CBSE Class 10 Mathematics Polynomials MCQs Set K

Question. If \( x + \frac{1}{x} = 5 \), then the value of \( x^3 + \frac{1}{x^3} \) is
(a) 110
(b) 90
(c) 80
(d) 50
Answer: (a) 110

Question. If \( x^3 - (x + 1)^2 = 2001 \) then the value of \( x \) is
(a) 14
(b) 13
(c) 10
(d) None of the options
Answer: (b) 13

Question. The square root of \( \frac{x^2}{y^2} + \frac{y^2}{4x^2} - \frac{x}{y} + \frac{y}{2x} - \frac{3}{4} \) is
(a) \( \frac{x}{y} - \frac{1}{2} - \frac{y}{2x} \)
(b) \( \frac{x}{y} + \frac{1}{2} - \frac{y}{2x} \)
(c) \( \frac{x}{y} + \frac{1}{2} + \frac{y}{2x} \)
(d) \( \frac{x}{y} - \frac{1}{4} - \frac{y}{2x} \)
Answer: (a) \( \frac{x}{y} - \frac{1}{2} - \frac{y}{2x} \)

Question. If the zeros of the polynomial \( ax^2 + bx + c \) be in the ratio \( m : n \), then
(a) \( b^2 mn = (m^2 + n^2) ac \)
(b) \( (m + n)^2 ac = b^2 mn \)
(c) \( b^2 (m^2 + n^2) = mnac \)
(d) None of the options
Answer: (b) \( (m + n)^2 ac = b^2 mn \)

Question. If \( \alpha \neq \beta \) and the difference between the roots of the polynomials \( x^2 + ax + b \) and \( x^2 + bx + a \) is the same, then
(a) \( a + b + 4 = 0 \)
(b) \( a + b - 4 = 0 \)
(c) \( a - b + 4 = 0 \)
(d) \( a - b - 4 = 0 \)
Answer: (a) \( a + b + 4 = 0 \)

Question. If \( \alpha \neq \beta \) and \( \alpha^2 = 5\alpha - 3 \), \( \beta^2 = 5\beta - 3 \), then the polynomial whose zeros are \( \frac{\alpha}{\beta} \) and \( \frac{\beta}{\alpha} \) is :
(a) \( 3x^2 - 25x + 3 \)
(b) \( x^2 - 5x + 3 \)
(c) \( x^2 + 5x - 3 \)
(d) \( 3x^2 - 19x + 3 \)
Answer: (a) \( 3x^2 - 25x + 3 \)

Question. The factors of \( a^2(b^3 - c^3) + b^2(c^3 - a^3) + c^2(a^3 - b^3) \) are
(a) \( (a - b) (b - c) (c - a) (ab + bc + ca) \)
(b) \( (a + b) (b + c) (c + a) (ab + bc + ca) \)
(c) \( (a - b) (b - c) (c - a) (ab - bc - ca) \)
(d) None of the options
Answer: (a) \( (a - b) (b - c) (c - a) (ab + bc + ca) \)

Question. If \( p, q \) are zeros of \( x^2 + px + q \), then
(a) \( p = 1 \)
(b) \( p = 1 \) or \( 0 \)
(c) \( p = -2 \)
(d) \( p = -2 \) or \( 0 \)
Answer: (a) \( p = 1 \)

Question. On simplifying \( (a + b)^3 + (a - b)^3 + 6a(a^2 - b^2) \) we get
(a) \( 8a^2 \)
(b) \( 8a^2b \)
(c) \( 8a^3b \)
(d) \( 8a^3 \)
Answer: (d) \( 8a^3 \)

Question. Factors of \( (42 - x - x^2) \) are
(a) \( (x - 7)(x - 6) \)
(b) \( (x + 7)(x - 6) \)
(c) \( (x + 7)(6 - x) \)
(d) \( (x + 7)(x + 6) \)
Answer: (c) \( (x + 7)(6 - x) \)

Question. Factors of \( \left( x^2 + \frac{x}{6} - \frac{1}{6} \right) \) are
(a) \( \frac{1}{6}(2x+1)(3x+1) \)
(b) \( \frac{1}{6}(2x+1)(3x-1) \)
(c) \( \frac{1}{6}(2x-1)(3x-1) \)
(d) \( \frac{1}{6}(2x-1)(3x+1) \)
Answer: (b) \( \frac{1}{6}(2x+1)(3x-1) \)

