CBSE Class 10 Mathematics Quadratic Equations MCQs Set 12

Question. If the roots, \( x_1 \) and \( x_2 \), of the quadratic equation \( x^2 - 2x + c = 0 \) also satisfy the equation \( 7x_2 - 4x_1 = 47 \), then which of the following is true?
(a) c = –15
(b) \( x_1 = 5, x_2 = 3 \)
(c) \( x_1 = 4.5, x_2 = -2.5 \)
(d) None of the options
Answer: (a) c = –15

Question. The integral values of k for which the equation \( (k - 2) x^2 + 8x + k + 4 = 0 \) has both the roots real, distinct and negative is :
(a) 0
(b) 2
(c) 3
(d) – 4
Answer: (c) 3

Question. If the roots of the equation \( \frac{x^2 - bx}{ax - c} = \frac{m - 1}{m + 1} \) are equal and of opposite sign, then the value of m will be :
(a) \( \frac{a - b}{a + b} \)
(b) \( \frac{b - a}{a + b} \)
(c) \( \frac{a + b}{a - b} \)
(d) \( \frac{b + a}{b - a} \)
Answer: (a) \( \frac{a - b}{a + b} \)

Question. If \( \alpha, \beta \) are the roots of the equation \( x^2 + 2x + 4 = 0 \), then \( \frac{1}{\alpha^3} + \frac{1}{\beta^3} \) is equal to :
(a) \( -\frac{1}{2} \)
(b) \( \frac{1}{4} \)
(c) 32
(d) \( \frac{1}{32} \)
Answer: (b) \( \frac{1}{4} \)

Question. If \( \alpha, \beta \) are the roots of the equation \( x^2 + 7x + 12 = 0 \), then the equation whose roots are \( (\alpha + \beta)^2 \) and \( (\alpha - \beta)^2 \) is :
(a) \( x^2 + 50x + 49 = 0 \)
(b) \( x^2 - 50x + 49 = 0 \)
(c) \( x^2 - 50x - 49 = 0 \)
(d) \( x^2 + 12x + 7 = 0 \)
Answer: (b) \( x^2 - 50x + 49 = 0 \)

Question. The value of k (k > 0) for which the equations \( x^2 + kx + 64 = 0 \) and \( x^2 - 8x + k = 0 \) both will have real roots is :
(a) 8
(b) 16
(c) – 64
(d) None
Answer: (b) 16

Question. If \( \alpha, \beta \) are roots of the quadratic equation \( x^2 + bx - c = 0 \), then the equation whose roots are b and c is
(a) \( x^2 + ax - b = 0 \)
(b) \( x^2 - [(\alpha + \beta) + \alpha\beta] x - \alpha\beta (\alpha + \beta) = 0 \)
(c) \( x^2 + (\alpha\beta + \alpha + \beta) x + \alpha\beta (\alpha + \beta) = 0 \)
(d) \( x^2 + (\alpha\beta + \alpha + \beta) x - \alpha\beta (\alpha + \beta) = 0 \)
Answer: (c) \( x^2 + (\alpha\beta + \alpha + \beta) x + \alpha\beta (\alpha + \beta) = 0 \)

Question. Solve for x : \( x^6 - 26x^3 - 27 = 0 \)
(a) – 1, 3
(b) 1, 3
(c) 1, – 3
(d) –1, –3
Answer: (a) – 1, 3

Question. Solve : \( \sqrt{2x + 9} + x = 13 \) :
(a) 4, 16
(b) 8, 20
(c) 2, 8
(d) None of the options
Answer: (b) 8, 20

Question. Solve : \( \sqrt{2x + 9} - \sqrt{x - 4} = 3 \)
(a) 4, 16
(b) 8, 20
(c) 2, 8
(d) None
Answer: (b) 8, 20

Question. Solve for x : \( 2 \left[ x^2 + \frac{1}{x^2} \right] - 9 \left[ x + \frac{1}{x} \right] + 14 = 0 \) :
(a) \( \frac{1}{2}, 1, 2 \)
(b) \( 2, 4, \frac{1}{3} \)
(c) \( \frac{1}{3}, 4, 1 \)
(d) None
Answer: (a) \( \frac{1}{2}, 1, 2 \)

