CBSE Class 10 Mathematics Polynomials MCQs Set E

Question. If α,β are the zeros of the polynomial x² − px + q, then α²/β² +β²/α² is equal to −
(a) p⁴/q² +2 −4p²/q
(b) p⁴/q² −2 +4p²/q
(c) p⁴/q²+2q² −4p²/q
(d) None of the options

Answer: A

Question. If a,b are the zeros of the polynomial x² − px + 36 and α² + β² = 9, then p =
(a) ± 6
(b) ± 3
(c) ± 8
(d) ± 9

Answer: D

Question. If α,β are the roots of ax² + bx + c and a + k, b + k are the roots of px² + qx + r, then k =
(a) −1/2[a/b−p/q]
(b) [a/b−p/q]
(c) 1/2[b/a−q/p]
(d) (ab − pq)

Answer: C

Question. The condition that x³ − ax² + bx − c = 0 may have two of the roots equal to each other but of opposite signs is :
(a) ab = c
(b) 2/3,a
(c) a²b = c
(d) None of the options

Answer: A

Question. If the sum of zeros of the polynomial p(x) = kx³ − 5x² − 11x − 3 is 2, then k is equal to
(a) k = − 5/ 2
(b) k = 2/ 5
(c) k = 10
(d) k = 5/ 2

Answer: D

Question. If y = f(x) = mx + c; then f(y) in terms of x is
(a) mx + m + c
(b) m + mc + c
(c) m²x + mc + c
(d) m²x + m²c

Answer: C

Question. If a, b and g are the zeros of the polynomial f(x) = x³ + px² − pqrx + r, then 1/αβ + 1/βγ +1/γα =
(a) r/p
(b) p/ r
(c) − p/ r
(d) − r/ p

Answer: B

Question. The coefficient of x in x² + px + q was taken as 17 in place of 13 and it's zeros were found to be − 2 and − 15. The zeros of the original polynomial are
(a) 3, 7
(b) − 3, 7
(c) − 3, − 7
(d) −3, − 10

Answer: D

Question. If a, b and g are the zeros of the polynomial 2x³ − 6x² − 4x + 30, then the value of (ab + bg + ga) is
(a) − 2
(b) 2
(c) 5
(d) − 30

Answer: A

Question. If ax³ + 9x² + 4x − 1 is divided by (x + 2), the remainder is −6; then the value of 'a' is
(a) −3
(b) −2
(c) 0
(d) 33 8

Answer: D

Question. The graph of y = – 3x² + 2x– 1 cuts the x-axis at :
(a) 1/ 3 and 0
(b) 1/ 3 and 1/ 3
(c) do not cut
(d) none of the options

Answer: C

Question. The two zeroes of f(x) = x⁴– 6x³ – 26x² + 138x – 35 are 2 ± √3 , the other two zeroes are :
(a) 7 and –5
(b) –7 and 5
(c) –7 and – 5
(d) none of the options

Answer: A

Question. The quadratic polynomial, sum of whose zeroes is 8 and their product is 12, is given by
(a) x² – 8x + 12
(b) x² + 8x – 12
(c) x² – 5x + 7
(d) x² + 5x – 7

Answer: A

Question. A quadratic polynomial whose product and sum of zeroes are −13/ 5 and 3/5 , respectively.
(a) k(x² + 12x + 5)
(b) k[x² – (8x) + (–9))
(c) k [x² − (1/2 x) + ( −7/5)]
(d) k [x² − (3/5 x) + (−13/5)] 

Answer: D

Question. If the L.C.M. of two polynomials p(x) and q(x) is (x + 3)(x − 2)²(x − 6) and their H.C.F. is (x − 2).If p(x) = (x + 3)(x − 2)², then q(x) =________
(a) (x + 3)((x − 2) 
(b) x2 − 3x − 18
(c) x2 − 8x + 12
(d) none of the options

Answer: C

Question. If α,β,γ are the zeros of the polynomial x³ + 4x + 1, then (α + β)⁻¹ + (β + γ)⁻¹ + (γ + α)⁻¹ =
(a) 2
(b) 3
(c) 4
(d) 5

Answer: C

Question. f(x) = 3x⁵ + 11x4 + 90x² − 19x + 53 is divided by x + 5 then the remainder is ______.
(a) 100 
(b) −100
(c) −102
(d) 102

Answer: C

Question. If f(x) = 4x³ − 6x² + 5x − 1 and α,β and γ are its zeros, then αβγ =
(a) 3/ 2
(b) 5/ 4
(c) −3/2
(d) 1/ 4

Answer: D

Question. Find the remainder obtained when x2007 is divisible by x² − 1.
(a) x²
(b) x
(c) x + 1
(d) −x

Answer: B

Question. If a polynomial 2x³ − 9x² + 15x + p, when divided by (x − 2), leaves −p as remainder, then p is equal to
(a) −16
(b) −5
(c) 20
(d) 10

Answer: B

Question. The zeroes of the quadratic equation 4s² – 4s + 1 are
(a) 1/2 , 1/4
(b) 1/2 , 1/2
(c) 1/4 , 1/14
(d) 1/3 , 1/4

Answer: B

Question. A quadratic polynomial whose zeroes are 1 and –3 is
(a) x² + 3x – 2
(b) x² + 5x – 5
(c) x² + 2x – 3
(d) none of the options