Question. Value of \( \frac{a^3 + b^3 + c^3 - 3abc}{ab + bc + ca - a^2 - b^2 - c^2} \), when \( a = -5, b = -6, c = 10 \) is
(a) 1
(b) -1
(c) 2
(d) -2
Answer: (b) -1

Question. If \( (x + y + z) = 1 \), \( xy + yz + zx = -1 \), \( xyz = -1 \), then the value of \( x^3 + y^3 + z^3 \) is
(a) -1
(b) 1
(c) 2
(d) -2
Answer: (c) 2

Question. In method of factorization of an algebraic expression, Which of the following statements is false?
(a) Taking out a common factor from two or more terms
(b) Taking out a common factor from a group of terms
(c) By using remainder theorem
(d) By using standard identities
Answer: (c) By using remainder theorem

Question. Factors of \( (a + b)^3 - (a - b)^3 \) are
(a) \( 2ab(3a^2 + b^2) \)
(b) \( ab(3a^2 + b^2) \)
(c) \( 2b(3a^2 + b^2) \)
(d) \( 3a^2 + b^{20} \)
Answer: (c) \( 2b(3a^2 + b^2) \)

Question. The homogeneous function of the second degree in \( x \) and \( y \) having \( 2x - y \) as a factor, taking the value 2 when \( x = y = 1 \) and vanishing if \( x = -1, y = 1 \) is
(a) \( 2x^2 + xy - y^2 \)
(b) \( 3x^2 - 2xy + y^2 \)
(c) \( x^2 + xy - 2y^2 \)
(d) None of the options
Answer: (a) \( 2x^2 + xy - y^2 \)

Question. The common quantity that must be added to each term of \( a^2 : b^2 \) to make it equal to \( a : b \) is
(a) \( ab \)
(b) \( a + b \)
(c) \( a - b \)
(d) \( \frac{a}{b} \)
Answer: (a) \( ab \)

Question. If the polynomial \( 16x^4 - 24x^3 + 41x^2 - mx + 16 \) be a perfect square, then the value of "m" is
(a) 12
(b) -12
(c) 24
(d) -24
Answer: (c) 24

Question. If \( a - b = 3 \), \( a + b + x = 2 \), then the value of \( (a - b)[x^3 - 2ax^2 + a^2x - (a + b)b^2] \) is
(a) 84
(b) 48
(c) 32
(d) 36
Answer: (b) 48

Question. If \( abx^2 = (a - b)^2(x + 1) \), then the value of \( 1 + \frac{4}{x} + \frac{4}{x^2} \) is:-
(a) \( \left( \frac{a - b}{a + b} \right)^2 \)
(b) \( \left( \frac{a + b}{a - b} \right)^2 \)
(c) \( \left( \frac{a}{a + b} \right)^2 \)
(d) \( \left( \frac{b}{a + b} \right)^2 \)
Answer: (b) \( \left( \frac{a + b}{a - b} \right)^2 \)

Question. Let \( \alpha, \beta \) be the zeros of the polynomial \( (x - a)(x - b) - c \) with \( c \neq 0 \). Then the zeros of the polynomial \( (x - \alpha)(x - \beta) + c \) are
(a) \( a, c \)
(b) \( b, c \)
(c) \( a, b \)
(d) \( a + c, b + c \)
Answer: (c) \( a, b \)

Question. A homogeneous expression of second degree in \( x \) & \( y \) is
(a) \( ax^2 + bx + c \)
(b) \( ax^2 + bx + cy \)
(c) \( ax^2 + bx + cy^2 \)
(d) \( ax^2 + bxy + cy^2 \)
Answer: (d) \( ax^2 + bxy + cy^2 \)

Question. If the sum of the zeros of the polynomial \( x^2 + px + q \) is equal to the sum of their squares, then
(a) \( p^2 - q^2 = 0 \)
(b) \( p^2 + q^2 = 2q \)
(c) \( p^2 + p = 2q \)
(d) None of the options
Answer: (c) \( p^2 + p = 2q \)

Question. The G.C.D of \( x^2 - 3x + 2 \) and \( x^2 - 4x + 4 \) is
(a) \( x - 2 \)
(b) \( (x - 2)(x - 1) \)
(c) \( (x - 2)^2 \)
(d) \( (x - 2)^3(x - 1) \)
Answer: (a) \( x - 2 \)

