JEE Mathematics Equation and Expression MCQs Set 02

Question. The solution set of the inequation \( \log_{1/3}(x^2 + x + 1) + 1 > 0 \) is
(a) (-∞, -2) \( \cup \) (1, +∞)
(b) [-1, 2]
(c) (-2, 1)
(d) (-∞, +∞)
Answer: (c) (-2, 1)

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

Question. If \( 3^{x/2} + 2^x > 25 \) then the solution set is
(a) R
(b) (2, +∞)
(c) (4, +∞)
(d) None of the options
Answer: (c) (4, +∞)

Question. If \( \sin^x \alpha + \cos^x \alpha \ge 1 \), \( 0 < \alpha < \frac{\pi}{2} \), then
(a) \( x \in [2, +∞) \)
(b) \( x \in (-∞, 2) \)
(c) \( x \in [-1, 1] \)
(d) None of the options
Answer: (b) \( x \in (-∞, 2) \)

Question. The solution set of \( x^2 + 2 \le 3x \le 2x^2 - 5 \) is
(a) \( \phi \)
(b) [1, 2]
(c) (-∞, -1] \( \cup \) [5/2, +∞)
(d) None of the options
Answer: (a) \( \phi \)

Question. The solution set of \( \frac{x^2 - 3x + 4}{x + 1} > 1 \), \( x \in R \), is
(a) (3, +∞)
(b) (-1, 1) \( \cup \) (3, +∞)
(c) [-1, 1] \( \cup \) [3, +∞)
(d) None of the options
Answer: (b) (-1, 1) \( \cup \) (3, +∞)

Question. The number of integral solutions of \( \frac{x + 2}{x^2 + 1} > \frac{1}{2} \) is
(a) 4
(b) 5
(c) 3
(d) None of the options
Answer: (c) 3

Question. If \( a, b, c \) are nonzero, unequal rational numbers then the roots of the equation \( abc^2x^2 + (3a^2 + b^2)cx - 6a^2 - ab + 2b^2 = 0 \) are
(a) rational
(b) imaginary
(c) irrational
(d) None of the options
Answer: (a) rational

Question. If \( l, m \) are real and \( l \neq m \) then the roots of the equation \( (l - m)x^2 - 5(l + m)x - 2(l - m) = 0 \) are
(a) real and equal
(b) nonreal complex
(c) real and unequal
(d) None of the options
Answer: (c) real and unequal

Question. If \( a, b, c, d \) are four consecutive terms of an increasing AP then the roots of the equation \( (x - a)(x - c) + 2(x - b)(x - d) = 0 \) are
(a) real and distinct
(b) nonreal complex
(c) real and equal
(d) integers
Answer: (a) real and distinct

Question. If \( a, b, c \) are three distinct positive real number then the number of real roots of \( ax^2 + 2b |x| - c = 0 \) is
(a) 4
(b) 2
(c) 0
(d) None of the options
Answer: (b) 2

Question. The equation \( x^2 - 6x + 8 + \lambda(x^2 - 4x + 3) = 0, \lambda \in R \), has
(a) real and unequal roots for all \( \lambda \)
(b) real roots for \( \lambda < 0 \) only
(c) real roots for \( \lambda > 0 \) only
(d) real and unequal roots for \( \lambda = 0 \) only
Answer: (a) real and unequal roots for all \( \lambda \)

Question. If \( \cos \theta, \sin \phi, \sin \theta \) are in GP then roots of \( x^2 + 2 \cot \phi \cdot x + 1 = 0 \) are always
(a) equal
(b) real
(c) imaginary
(d) greater than 1
Answer: (b) real

Question. The roots of \( ax^2 + bx + c = 0 \), where \( a \neq 0 \) and coefficients are real, are nonreal complex and \( a + c < b \). Then
(a) \( 4a + c > 2b \)
(b) \( 4a + c < 2b \)
(c) \( 4a + c = 2b \)
(d) None of the options
Answer: (b) \( 4a + c < 2b \)

