CBSE Class 10 Mathematics Polynomials MCQs Set 12

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: (c) \( a + b = 0 \)

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 of the options
Answer: (d) None of the options

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 of the options
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 of the options
Answer: (c) 10

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 the options
Answer: (b) An even positive integer

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

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 of the options
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 the options
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 the options
Answer: (d) None of the options

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) \( \frac{33}{8} \)
Answer: (b) -2

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) \( \frac{490}{9} \)
(b) \( \frac{-490}{9} \)
(c) \( \frac{470}{9} \)
(d) None of the options
Answer: (b) \( \frac{-490}{9} \)

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: (d) -2

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: (c) 3

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

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

Question. Let \( \alpha, \beta \) be the zeros of the polynomial \( x^2 - px + r \) and \( \frac{\alpha}{2}, 2\beta \) be the zeros of \( x^2 - qx + r \). Then the value of r is –
(a) \( \frac{2}{9}(p - q)(2q - p) \)
(b) \( \frac{2}{9}(q - p)(2p - q) \)
(c) \( \frac{2}{9}(q - 2p)(2q - p) \)
(d) \( \frac{2}{9}(2p - q)(2q - p) \)
Answer: (d) \( \frac{2}{9}(2p - q)(2q - p) \)

Question. When \( x^{200} + 1 \) is divided by \( x^2 + 1 \), the remainder is equal to –
(a) \( x + 2 \)
(b) \( 2x - 1 \)
(c) 2
(d) -1
Answer: (c) 2

Question. If \( a(p + q)^2 + 2bpq + c = 0 \) and also \( a(q + r)^2 + 2bqr + c = 0 \) then \( pr \) is equal to –
(a) \( p^2 + \frac{a}{c} \)
(b) \( q^2 + \frac{c}{a} \)
(c) \( p^2 + \frac{a}{b} \)
(d) \( q^2 + \frac{a}{c} \)
Answer: (b) \( q^2 + \frac{c}{a} \)

Question. If a, b and c are not all equal and \( \alpha \) and \( \beta \) be the zeros of the polynomial \( ax^2 + bx + c \), then value of \( (1 + \alpha + \alpha^2) (1 + \beta + \beta^2) \) is :
(a) 0
(b) positive
(c) negative
(d) non-negative
Answer: (b) positive

Question. If 2 and 3 are the zeros of \( f(x) = 2x^3 + mx^2 - 13x + n \), then the values of m and n are respectively –
(a) -5, -30
(b) -5, 30
(c) 5, 30
(d) 5, -30
Answer: (b) -5, 30

Question. If \( \alpha, \beta \) are the zeros of the polynomial \( 6x^2 + 6px + p^2 \), then the polynomial whose zeros are \( (\alpha + \beta)^2 \) and \( (\alpha - \beta)^2 \) is –
(a) \( 3x^2 + 4p^2x + p^4 \)
(b) \( 3x^2 + 4p^2x - p^4 \)
(c) \( 3x^2 - 4p^2x + p^4 \)
(d) None of the options
Answer: (c) \( 3x^2 - 4p^2x + p^4 \)

Question. If \( c, d \) are zeros of \( x^2 - 10ax - 11b \) and \( a, b \) are zeros of \( x^2 - 10cx - 11d \), then value of \( a + b + c + d \) is
(a) 1210
(b) -1
(c) 2530
(d) -11
Answer: (a) 1210

Question. If the ratio of the roots of polynomial \( x^2 + bx + c \) is the same as that of the ratio of the roots of \( x^2 + qx + r \), then
(a) \( br^2 = qc^2 \)
(b) \( cq^2 = rb^2 \)
(c) \( q^2c^2 = b^2r^2 \)
(d) \( bq = rc \)
Answer: (b) \( cq^2 = rb^2 \)

Question. The quadratic polynomial whose zeros are twice the zeros of \( 2x^2 - 5x + 2 = 0 \) is –
(a) \( 8x^2 - 10x + 2 \)
(b) \( x^2 - 5x + 4 \)
(c) \( 2x^2 - 5x + 2 \)
(d) \( x^2 - 10x + 6 \)
Answer: (b) \( x^2 - 5x + 4 \)

Question. If \( \alpha, \beta, \gamma \) are the zeros of the polynomial \( x^3 - 3x + 11 \), then the polynomial whose zeros are \( (\alpha+\beta) \), \( (\beta+\gamma) \) and \( (\gamma+\alpha) \) is –
(a) \( x^3 + 3x + 11 \)
(b) \( x^3 - 3x + 11 \)
(c) \( x^3 + 3x - 11 \)
(d) \( x^3 - 3x - 11 \)
Answer: (d) \( x^3 - 3x - 11 \)

Question. If \( \alpha, \beta, \gamma \) are such that \( \alpha + \beta + \gamma = 2 \), \( \alpha^2 + \beta^2 + \gamma^2 = 6 \), \( \alpha^3 + \beta^3 + \gamma^3 = 8 \), then \( \alpha^4 + \beta^4 + \gamma^4 \) is equal to
(a) 10
(b) 12
(c) 18
(d) None of the options
Answer: (c) 18