Question. The L.C.M. of \( 22x(x + 1)^2 \) and \( 36x^2(2x^2 + 3x + 1) \) is
(a) \( 2x(x + 1) \)
(b) \( 396x^2(x + 1)^2(2x + 1) \)
(c) \( 792x^3(x + 1)^2 (2x^2 + 3x + 1) \)
(d) None of the options
Answer: (b) \( 396x^2(x + 1)^2(2x + 1) \)

Question. The L.C.M of \( x^3 - 8 \) and \( x^2 - 5x + 6 \) is
(a) \( x - 2 \)
(b) \( x^2 + 2x + 4 \)
(c) \( (x - 2)(x^2 + 2x + 4) \)
(d) \( (x - 2)(x - 3)(x^2 + 2x + 4) \)
Answer: (d) \( (x - 2)(x - 3)(x^2 + 2x + 4) \)

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

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) \( x^2 - 3x - 18 \)
(c) \( x^2 - 8x + 12 \)
(d) None of the options
Answer: (c) \( x^2 - 8x + 12 \)

Question. The G.C.D. of two polynomials is \( (x - 1) \) and their L.C.M. is \( x^6 - 1 \). If one of the polynomials is \( x^3 - 1 \), then the other polynomial is
(a) \( x^3 - 1 \)
(b) \( x^4 - x^3 + x - 1 \)
(c) \( x^2 - x + 1 \)
(d) None of the options
Answer: (b) \( x^4 - x^3 + x - 1 \)

Question. The L.C.M. of \( 2x \) and 8 is
(a) \( 2x \)
(b) \( 4x \)
(c) \( 8x \)
(d) \( 16x \)
Answer: (c) \( 8x \)

Question. If \( x^2 + \frac{1}{x^2} = 38 \), then the value of \( x - \frac{1}{x} \) is
(a) 6
(b) 4
(c) 0
(d) None of the options
Answer: (a) 6

Question. The simplest form of \( (2x + 3)^3 - (2x - 3)^3 \) is
(a) \( 54 + 72x^2 \)
(b) \( 72 + 54x^2 \)
(c) \( 54 + 54x^2 \)
(d) None of the options
Answer: (a) \( 54 + 72x^2 \)

Question. The simplest form of \( (p - q)^3 + (q - r)^3 + (r - p)^3 \) is
(a) \( 4(p - q)(q - r)(r - p) \)
(b) \( 2 (p - q)(q - r)(r - p) \)
(c) \( 3 (p - q)(q - r)(r - p) \)
(d) None of the options
Answer: (c) \( 3 (p - q)(q - r)(r - p) \)

Question. The square root of \( x^4 + 6x^3 + 17x^2 + 24x + 16 \) is
(a) \( x^2 + 3x + 4 \)
(b) \( 2x^2 + 3x + 4 \)
(c) \( 3x^2 + 3x + 4 \)
(d) None of the options
Answer: (a) \( x^2 + 3x + 4 \)

Question. The square root of \( x^4 - 2x^3 + 3x^2 - 2x + 1 \) is
(a) \( x^2 + x + 1 \)
(b) \( x^2 - x + 1 \)
(c) \( x^2 + x - 1 \)
(d) \( x^2 - x - 1 \)
Answer: (b) \( x^2 - x + 1 \)

Question. The value of \( \lambda \) for which one zero of \( 3x^2 - (1 + 4\lambda) x + \lambda^2 + 2 \) may be one-third of the other is
(a) 4
(b) \( \frac{33}{8} \)
(c) \( \frac{17}{4} \)
(d) \( \frac{31}{8} \)
Answer: (b) \( \frac{33}{8} \)

Question. The factors of \( a^3(b - c) + b^3(c - a) + c^3(a - b) \) are
(a) \( (a + b + c) (a - b) (b - c) (c - a) \)
(b) \( -(a + b + c) (a - b) (b - c) (c - a) \)
(c) \( 2 (a + b + c) (a - b) (b - c) (c - a) \)
(d) \( -2 (a + b + c) (a - b) (b - c) (c - a) \)
Answer: (b) \( -(a + b + c) (a - b) (b - c) (c - a) \)

Question. The value of 'a', for which one root of the quadratic polynomial \( (a^2 - 5a + 3) x^2 + (3a - 1) x + 2 \) is twice as large as the other, is
(a) \( -\frac{1}{3} \)
(b) \( \frac{2}{3} \)
(c) \( -\frac{2}{3} \)
(d) \( \frac{1}{3} \)
Answer: (d) \( \frac{1}{3} \)