Question. The equation \( (a + 2)x^2 + (a - 3)x = 2a - 1, a \neq -2 \) has roots rational for
(a) all rational values of except \( a = -2 \)
(b) all real values of a except \( a = -2 \)
(c) rational values of \( a > 1/2 \)
(d) None of the options
Answer: (a) all rational values of except \( a = -2 \)

Question. If \( a \cdot 3^{\tan x} + a \cdot 3^{-\tan x} - 2 = 0 \) has real solutions, \( x \neq \frac{\pi}{2}, 0 \le x \le \pi \), then the set of possible values of the parameter \( a \) is
(a) [-1, 1]
(b) [-1, 0)
(c) (0, 1]
(d) (0, +∞)
Answer: (c) (0, 1]

Question. If \( a > 1 \), roots of the equation \( (1 - a)x^2 + 3ax - 1 = 0 \) are
(a) one positive and one negative
(b) both negative
(c) both positive
(d) both nonreal complex
Answer: (a) one positive and one negative

Question. If \( a \in R, b \in R \) then the equation \( x^2 - abx - a^2 = 0 \) has
(a) one positive root and one negative root
(b) both roots positive
(c) both roots negative
(d) nonreal roots
Answer: (a) one positive root and one negative root

Question. If the roots of the equation \( x^2 - 2ax + a^2 + a - 3 = 0 \) are less than 3 then
(a) \( a < 2 \)
(b) \( 2 \le a \le 3 \)
(c) \( 3 < a \le 4 \)
(d) \( a > 4 \)
Answer: (a) \( a < 2 \)

Question. If \( \alpha, \beta \) are the roots of \( x^2 - 3x + a = 0, a \in R \) and \( \alpha < 1 < \beta \) then
(a) \( a \in (-∞, 2) \)
(b) \( a \in (-∞, 9/4] \)
(c) \( a \in (2, 9/4] \)
(d) None of the options
Answer: (a) \( a \in (-∞, 2) \)

Question. If \( \alpha, \beta \) be the roots of \( 4x^2 - 16x + \lambda = 0, \lambda \in R \) such that \( 1 < \alpha < 2 \) and \( 2 < \beta < 3 \) then the number of integral solutions of \( \lambda \) is
(a) 5
(b) 6
(c) 2
(d) 3
Answer: (d) 3

Question. The number of integer values of \( a \) for which \( x^2 - (a - 1)x + 3 = 0 \) has both roots positive and \( x^2 + 3x + 6 - a = 0 \) has both roots negative is
(a) 0
(b) 1
(c) 2
(d) infinite
Answer: (b) 1

Question. If \( X \) denotes the set of real numbers \( p \) for which the equation \( x^2 = p(x + p) \) has its roots greater than \( p \) then \( X \) is equal to
(a) \( (-2, -1/2) \)
(b) \( (-1/2, 1/4) \)
(c) null set \( \phi \)
(d) (-∞, 0)
Answer: (b) \( (-1/2, 1/4) \)

Question. If \( \cos^4 x + \sin^2 x - p = 0 \), \( p \in R \) has real solutions then
(a) \( p \le 1 \)
(b) \( 3/4 \le p \le 1 \)
(c) \( p \ge 3/4 \)
(d) None of the options
Answer: (b) \( 3/4 \le p \le 1 \)

Question. If one root of the equation \( (k^2 + 1)x^2 + 13x + 4k = 0 \) is reciprocal of the other then \( k \) has the value
(a) \( -2 + \sqrt{3} \)
(b) \( 2 - \sqrt{3} \)
(c) 1
(d) None of the options
Answer: (b) \( 2 - \sqrt{3} \)

Question. If the ratio of the roots of \( \lambda x^2 + \mu x + \nu = 0 \) is equal to the ratio of the roots of \( x^2 + x + 1 = 0 \) then \( \lambda, \mu, \nu \) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (b) GP