Question. Solve for x : \( \sqrt{x^2 + x - 6} - x + 2 = \sqrt{x^2 - 7x + 10} \), \( x \in R \) :
(a) \( 2, 6, -\frac{10}{3} \)
(b) 2, 6
(c) –2, –6
(d) None of the options
Answer: (b) 2, 6

Question. The number of real solutions of \( x - \frac{1}{x^2 - 4} = 2 - \frac{1}{x^2 - 4} \) is :
(a) 0
(b) 1
(c) 2
(d) Infinite
Answer: (a) 0

Question. The equation \( \sqrt{x + 1} - \sqrt{x - 1} = \sqrt{4x - 1} \) has :
(a) No solution
(b) One solution
(c) Two solutions
(d) More than two solutions
Answer: (a) No solution

Question. The number of real roots of the equation \( (x - 1)^2 + (x - 2)^2 + (x - 3)^2 = 0 \) :
(a) 0
(b) 2
(c) 3
(d) 6
Answer: (a) 0

Question. If the equation \( (3x)^2 + (27 \times 3^{1/k} - 15) x + 4 = 0 \) has equal roots, then k =
(a) – 2
(b) \( -\frac{1}{2} \)
(c) \( \frac{1}{2} \)
(d) 0
Answer: (b) \( -\frac{1}{2} \)

Question. Equation \( ax^2 + 2x + 1 \) has one double root if :
(a) a = 0
(b) a = – 1
(c) a = 1
(d) a = 2
Answer: (c) a = 1

Question. Solve for x : (x + 2) (x – 5) (x – 6) (x + 1) = 144
(a) –1, –2, –3
(b) 7, – 3, 2
(c) 2, – 3, 5
(d) None of the options
Answer: (b) 7, – 3, 2

Question. Consider a polynomial \( ax^2 + bx + c \) such that zero is one of its roots then
(a) \( c = 0, x = \frac{-b}{a} \) satisfies the polynomial equation
(b) \( c \neq 0, x = \frac{-a}{b} \) satisfies the polynomial equation
(c) \( x = \frac{-b}{a} \) satisfies the polynomial equation.
(d) Polynomial has equal roots.
Answer: (a) \( c = 0, x = \frac{-b}{a} \) satisfies the polynomial equation

Question. Consider a quadratic polynomial \( f(x) = ax^2 - x + c \) such that ac > 1 and its graph lies below x-axis then:
(a) a < 0, c > 0
(b) a < 0, c < 0
(c) a > 0, c > 0
(d) a > 0, c < 0
Answer: (b) a < 0, c < 0

Question. If \( \alpha, \beta \) are the roots of a quadratic equation \( x^2 - 3x + 5 = 0 \) then the equation whose roots are \( (\alpha^2 - 3\alpha + 7) \) and \( (\beta^2 - 3\beta + 7) \) is :
(a) \( x^2 + 4x + 1 = 0 \)
(b) \( x^2 - 4x + 4 = 0 \)
(c) \( x^2 - 4x - 1 = 0 \)
(d) \( x^2 + 2x + 3 = 0 \)
Answer: (b) \( x^2 - 4x + 4 = 0 \)

Question. The expression \( a^2x^2 + bx + 1 \) will be positive for all \( x \in R \) if :
(a) \( b^2 > 4a^2 \)
(b) \( b^2 < 4a^2 \)
(c) \( 4b^2 > a^2 \)
(d) \( 4b^2 < a^2 \)
Answer: (b) \( b^2 < 4a^2 \)

Question. For what value of a the curve \( y = x^2 + ax + 25 \) touches the x-axis :
(a) 0
(b) \( \pm 5 \)
(c) \( \pm 10 \)
(d) None
Answer: (c) \( \pm 10 \)

Question. The value of the expression \( x^2 + 2bx + c \) will be positive for all real x if :
(a) \( b^2 - 4c > 0 \)
(b) \( b^2 - 4c < 0 \)
(c) \( c^2 < b \)
(d) \( b^2 < c \)
Answer: (d) \( b^2 < c \)

Question. If the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are imaginary then for all values of a, b, c and \( x \in R \), the expression \( a^2x^2 + abx + ac \) is
(a) Positive
(b) Non-negative
(c) Negative
(d) May be positive, zero or negative
Answer: (a) Positive