Question. If \( \alpha, \beta \) are the roots of \( ax^2 + bx + c \) and \( \alpha + k \), \( \beta + k \) are the roots of \( px^2 + qx + r \), then \( k = \)
(a) \( -\frac{1}{2} \left[ \frac{a}{b} - \frac{p}{q} \right] \)
(b) \( \left[ \frac{a}{b} - \frac{p}{q} \right] \)
(c) \( \frac{1}{2} \left[ \frac{b}{a} - \frac{q}{p} \right] \)
(d) \( (ab - pq) \)
Answer: (c) \( \frac{1}{2} \left[ \frac{b}{a} - \frac{q}{p} \right] \)

Question. The condition that \( x^3 - ax^2 + bx - c = 0 \) may have two of the roots equal to each other but of opposite signs is :
(a) \( ab = c \)
(b) \( \frac{2}{3}a = bc \)
(c) \( a^2b = c \)
(d) None of the options
Answer: (a) \( ab = c \)

Question. If one zero of the polynomial \( ax^2 + bx + c \) is positive and the other negative then \( (a,b,c \in R, a \neq 0) \)
(a) a and b are of opposite signs.
(b) a and c are of opposite signs.
(c) b and c are of opposite signs.
(d) a,b,c are all of the same sign.
Answer: (b) a and c are of opposite signs.

Question. If \( \alpha, \beta \) are the zeros of the polynomial \( x^2 - px + q \), then \( \frac{\alpha^2}{\beta^2} + \frac{\beta^2}{\alpha^2} \) is equal to –
(a) \( \frac{p^4}{q^2} + 2 - \frac{4p^2}{q} \)
(b) \( \frac{p^4}{q^2} - 2 + \frac{4p^2}{q} \)
(c) \( \frac{p^4}{q^2} + 2q^2 - \frac{4p^2}{q} \)
(d) None of the options
Answer: (a) \( \frac{p^4}{q^2} + 2 - \frac{4p^2}{q} \)

Question. If \( \alpha, \beta \) are the zeros of the polynomial \( x^2 - px + 36 \) and \( \alpha^2 + \beta^2 = 9 \), then p =
(a) \( \pm 6 \)
(b) \( \pm 3 \)
(c) \( \pm 8 \)
(d) \( \pm 9 \)
Answer: (d) \( \pm 9 \)

Question. If \( \alpha, \beta \) are zeros of \( ax^2 + bx + c \), \( ac \neq 0 \), then zeros of \( cx^2 + bx + a \) are –
(a) \( -\alpha, -\beta \)
(b) \( \alpha, \frac{1}{\beta} \)
(c) \( \frac{1}{\beta}, \frac{1}{\alpha} \)
(d) \( \frac{1}{\alpha}, \frac{1}{\beta} \)
Answer: (d) \( \frac{1}{\alpha}, \frac{1}{\beta} \)

Question. A real number is said to be algebraic if it satisfies a polynomial equation with integral coefficients. Which of the following numbers is not algebraic :
(a) \( \frac{2}{3} \)
(b) 2
(c) 0
(d) \( \pi \)
Answer: (d) \( \pi \)

Question. The cubic polynomials whose zeros are 4, \( \frac{3}{2} \) and -2 is :
(a) \( 2x^3 + 7x^2 + 10x - 24 \)
(b) \( 2x^3 + 7x^2 - 10x - 24 \)
(c) \( 2x^3 - 7x^2 - 10x + 24 \)
(d) None of the options
Answer: (c) \( 2x^3 - 7x^2 - 10x + 24 \)

Question. If the sum of zeros of the polynomial \( p(x) = kx^3 - 5x^2 - 11x - 3 \) is 2, then k is equal to
(a) \( k = -\frac{5}{2} \)
(b) \( k = \frac{2}{5} \)
(c) \( k = 10 \)
(d) \( k = \frac{5}{2} \)
Answer: (d) \( k = \frac{5}{2} \)

Question. If \( f(x) = 4x^3 - 6x^2 + 5x - 1 \) and \( \alpha, \beta \) and \( \gamma \) are its zeros, then \( \alpha\beta\gamma = \)
(a) \( \frac{3}{2} \)
(b) \( \frac{5}{4} \)
(c) \( -\frac{3}{2} \)
(d) \( \frac{1}{4} \)
Answer: (d) \( \frac{1}{4} \)

Question. Consider \( f(x) = 8x^4 - 2x^2 + 6x - 5 \) and \( \alpha, \beta, \gamma, \delta \) are it's zeros then \( \alpha + \beta + \gamma + \delta = \)
(a) \( \frac{1}{4} \)
(b) \( -\frac{1}{4} \)
(c) \( -\frac{3}{2} \)
(d) None of the options​​​​​​​
Answer: (d) None of the options

Question. If \( x^2 - ax + b = 0 \) and \( x^2 - px + q = 0 \) have a root in common and the second equation has equal roots, then
(a) \( b + q = 2ap \)
(b) \( b + q = \frac{ap}{2} \)
(c) \( b + q = ap \)
(d) None of the options
Answer: (b) \( b + q = \frac{ap}{2} \)