CBSE Class 10 Polynomials Sure Shot Questions Set E

Question. The polynomial \( f(x) \) has roots of equations 3, –3, –k. Given that the coefficient of \( x^3 \) is 2, and that \( f(x) \) has a remainder of 8 when divided by \( x + 1 \), the value of k is
(a) 1/2
(b) 1/4
(c) 1/5
(d) 2
Answer: (a) 1/2

Question. One of the factors of \( x^3 + 3x^2 – x – 3 \) is
(a) \( x + 1 \)
(b) \( x + 2 \)
(c) \( x – 2 \)
(d) \( x – 3 \)
Answer: (a) \( x + 1 \)

Question. If \( ax^2 + 2a^2x + b^3 \) is divisible by \( x + a \), then _____.
(a) \( a = b \)
(b) \( a + b = 0 \)
(c) \( a^2 – ab + b^2 = 0 \)
(d) \( a^2 + 2ab + b^2 = 0 \)
Answer: (a) \( a = b \)

Question. If \( x^3 + 2x^2 + ax + b \) is exactly divisible by \( (x + a) \) and \( (x – 1) \), then _____.
(a) \( a = –2 \)
(b) \( b = –1 \)
(c) \( a = –1 \)
(d) \( b = 1 \)
Answer: (c) \( a = –1 \)

Question. If \( f(x) = ax^2 + bx + c \) is divided by \( (bx + c) \), then the remainder is_____.
(a) \( \frac{c^2}{b^2} \)
(b) \( \frac{ac^2}{b^2} + 2c \)
(c) \( f\left(-\frac{c}{b}\right) \)
(d) \( \frac{ac^2 + 2b^2c}{b^2} \)
Answer: (c) \( f\left(-\frac{c}{b}\right) \)

Question. \( ax^4 + bx^3 + cx^2 + dx + e \) is exactly divisible by \( x^2 – 1 \), when:
(a) \( a + b + c + e = 0 \)
(b) \( a + c + e = 0 \)
(c) \( a + b = 0 \)
(d) \( a + c + e = b + d = 1 \)
Answer: (b) \( a + c + e = 0 \)

Question. The remainder of \( x^4 + x^3 – x^2 + 2x + 3 \) when divided by \( x – 3 \) is
(a) 105
(b) 108
(c) 10
(d) None
Answer: (b) 108

Question. If \( x – 3 \) is a factor of \( x^3 + 3x^2 + 3x + p \), then the value of p is
(a) 0
(b) –63
(c) 10
(d) None
Answer: (b) –63

Question. The value of \( ax^2 + bx + c \) when \( x = 0 \) is 6. The remainder when dividing by \( x + 1 \) is 6. The remainder when dividing by \( x + 2 \) is 8. Then the sum of a, b and c is
(a) 0
(b) –1
(c) 10
(d) None
Answer: (a) 0

Question. \( x^n – y^n \) is divisible by \( x + y \), when n is_______.
(a) An odd positive integer
(b) An even positive integer
(c) An integer
(d) None of these
Answer: (c) An integer

Question. If \( \alpha, \beta \) are the zeros of the quadratic polynomial \( 4x^2 – 4x + 1 \), then \( \alpha^3 + \beta^3 \) is –
(a) 1/4
(b) 1/8
(c) 16
(d) 32
Answer: (a) 1/4

Question. If \( \alpha, \beta, \gamma \) are the zeros of the polynomial \( x^3 + 4x + 1 \), then \( (\alpha + \beta)^{-1} + (\beta + \gamma)^{-1} + (\gamma + \alpha)^{-1} = \)
(a) 2
(b) 3
(c) 4
(d) 5
Answer: (c) 4

Question. The remainder when \( x^{1999} \) is divided by \( x^2 – 1 \) is
(a) – x
(b) 3x
(c) x
(d) None
Answer: (c) x

Question. For the expression \( f(x) = x^3 + ax^2 + bx + c \), if \( f(1) = f(2) = 0 \) and \( f(4) = f(0) \). 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 these
Answer: (a) \( a = – 9, b = 20, c = – 12 \)

Question. If \( x + 1 \) is a factor of \( ax^4 + bx^3 + cx^2 + dx + e = 0 \) then ____
(a) \( a + c + e = b + d \)
(b) \( a + b = c + d \)
(c) \( a + b + c + d + e = 0 \)
(d) \( a + c + b = d + e \)
Answer: (a) \( a + c + e = b + d \)

Question. If \( (x – 3) \) is the factor of \( 3x^3 – x^2 + px + q \) then___
(a) \( p + q = 72 \)
(b) \( 3p + q = 72 \)
(c) \( 3p + q = –72 \)
(d) \( q – 3p = 72 \)
Answer: (c) \( 3p + q = –72 \)

Question. For what values of n, \( (x + y) \) is a factor of \( (x – y)^n \).
(a) for all values of n
(b) 1
(c) only for odd numbers
(d) none of these
Answer: (d) none of these

Question. \( f(x) = 3x^5 + 11x^4 + 90x^2 – 19x + 53 \) is divided by \( x + 5 \) then the remainder is ______.
(a) 100
(b) –100
(c) –102
(d) 102
Answer: (c) –102

Question. If \( (x – 3), (x – 3) \) are factors of \( x^3 – 4x^2 – 3x + 18 \); then the other factor is
(a) \( x + 2 \)
(b) \( x + 3 \)
(c) \( x – 2 \)
(d) \( x + 6 \)
Answer: (a) \( x + 2 \)

