Class 11 Mathematics Complex Numbers and Quadratic Equation MCQs Set L

Question. The set of real values of x satisfying \( |x-1| \leq 3 \) and \( |x-1| \geq 1 \)
(a) \( [2, 4] \)
(b) \( (-\infty, 2) \cup (4, \infty) \)
(c) \( [-2, 0] \cup [2, 4] \)
(d) \( [0, 2] \)
Answer: (c) \( [-2, 0] \cup [2, 4] \)

Question. If \( \sin\theta, \sin k\theta \) are the roots of \( 4x^2 + 2x - 1 = 0 \), then k =
(a) -2
(b) -3
(c) 3
(d) 2
Answer: (b) -3

Question. If \( \alpha, \beta \) are the roots of \( x^2 - 3x + a = 0 \) and \( \gamma, \delta \) that of \( x^2 - 12x + b = 0 \) and \( \alpha, \beta, \gamma, \delta \) form an increasing G.P. then
(a) \( a = 3, b = 12 \)
(b) \( a = 4, b = 16 \)
(c) \( a = 2, b = 32 \)
(d) \( a = 12, b = 3 \)
Answer: (c) \( a = 2, b = 32 \)

Question. Let \( f(x) \) be a polynomial for which the remainders when divided by \( x-1, x-2, x-3 \) respectively 3, 7, 13. Then the remainder of \( f(x) \) when divided by \( (x-1)(x-2)(x-3) \) is
(a) \( f(x) \)
(b) \( x^2 + x + 1 \)
(c) \( x^2 + 1 \)
(d) \( x + 2 \)
Answer: (b) \( x^2 + x + 1 \)

Question. Let a, b, c be the real numbers, \( a \neq 0 \). If \( \alpha \) is a root of \( a^2x^2 + bx + c = 0 \), \( \beta \) is a root of \( a^2x^2 - bx - c = 0 \) and \( 0 < \alpha < \beta \), then the equation \( a^2x^2 + 2bx + 2c = 0 \), has a root \( \gamma \) that always satisfies
(a) \( \gamma = (\alpha+\beta)/2 \)
(b) \( \gamma = (\alpha+\beta/2) \)
(c) \( \gamma = \alpha \)
(d) \( \alpha < \gamma < \beta \)
Answer: (d) \( \alpha < \gamma < \beta \)

Question. The equation \( (x-a)^3 + (x-b)^3 + (x-c)^3 = 0 \) has
(a) all the roots are equal
(b) one real and two imaginary
(c) 3 real roots namely \( x = a, x = b, x = c \)
(d) no real roots
Answer: (b) one real and two imaginary

Question. The value of 'a' for which the quadratic equation \( 3x^2 + 2(a^2+1)x + (a^2-3a+2) = 0 \) possesses roots of opposite signs lies on
(a) \( (-\infty, 1) \)
(b) \( (-\infty, 0) \)
(c) \( (1, 2) \)
(d) \( [1, 2] \)
Answer: (c) \( (1, 2) \)

Question. If the roots of the equation \( ax^2 + bx + c = 0 \) are the reciprocals of the roots of the equation \( px^2 + qx + r = 0 \) then
(a) \( acq^2 = b^2pr \)
(b) \( ac = pr \)
(c) \( b^2ac = q^2pr \)
(d) \( ab = pq \)
Answer: (a) \( acq^2 = b^2pr \)

Question. If \( y = \tan x \cot 3x, \, x \in R \) then
(a) \( \frac{1}{3} < y < 1 \)
(b) \( \frac{1}{3} \leq y \leq 1 \)
(c) \( \frac{1}{3} \leq y \leq 3 \)
(d) \( y \leq \frac{1}{3} \text{ or } y \geq 3 \)
Answer: (d) \( y \leq \frac{1}{3} \text{ or } y \geq 3 \)

Question. If \( 5^x + (2\sqrt{3})^{2x} \geq 13^x \) then the solution set for x is
(a) \( [2, +\infty) \)
(b) \( \{2\} \)
(c) \( (-\infty, 2] \)
(d) \( [0, 2] \)
Answer: (c) \( (-\infty, 2] \)

Question. If \( x^2 + 6x - 27 > 0; \, -x^2 + 3x + 4 > 0 \) then x lies in the interval
(a) \( (3, 4) \)
(b) \( \{3, 4\} \)
(c) \( (-\infty, 3) \cup (4, \infty) \)
(d) \( (-9, 4) \)
Answer: (a) \( (3, 4) \)

Question. The range of values of x which satisfy \( 5x + 2 < 3x + 8 \) and \( \frac{x+2}{x-1} < 4 \) are
(a) \( (2, 3) \)
(b) \( (-\infty, 1) \cup (2, 3) \)
(c) \( (2, \infty) \)
(d) R
Answer: (a) \( (2, 3) \)

Question. The set of exhaustive values of 'a' for which the inequality \( (a^2+3)x^2 + (a+2)x - 4 < 2 \) holds of at least one negative x, is
(a) \( a > 0 \)
(b) \( a < 0 \)
(c) \( -\infty < a < \infty \)
(d) \( a \in \phi \)
Answer: (c) \( -\infty < a < \infty \)