Question. The value of k, so that the equations \( 2x^2 + kx - 5 = 0 \) and \( x^2 - 3x - 4 = 0 \) have one root in common is :
(a) – 2, – 3
(b) \( -3, -\frac{27}{4} \)
(c) – 5, – 6
(d) None of the options
Answer: (b) \( -3, -\frac{27}{4} \)

Question. If the expression \( x^2 - 11x + a \) and \( x^2 - 14x + 2a \) must have a common factor and \( a \neq 0 \), then the common factor is :
(a) (x – 3)
(b) (x – 6)
(c) (x – 8)
(d) None
Answer: (c) (x – 8)

Question. The value of m for which one of the roots of \( x^2 - 3x + 2m = 0 \) is double of one of the roots of \( x^2 - x + m = 0 \) is :
(a) 0, 2
(b) 0, – 2
(c) 2, – 2
(d) None
Answer: (b) 0, – 2

Question. If the equations \( x^2 + bx + c = 0 \) and \( x^2 + cx + b = 0 \), (\( b \neq c \)) have a common root then :
(a) b + c = 0
(b) b + c = 1
(c) b + c + 1 = 0
(d) None of the options
Answer: (c) b + c + 1 = 0

Question. If both the roots of the equations \( k(6x^2 + 3) + rx + 2x^2 - 1 = 0 \) and \( 6k (2x^2 + 1) + px + 4x^2 - 2 = 0 \) are common, then 2r – p is equal to :
(a) 1
(b) – 1
(c) 2
(d) 0
Answer: (d) 0

Question. If \( x^2 - ax - 21 = 0 \) and \( x^2 - 3ax + 35 = 0 \) ; a > 0 have a common root, then a is equal to :
(a) 1
(b) 2
(c) 4
(d) 5
Answer: (c) 4

Question. The values of a for which the quadratic equation \( (1 - 2a) x^2 - 6ax - 1 = 0 \) and \( ax^2 - x + 1 = 0 \) have at least one root in common are :
(a) \( \frac{1}{2}, \frac{2}{9} \)
(b) \( 0, \frac{1}{2} \)
(c) \( \frac{2}{9} \)
(d) \( 0, \frac{1}{2}, \frac{2}{9} \)
Answer: (c) \( \frac{2}{9} \)

Question. If the quadratic equation \( 2x^2 + ax + b = 0 \) and \( 2x^2 + bx + a = 0 \) (\( a \neq b \)) have a common root, the value of a + b is :
(a) – 3
(b) – 2
(c) – 1
(d) 0
Answer: (b) – 2

Question. If the equation \( x^2 + bx + ca = 0 \) and \( x^2 + cx + ab = 0 \) have a common root and \( b \neq c \), then their other roots will satisfy the equation :
(a) \( x^2 - (b + c) x + bc = 0 \)
(b) \( x^2 - ax + bc = 0 \)
(c) \( x^2 + ax + bc = 0 \)
(d) None of the options
Answer: (c) \( x^2 + ax + bc = 0 \)

Question. If both the roots of the equations \( x^2 + mx + 1 = 0 \) and \( (b - c) x^2 + (c - a) x + (a - b) = 0 \) are common, then :
(a) m = – 2
(b) m = – 1
(c) m = 0
(d) m = 1
Answer: (a) m = – 2

Question. For the equation \( 3x^2 + px + 3 = 0 \), p > 0, if one of the roots is square of the other, then p =
(a) \( \frac{1}{3} \)
(b) 1
(c) 3
(d) \( \frac{2}{3} \)
Answer: (c) 3

Question. The roots of the equation \( |x^2 - x - 6| = x + 2 \) are
(a) – 2, 1, 4
(b) 0, 2, 4
(c) 0, 1, 4
(d) –2, 2, 4
Answer: (d) –2, 2, 4

Question. The equation \( x - \frac{2}{x - 1} = 1 - \frac{2}{x - 1} \) has
(a) Two roots
(b) Infinitely many roots
(c) Only one root
(d) No root
Answer: (d) No root

Question. The value of x which satisfy the expression : \( (5 + 2\sqrt{6})^{x^2 - 3} + (5 - 2\sqrt{6})^{x^2 - 3} = 10 \)
(a) \( \pm 2, \pm \sqrt{3} \)
(b) \( \pm \sqrt{2}, \pm 4 \)
(c) \( \pm 2, \pm \sqrt{2} \)
(d) \( 2, \sqrt{2}, \sqrt{3} \)
Answer: (c) \( \pm 2, \pm \sqrt{2} \)