Answer: C

Question. The value of ax² + bx + c when x = 0 is 6. The remainder when dividing by x + 1 is 6. The remainderwhen dividing by x + 2 is 8. Then the sum of a, b and c is
(a) 0
(b) −1
(c) 10
(d) None of the options

Answer: A

Question. If the G.C.D. of the polynomials x³ − 3x² + px + 24 and x² −7x + q is (x − 2), then the value of (p + q) is:
(a) 0
(b) 20
(c) −20
(d) 40

Answer: A

Question. If (x − 1), (x + 1) and (x − 2) are factors of x⁴ + (p − 3)x³ − (3p − 5)x² + (2p − 9) x + 6 then the value of p is
(a) 1
(b) 2
(c) 3
(d) 4

Answer: D

Question. If f (−3/4)=0; then for f(x), which of the following is a factor?
(a) 3x − 4
(b) 4x + 3
(c) −3x + 4
(d) 4x − 3

Answer: B

Question. The zeroes of f (x) = 4 √3x² +5x − 2 √3 are :
(a) 2/√3 and −√3/4
(b) −/√32 and −√3/4 
(c) −2/√3 and √3/4
(d) none of the options

Answer: D

Question. The solution of x² + 6x + 9 = 0 is
(a) –1
(b) 3
(c) –3
(d) 1

Answer: C

Question. The remainder of x⁴ + x³ − x² + 2x + 3 when divided by x − 3 is
(a) 105
(b) 108
(c) 10
(d) None of the options

Answer: B

Question. The remainder when x1999 is divided by x² − 1 is
(a) − x
(b) 3x
(c) x
(d) Noneo f the options

Answer: C

Question. If a,b and c are not all equal and a and b be the zeros of the polynomial ax² + bx + c, then value of (1 + a + a²) (1 + b + b²) is :
(a) 0
(b) positive
(c) negative
(d) non-negative

Answer: D

Question. f(x) = 16x² + 51x + 35 then one of the factors of f(x) is
(a) x − 1
(b) x + 3
(c) x − 3
(d) x + 1

Answer: D

Question. For the expression f(x) = x³ + ax² + bx + c, if f(a) = f(b) = 0 and f(d) = f(0). The values of a, b & c are
(a) a = − 9, b = 20, c = − 12
(b) a = 9, b = 20, c = 12
(c) a = − 1, b = 2, c = − 3
(d) none of the options

Answer: A

Question. If x² − ax + b = 0 and x² − px + q = 0 have a root in common and the second equation has equal roots, then
(a) b + q = 2ap
(b) b + q = ap /2
(c) b + q = ap
(d) none of the options

Answer: B

Question. If a, b are the zeroes of the polynomial 2x² – 5x + 7, then a polynomial whose zeroes are 2α + 3β, 3α + 2β is
(a) k(x² − 3/5 x + 21)
(b) k (x² − 25/2 x + 41)
(c) k (x² +9/2 x − 45 )
(d) none of the options

Answer: B

Question. The remainder when f (x) = – 3x² + x⁴ + 4x + 5 is divided by –x² + 2 is
(a) 6
(b) 8
(c) 12
(d) none of the options

Answer: D

Question. The zeroes of y = x² + 7x + 12 are :
(a) 4 and – 3
(b) –4 and –3
(c) –4 and 3
(d) 4 and 3

Answer: B

Question. If the zeroes of f(x) = x³ – 3x² + x + 1 are a – b, a, a + b, then value of a and b is :
(a) a =1, b = −√ 2
(b) a =1, b = −√2
(c) a =1, b =√2
(d) none of the options

Answer: B

Assertion and Reason:
(a) If both Assertion and Reason are correct and Reason is the correct explanation of Assertion.
(b) If both Assertion and Reason are correct, but Reason is not the correct explanation of Assertion.
(c) If Assertion is correct but Reason is incorrect.
(d) If Assertion is incorrect but Reason is correct.

Question. Assertion : x³ + x has only one real zero.
Reason : A polynomial of nth degree must have n real zeroes.

Answer: C

Question. Assertion : Degree of a zero polynomial is not defined.
Reason: Degree of a non-zero constant polynomial is ‘0’.

Answer: B

Question. Assertion : Zeroes of f(x) = x² – 4x – 5 are 5, – 1.
Reason : The polynomial whose zeroes are 2 + √3, 2 –√3 is x² – 4x + 7.

Answer: C

Question. Assertion : x² + 4x – 5 has two zeroes.
Reason : A quadratic polynomial can have at the most two zeroes.

Answer: D

Question. Assertion : If one zero of polynomial p(x) = (k ² + 4) x² + 13x + 4k is reciprocal of other, then k = 2.
Reason : If (x – α) is a factor of p(x), then p(α) = 0 i.e. α is a zero of p(x).

Answer: B

One word Question :

Question. Find the common zero of x² –1, x⁴ –1 and (x –1)² ?
Answer: x + 1

Question. Write a polynomial whose sum and product of zeroes are 2 and –9 respectively.
Answer: x² – 2x – 9

Question. What should be added to the polynomial p(x) = x² – 5x + 4, so that 2 is a zero of p(x)?
Answer: 2

Question. If α and β are the zeroes of the polynomial x2 – 5x + 6, find the value of 1/α +1/β .
Answer: 5/ 6