Question. If the expression \( 3x^2 + 2pxy + 2y^2 + 2ax - 4y + 1 \) can be resolved into two linear factors then p must be a root of the equation
(a) \( x^2 + ax + 6 = 0 \)
(b) \( x^2 + 4ax + 6 = 0 \)
(c) \( x^2 + 4ax + 2a^2 + 6 = 0 \)
(d) \( x^2 - 4ax + 6 = 0 \)
Answer: (c) \( x^2 + 4ax + 2a^2 + 6 = 0 \)

Question. The set of values of x for which the inequalities \( x^2 - 3x - 10 < 0 \), \( 10x - x^2 - 16 > 0 \) hold simultaneously is (EAM - 2007)
(a) \( (-2, 5) \)
(b) \( (2, 8) \)
(c) \( (-2, 8) \)
(d) \( (2, 5) \)
Answer: (d) \( (2, 5) \)

Question. If p and q are distinct prime numbers and if the equation \( x^2 - px + q = 0 \) has positive integers as its roots, then the roots of the equation are (Eamcet-2014)
(a) 1, -1
(b) 2, 3
(c) 1, 2
(d) 3, 1
Answer: (c) 1, 2

Question. Let \( f(x) = x^2 + ax + b \), where \( a, b \in R \). If \( f(x) = 0 \) has all its roots imaginary then the roots of \( f(x) + f'(x) + f''(x) = 0 \) are (EAM - 2009)
(a) real and distinct
(b) imaginary
(c) equal
(d) rational and equal
Answer: (b) imaginary

Question. For \( x \in R \), the least value of \( \frac{x^2-6x+5}{x^2+2x+1} \) is (EAM - 2010)
(a) -1
(b) -1/2
(c) -1/4
(d) -1/3
Answer: (d) -1/3

Question. The product of real roots of the equation \( |x|^{6/5} - 26|x|^{3/5} - 27 = 0 \) is (EAM - 2012)
(a) \( -3^{10} \)
(b) \( -3^{12} \)
(c) \( -3^{12/5} \)
(d) \( -3^{21/5} \)
Answer: (a) \( -3^{10} \)

Question. The set of solutions satisfying both \( x^2 + 5x + 6 \geq 0 \) and \( x^2 + 3x - 4 < 0 \) is (EAM - 2013)
(a) \( (-4, 1) \)
(b) \( (-4, -3] \cup [-2, 1) \)
(c) \( (-4, -3) \cup (-2, 1) \)
(d) \( [-4, -3] \cup [-2, 1] \)
Answer: (b) \( (-4, -3] \cup [-2, 1) \)

Question. If \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 - 2x + 4 = 0 \), then \( \alpha^9 + \beta^9 \) is equal to (EAM - 2013)
(a) \( -2^8 \)
(b) \( 2^9 \)
(c) \( -2^{10} \)
(d) \( 2^{10} \)
Answer: (c) \( -2^{10} \)

Question. If \( 0 < \alpha < \beta < \gamma < \frac{\pi}{2} \), then the equation \( \frac{1}{x - \sin \alpha} + \frac{1}{x - \sin \beta} + \frac{1}{x - \sin \gamma} = 0 \) has
(a) imaginary roots
(b) real and equal roots
(c) real and uequal roots
(d) rational roots
Answer: (c) real and uequal roots

Question. If n is a multiple of 6 and \( \alpha, \beta \) are the roots of \( x^2 + x + 1 = 0 \) then \( (1 + \alpha)^{-n} + (1 + \beta)^{-n} = \)
(a) 2
(b) -2
(c) 0
(d) -1
Answer: (a) 2

Question. If \( \alpha, \beta \) are real and \( \alpha^2, -\beta^2 \) are the roots of the equation \( a^2x^2 + x + (1 - a^2) = 0 (a > 1) \) then \( \beta^2 = \)
(a) \( a^2 \)
(b) 1
(c) \( 1 - a^2 \)
(d) \( 1 + a^2 \)
Answer: (b) 1

Question. If \( \alpha, \beta \) are the roots of \( x^2 + px - q = 0 \) and \( \gamma, \delta \) are that of \( x^2 + px + r = 0 \) then \( (\alpha - \gamma)(\beta - \gamma)(\alpha - \delta)(\beta - \delta) = \)
(a) \( (q - r)^2 \)
(b) \( (q + r)^2 \)
(c) \( -(q + r)^2 \)
(d) \( -(q - r)^2 \)
Answer: (b) \( (q + r)^2 \)

Question. If \( \alpha, \beta \) are the roots of \( ax^2 + bx + c = 0 \) and \( S_n = \alpha^n + \beta^n \) then \( aS_{n+1} + bS_n + cS_{n-1} = \)
(a) 0
(b) 1
(c) 2
(d) 5
Answer: (a) 0

Question. \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 + px + p^3 = 0, (p \neq 0) \). If the point \( (\alpha, \beta) \) lies on the curve \( x = y^2 \) then the roots of the given equation are
(a) 4, -2
(b) 4, 2
(c) 1, -1
(d) 1, 1
Answer: (a) 4, -2