Question. Find all the integral values of a for which the quadratic equation (x – a) (x – 10) + 1 = 0 has integral roots :
(a) 12, 8
(b) 4, 6
(c) 2, 0
(d) None
Answer: (a) 12, 8

Question. If \( x^2 - (a + b) x + ab = 0 \), then the value of \( (x - a)^2 + (x - b)^2 \) is
(a) \( a^2 + b^2 \)
(b) \( (a + b)^2 \)
(c) \( (a - b)^2 \)
(d) \( a^2 - b^2 \)
Answer: (c) \( (a - b)^2 \)

Question. The sum of the roots of \( \frac{1}{x + a} + \frac{1}{x + b} = \frac{1}{c} \) is zero. The product of the roots is
(a) 0
(b) \( \frac{1}{2} (a + b) \)
(c) \( -\frac{1}{2} (a^2 + b^2) \)
(d) \( 2(a^2 + b^2) \)
Answer: (c) \( -\frac{1}{2} (a^2 + b^2) \)

Question. If the roots of the equations \( (c^2 - ab)x^2 - 2(a^2 - bc)x + (b^2 - ac) = 0 \) for \( a \neq 0 \) are real and equal, then the value of \( a^3 + b^3 + c^3 \) is
(a) abc
(b) 3abc
(c) zero
(d) None of the options
Answer: (b) 3abc

Question. If, \( \alpha, \beta \) are the roots of \( X^2 - 8X + P = 0 \) and \( \alpha^2 + \beta^2 = 40 \). then the value of P is
(a) 8
(b) 10
(c) 12
(d) 14
Answer: (c) 12

Question. If, l, m, n are real and l=m, then the roots of the equations \( (l - m)x^2 - 5(l + m)x - 2(l - m) = 0 \) are
(a) Real and Equal
(b) Complex
(c) Real and Unequal
(d) None of the options
Answer: (c) Real and Unequal

Question. In a family, eleven times the number of children is greater than twice the square of the number of children by 12. How many children are there ?
(a) 3
(b) 4
(c) 2
(d) 5
Answer: (b) 4

Question. The sum of all the real roots of the equation \( |x - 2|^2 + |x - 2| - 2 = 0 \) is
(a) 2
(b) 3
(c) 4
(d) None of the options
Answer: (c) 4

Question. If the ratio between the roots of the equation \( lx^2 + mx + n = 0 \) is p:q, then the value of \( \sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} + \sqrt{\frac{n}{l}} \) is
(a) 4
(b) 3
(c) 0
(d) –1
Answer: (a) 4

Question. Find the root of the quadratic equation \( bx^2 - 2ax + a = 0 \)
(a) \( \frac{\sqrt{b}}{\sqrt{b} \pm \sqrt{a - b}} \)
(b) \( \frac{\sqrt{a}}{\sqrt{b} \pm \sqrt{a - b}} \)
(c) \( \frac{\sqrt{a}}{\sqrt{a} \pm \sqrt{a - b}} \)
(d) \( \frac{\sqrt{a}}{\sqrt{a} \pm \sqrt{a + b}} \)
Answer: (c) \( \frac{\sqrt{a}}{\sqrt{a} \pm \sqrt{a - b}} \)

Question. If 4 is a solution of the equation \( x^2 + 3x + k = 10 \), where k is a constant, what is the other solution ?
(a) –18
(b) –7
(c) –28
(d) None of the options
Answer: (b) –7

Question. The coefficient of x in the equation \( x^2 + px + p = 0 \) was wrongly written as 17 in place of 13 and the roots thus found were –2 and –15. The roots of the correct equation would be
(a) –4, –9
(b) –3, –10
(c) –3, –9
(d) –4, –10
Answer: (b) –3, –10

Question. If \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( ax^2 + bx + c = 0 \), then the value of \( \frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha} \) is
(a) \( \frac{2bc - a^3}{b^2c} \)
(b) \( \frac{3abc - b^3}{a^2c} \)
(c) \( \frac{3abc - b^2}{a^3c} \)
(d) \( \frac{ab - b^2c}{2b^2c} \)
Answer: (b) \( \frac{3abc - b^3}{a^2c} \)