Question. If the polynomial \( x^{19} + x^{17} + x^{13} + x^{11} + x^7 + x^5 + x^3 \) is divided by \( (x^2 + 1) \), then the remainder is
(a) 1
(b) \( x^2 + 4 \)
(c) \( -x \)
(d) \( x \)
Answer: (c) \( -x \)

Question. If \( (x - 2) \) is a common factor of \( x^3 - 4x^2 + ax + b \) and \( x^3 - ax^2 + bx + 8 \), then the values of a and b are respectively
(a) 3 and 5
(b) 2 and -4
(c) 4 and 0
(d) 0 and 4
Answer: (d) 0 and 4

Question. If the expressions \( ax^3 + 3x^2 - 3 \) and \( 2x^3 - 5x + a \) on dividing by \( x - 4 \) leave the same remainder, then the value of a is
(a) 1
(b) 0
(c) 2
(d) -1
Answer: (a) 1

Question. If the polynomial \( x^6 + px^5 + qx^4 - x^2 - x - 3 \) is divisible by \( x^4 - 1 \), then the value of \( p^2 + q^2 \) is
(a) 1
(b) 5
(c) 10
(d) 13
Answer: (c) 10

Question. If \( 3x^3 + 2x^2 - 3x + 4 = (Ax + B)(x - 1)(x + 2) + C(x - 1) + D \) for all values of \( x \), then \( A + B + C + D \) is
(a) 0
(b) 14
(c) 10
(d) All
Answer: (b) 14

Question. The expression \( x^3 + gx^2 + hx + k \) is divisible by both \( x \) and \( x - 2 \) but leaves a remainder of 24 when divided by \( x + 2 \) then the values of g, h and k are
(a) \( g = 10, h = -3, k = 0 \)
(b) \( g = 3, h = -10, k = 0 \)
(c) \( g = 10, h = -2, k = 3 \)
(d) None of the options
Answer: (b) \( g = 3, h = -10, k = 0 \)

Question. The value of m if \( 2x^m + x^3 - 3x^2 - 26 \) leaves a remainder of 226 when it is divided by \( x - 2 \).
(a) 0
(b) 7
(c) 10
(d) All of these
Answer: (b) 7

Question. The expression \( Ax^3 + x^2 + Bx + C \) leaves remainder of \( \frac{21}{4} \) when divided by \( 1 - 2x \) and 18 when divided by \( x \). Given also the expression has a factor of \( (x - 2) \), the values of A, B and C are
(a) \( A = 5, B = -9, C = 3 \)
(b) \( A = 27, B = -18, C = 4 \)
(c) \( A = 4, B = -27, C = 18 \)
(d) None of the options
Answer: (c) \( A = 4, B = -27, C = 18 \)

Question. If \( h(x) = 2x^3 + (6a^2 - 10) x^2 + (6a + 2) x - 14a - 2 \) is exactly divisible by \( x - 1 \) but not by \( x + 1 \), then the value of a is
(a) 0
(b) -1
(c) 10
(d) 2
Answer: (d) 2

Question. Given the polynomial is exactly divided by \( x + 1 \), and when it is divided by \( 3x - 1 \), the remainder is 4. The polynomial gives a remainder \( hx + k \) when divided by \( 3x^2 + 2x - 1 \) then the values of h and k are
(a) \( h = 2, k = 3 \)
(b) \( h = 3, k = 3 \)
(c) \( h = 3, k = 2 \)
(d) None of the options
Answer: (c) \( h = 3, k = 2 \)

Question. The remainder when \( f(x) = (x^4 - x^3 + 2x - 3) g(x) \) is divided by \( x - 3 \), given that \( x - 3 \) is a factor of \( g(x) + 3 \), where \( g(x) \) is a polynomial is
(a) 0
(b) -171
(c) 10
(d) 2
Answer: (b) -171

Question. If \( x^3 - hx^2 + kx - 9 \) has a factor of \( x^2 + 3 \), then the values of h and k are
(a) \( h = 3, k = 3 \)
(b) \( h = 2, k = 2 \)
(c) \( h = 2, k = 1 \)
(d) None of the options
Answer: (a) \( h = 3, k = 3 \)

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: (c) \( x - 2 \)

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: (c) \( a^2 - ab + b^2 = 0 \)

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) \( f \left( -\frac{c}{b} \right) \)