Question. If a and b are distinct positive real numbers such that \( a, a_1, a_2, a_3, a_4, a_5, b \) are in A.P; \( a, b_1, b_2, b_3, b_4, b_5, b \) are in GP; and \( a, c_1, c_2, c_3, c_4, c_5, b \) are in HP; Then the roots of \( a_3x^2 + b_3x + c_3 = 0 \) are
(a) real and distinct
(b) real and equal
(c) imaginary
(d) rational
Answer: (c) imaginary

Question. p, q, r and s are integers. If the A.M. of the roots of \( x^2 - px + q^2 = 0 \) and G.M. of the roots of \( x^2 - rx + s^2 = 0 \) are equal then
(a) q is an odd integer
(b) r is an even integer
(c) p is an even integer
(d) s is an odd integer
Answer: (c) p is an even integer

Question. If \( x^2 + x + 1 = 0 \) then the value of \( \left( x + \frac{1}{x} \right) + \left( x^2 + \frac{1}{x^2} \right) + \ldots + \left( x^{52} + \frac{1}{x^{52}} \right) \) equals
(a) 1
(b) 0
(c) -1
(d) -52
Answer: (c) -1

Question. The value of the continued fraction \( 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \dots \infty}}} \) is
(a) \( \frac{\sqrt{5} - 1}{2} \)
(b) \( \frac{\sqrt{5} + 1}{2} \)
(c) \( \frac{\sqrt{5} - 1}{4} \)
(d) \( \frac{\sqrt{5} + 1}{4} \)
Answer: (b) \( \frac{\sqrt{5} + 1}{2} \)

Question. Number of real roots of the equation \( \sqrt{x} + \sqrt{x - \sqrt{1 - x}} = 1 \) is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (b) 1

Question. The equation \( 2 \cos^2 \frac{x}{2} \sin^2 x = x^2 + \frac{1}{x^2}, 0 \leq x \leq \frac{\pi}{2} \) has
(a) One real solution
(b) No real solution
(c) More than one real solution
(d) two real solutions
Answer: (b) No real solution

Question. The quadratic equations \( x^2 - 6x + a = 0 \) and \( x^2 - cx + 6 = 0 \) have one root common. The other roots of the first and second equations are integers in the ratio \( 4:3 \). Then the common root is (AIEEE 2008)
(a) 3
(b) 2
(c) 1
(d) 4
Answer: (b) 2

Question. If the two equations \( x^2 - cx + d = 0 \) and \( x^2 - ax + b = 0 \) have a common root and the second equation has equal roots then
(a) \( b + d = ac \)
(b) \( 2(b + d) = ac \)
(c) \( b + d = 2ac \)
(d) \( (b + d)^2 = a + c \)
Answer: (b) \( 2(b + d) = ac \)

Question. If \( x^2 + 2ax + a < 0 \ \forall x \in [1, 2] \) then
(a) \( a \in \left(-\infty, -\frac{4}{5}\right) \)
(b) \( a \in (0, \infty) \)
(c) \( a \in \left(-\frac{4}{5}, 0\right) \)
(d) \( a \in \left(\frac{4}{5}, \infty\right) \)
Answer: (a) \( a \in \left(-\infty, -\frac{4}{5}\right) \)

Question. If \( f(x) \) is a quadratic expression which is positive for all real vaues of x and \( g(x) = f(x) + f'(x) + f''(x) \) then for any real value of x
(a) \( g(x) < 0 \)
(b) \( g(x) > 0 \)
(c) \( g(x) = 0 \)
(d) \( g(x) = 2 \)
Answer: (b) \( g(x) > 0 \)

Question. If the roots of the equation \( x^2 + 2ax + b = 0 \) are real and distinct and they differ by atmost 2m then 'b' lies in the interval
(a) \( (a^2 - m^2, a^2) \)
(b) \( [a^2 - m^2, a^2) \)
(c) \( (a^2, a^2 + m^2) \)
(d) \( (a^2 + m^2, a^2) \)
Answer: (b) \( [a^2 - m^2, a^2) \)

Question. The solution of the equation \( (3|x| - 3)^2 = |x| + 7 \) which belongs to the domain of \( \sqrt{x(x - 3)} \) are given by
(a) \( \pm \frac{1}{9}, \pm 2 \)
(b) \( \frac{1}{9}, -2 \)
(c) \( -\frac{1}{9}, -2 \)
(d) \( -\frac{1}{9}, 2 \)
Answer: (c) \( -\frac{1}{9}, -2 \)

Question. For \( a < 0 \), the roots of the equation \( x^2 - 2a|x - a| - 3a^2 = 0 \) are ......... IIT-1987
(a) \( (1 - \sqrt{2})a, (-1 + \sqrt{6})a \)
(b) \( (1 + \sqrt{2})a, (-1 + \sqrt{6})a \)
(c) \( (1 + \sqrt{2})a, (-1 - \sqrt{6})a \)
(d) \( (1 - \sqrt{2})a, (-1 - \sqrt{6})a \)
Answer: (a) \( (1 - \sqrt{2})a, (-1 + \sqrt{6})a \)