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Polynomials Class 10 MCQs Questions with Answers
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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 x101 + 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) = (-1101) + 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 Ax2 + 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 x3 + ax2 + 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) x2 + 4x + 12
(b) x2 – 4x – 12
(c) x2 + 4x – 12
(d) x2 – 4x + 12
Answer
Answer: (c) x2 + 4x – 12
Question 44.
If 5 is a zero of the quadratic polynomial, x2 – 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 x2 + 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 x2 + (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 x2 +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 x2 + 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 3x2 – 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 x2 + 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 x2-6x+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 5x2 – 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 x2 + 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 x2 + 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 x2 + 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) = ax2 + bx +c (a ≠ 0) | Cubic
Polynomial |
2 Zeroes | Sum of the zeroes = 0 |
| (c) r(x) = ax3 + bx2 + 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) = ax2 + bx +c (a ≠ 0) | Quadratic
Polynomial |
2 Zeroes | α + s = –\(\frac{b}{a}\) |
| (c) r(x) = ax3 + bx2 + 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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