\(\frac{b}{a}\) |

(c) r(x) = ax^{3} + bx^{2 }+ cx + d(a≠ 0) |
Cubic
Polynomial |
3 Zeroes | αsv= – \(\frac{d}{a}\) |

No. of Zeroes = 3 | Meets X-axis at 3 | Sum of the zeroes = 0 |

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