Question. If \( f\left(-\frac{3}{4}\right)=0 \); 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) \( 4x + 3 \)

Question. \( f(x) = 16x^2 + 51x + 35 \) then one of the factors of \( f(x) \) is
(a) \( x – 1 \)
(b) \( x + 3 \)
(c) \( x – 3 \)
(d) \( x + 1 \)
Answer: (d) \( x + 1 \)

Question. If \( ax^3 + 9x^2 + 4x – 1 \) 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) 33/8

Question. If \( a^3 –3a^2b + 3ab^2 – b^3 \) is divided by \( (a – b) \), then the remainder is
(a) \( a^2 – ab + b^2 \)
(b) \( a^2 + ab + b^2 \)
(c) 1
(d) 0
Answer: (d) 0

Question. If \( \alpha + \beta = 4 \) and \( \alpha^3 + \beta^3 = 44 \), then \( \alpha, \beta \) are the zeros of the polynomial.
(a) \( 2x^2 – 7x + 6 \)
(b) \( 3x^2 + 9x + 11 \)
(c) \( 9x^2 – 27x + 20 \)
(d) \( 3x^2 – 12x + 5 \)
Answer: (d) \( 3x^2 – 12x + 5 \)

Question. If \( y = f(x) = mx + c \); then \( f(y) \) in terms of x is
(a) \( mx + m + c \)
(b) \( m + mc + c \)
(c) \( m^2x + mc + c \)
(d) \( m^2x + m^2c \)
Answer: (c) \( m^2x + mc + c \)

Question. If \( 7 + 3x \) is a factor of \( 3x^3 + 7x \), then the remainder is
(a) 490/9
(b) –490/9
(c) 470/9
(d) None
Answer: (b) –490/9

Question. The remainder when \( f(x) = 3x^4 + 2x^3 - \frac{x^2}{3} - \frac{x}{9} + \frac{2}{27} \) is divided by \( g(x) = x + \frac{2}{3} \) is
(a) –1
(b) 1
(c) 0
(d) –2
Answer: (c) 0

Question. The remainder when \( 1 + x + x^2 + x^3 + ..........+ x^{2006} \) is divided by \( x – 1 \) is
(a) 2005
(b) 2006
(c) 2007
(d) 2008
Answer: (c) 2007

Question. If \( (x – 1), (x + 1) \) and \( (x – 2) \) are factors of \( x^4 + (p – 3)x^3 – (3p – 5)x^2 + (2p – 9)x + 6 \) then the value of p is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (d) 4

Question. If the remainder when the polynomial \( f(x) \) is divided by \( x – 1, x + 1 \) are 6, 8 respectively then the remainder when \( f(x) \) is divided by \( (x – 1)(x + 1) \) is
(a) \( 7 – x \)
(b) \( 7 + x \)
(c) \( 8 – x \)
(d) \( 8 + x \)
Answer: (a) \( 7 – x \)

Question. Find the remainder obtained when \( x^{2007} \) is divisible by \( x^2 – 1 \).
(a) \( x^2 \)
(b) \( x \)
(c) \( x + 1 \)
(d) \( –x \)
Answer: (b) \( x \)

Question. If a polynomial \( 2x^3 – 9x^2 + 15x + p \), when divided by \( (x – 2) \), leaves –p as remainder, then p is equal to
(a) –16
(b) –5
(c) 20
(d) 10
Answer: (b) –5

Question. If \( \alpha, \beta \) and \( \gamma \) are the zeros of the polynomial \( 2x^3 – 6x^2 – 4x + 30 \), then the value of \( (\alpha\beta + \beta\gamma + \gamma\alpha) \) is
(a) – 2
(b) 2
(c) 5
(d) – 30
Answer: (a) – 2

Question. If \( \alpha, \beta \) and \( \gamma \) are the zeros of the polynomial \( f(x) = ax^3 + bx^2 + cx + d \), then \( \frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} = \)
(a) \( -\frac{b}{a} \)
(b) \( \frac{c}{d} \)
(c) \( -\frac{c}{d} \)
(d) \( -\frac{c}{a} \)
Answer: (c) \( -\frac{c}{d} \)

Question. If \( \alpha, \beta \) and \( \gamma \) are the zeros of the polynomial \( f(x) = ax^3 – bx^2 + cx – d \), then \( \alpha^2 + \beta^2 + \gamma^2 = \)
(a) \( \frac{b^2 - ac}{a^2} \)
(b) \( \frac{b^2 + 2ac}{b^2} \)
(c) \( \frac{b^2 - 2ac}{a} \)
(d) \( \frac{b^2 - 2ac}{a^2} \)
Answer: (d) \( \frac{b^2 - 2ac}{a^2} \)

Question. If \( \alpha, \beta \) and \( \gamma \) are the zeros of the polynomial \( f(x) = x^3 + px^2 – pqrx + r \), then \( \frac{1}{\alpha\beta} + \frac{1}{\beta\gamma} + \frac{1}{\gamma\alpha} = \)
(a) \( \frac{r}{p} \)
(b) \( \frac{p}{r} \)
(c) \( -\frac{p}{r} \)
(d) \( -\frac{r}{p} \)
Answer: (b) \( \frac{p}{r} \)

Question. The coefficient of x in \( x^2 + px + q \) 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) –3, – 10