Question. \( p, q, r \) and \( s \) are integers. If the AM of the roots of \( x^2 - px + q^2 = 0 \) and GM 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 \( \alpha, \beta \) are roots of the equation \( (x - a)(x - b) = c, c \neq 0 \), then the roots of the equation \( (x - \alpha)(x - \beta) + c = 0 \) are
(a) \( a, c \)
(b) \( b, c \)
(c) \( a, b \)
(d) \( a + c, b + c \)
Answer: (c) \( a, b \)

Question. If the roots of \( 4x^2 + 5k = (5k + 1)x \) differ by unity then the negative value of \( k \) is
(a) -3
(b) -1/5
(c) -3/5
(d) None of the options
Answer: (b) -1/5

Question. The harmonic mean of the roots of the equation \( (5 + \sqrt{2})x^2 - (4 + \sqrt{5})x + 8 + 2\sqrt{5} = 0 \) is
(a) 2
(b) 4
(c) 6
(d) 8
Answer: (b) 4

Question. If the product of the roots of the equation \( x^2 - 5x + 4^{\log_2 \lambda} = 8 \) then \( \lambda \) is
(a) \( \pm 2\sqrt{2} \)
(b) \( 2\sqrt{2} \)
(c) 3
(d) None of the options
Answer: (b) \( 2\sqrt{2} \)

Question. If the roots of \( a_1x^2 + b_1x + c_1 = 0 \) are \( \alpha_1, \beta_1 \), and those of \( a_2x^2 + b_2x + c_2 = 0 \) are \( \alpha_2, \beta_2 \) such that \( \alpha_1 \alpha_2 = \beta_1 \beta_2 = 1 \) then
(a) \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)
(b) \( \frac{a_1}{c_2} = \frac{b_1}{b_2} = \frac{c_1}{a_2} \)
(c) \( a_1 a_2 = b_1 b_2 = c_1 c_2 \)
(d) None of the options
Answer: (b) \( \frac{a_1}{c_2} = \frac{b_1}{b_2} = \frac{c_1}{a_2} \)

Question. If \( \alpha, \beta \) are the roots of \( ax^2 + c = bx \) then the equation \( (a + cy)^2 = b^2 y \) in \( y \) has the roots
(a) \( \alpha^{-1}, \beta^{-1} \)
(b) \( \alpha^2, \beta^2 \)
(c) \( \alpha \beta^{-1}, \alpha^{-1} \beta \)
(d) \( \alpha^2, \beta^{-2} \)
Answer: (d) \( \alpha^2, \beta^{-2} \)

Question. If the roots of \( ax^2 - bx - c = 0 \) change by the same quantity then the expression in \( a, b, c \) that does not change is
(a) \( \frac{b^2 - 4ac}{a^2} \)
(b) \( \frac{b - 4c}{a} \)
(c) \( \frac{b^2 + 4ac}{a^2} \)
(d) None of the options
Answer: (c) \( \frac{b^2 + 4ac}{a^2} \)

Question. If \( \alpha, \beta \) are roots of \( x^2 - bx + c = 0 \) has equal integral roots then
(a) b and c are integers
(b) b and c are even integers
(c) b is an even integer and c is a perfect square of a positive integer
(d) None of the options
Answer: (a) b and c are integers
(c) b is an even integer and c is a perfect square of a positive integer

Question. Let \( A, G \) and \( H \) be the AM, GM and HM of two positive numbers \( a \) and \( b \). The quadratic equation whose roots are \( A \) and \( H \) is
(a) \( Ax^2 - (A^2 + G^2)x + AG^2 = 0 \)
(b) \( Ax^2 - (A^2 + H^2)x + AH^2 = 0 \)
(c) \( Hx^2 - (H^2 + G^2)x + HG^2 = 0 \)
(d) None of the options
Answer: (a) \( Ax^2 - (A^2 + G^2)x + AG^2 = 0 \)
(c) \( Hx^2 - (H^2 + G^2)x + HG^2 = 0 \)