JEE Mathematics Solutions of Equations and Inequations MCQs Set 02

Type 1: Choose the most appropriate option (a, b, c or d).

Question. The number of solutions of \( |\cos x| = \sin x, 0 \leq x \leq 4\pi \), is
(a) 8
(b) 4
(c) 2
(d) None of the options
Answer: (b) 4

Question. The number of solution of \( \cos x = |1 + \sin x|, 0 \leq x \leq 3\pi \), is
(a) 3
(b) 2
(c) 4
(d) None of the options
Answer: (a) 3

Question. Let [x] = the greatest integer less than or equal to x and let \( f(x) = \sin x + \cos x \). Then the most general solutions of \( [f(x)] = [f(\pi/10)] \) are
(a) \( 2n\pi + \frac{\pi}{2}, n \in Z \)
(b) \( n\pi, n \in Z \)
(c) \( 2n\pi, n \in Z \)
(d) None of the options
Answer: (d) None of the options

Question. The most general solution of \( 2^{1 + |\cos x| + \cos^2 x + |\cos^3 x| + \dots \text{ to } \infty} = 4 \) are given by
(a) \( n\pi \pm \frac{\pi}{3}, n \in Z \)
(b) \( 2n\pi \pm \frac{\pi}{3}, n \in Z \)
(c) \( 2n\pi \pm \frac{2\pi}{3}, n \in Z \)
(d) None of the options
Answer: (a) \( n\pi \pm \frac{\pi}{3}, n \in Z \)

Question. If \( x \neq \frac{n\pi}{2} \) and \( (\cos x)^{\sin^2 x - 3\sin x + 2} = 1 \) then all solutions of \( x \) are given by
(a) \( 2n\pi + \frac{\pi}{2} \)
(b) \( (2n + 1)\pi - \frac{\pi}{2} \)
(c) \( n\pi + (-1)^n \frac{\pi}{2} \)
(d) None of the options
Answer: (d) None of the options

Question. The solution of the equation \( (\sin x + \cos x)^{1 + \sin 2x} = 2, -\pi \leq x < \pi \), is
(a) \( \frac{\pi}{2} \)
(b) \( \pi \)
(c) \( \frac{\pi}{4} \)
(d) None of the options
Answer: (c) \( \frac{\pi}{4} \)

Question. The most general solutions \( 2^{\sin x} + 2^{\cos x} = 2^{1 - 1/\sqrt{2}} \) are
(a) \( n\pi - \frac{\pi}{4} \)
(b) \( n\pi + \frac{\pi}{4} \)
(c) \( n\pi + (-1)^n \frac{\pi}{4} \)
(d) \( 2n\pi \pm \frac{\pi}{4} \)
Answer: (b) \( n\pi + \frac{\pi}{4} \)

Question. The number of solutions of \( 16^{\sin^2 x} + 16^{\cos^2 x} = 10, 0 \leq x \leq 2\pi \), is
(a) 8
(b) 6
(c) 4
(d) 2
Answer: (a) 8

Question. If \( 3\sin^2\theta + 2\sin^2\phi = 1 \) and \( 3\sin 2\theta = 2\sin 2\phi, 0 < \theta < \frac{\pi}{2} \) and \( 0 < \phi < \frac{\pi}{2} \), then the value of \( \theta + 2\phi \) is
(a) \( \frac{\pi}{2} \)
(b) \( \frac{\pi}{4} \)
(c) 0
(d) None of the options
Answer: (a) \( \frac{\pi}{2} \)

Question. The most general values of \( \theta \) satisfying the equation \( (1 + 2 \sin \theta)^2 + (\sqrt{3} \tan \theta - 1)^2 = 0 \) are given by
(a) \( n\pi \pm \frac{\pi}{6} \)
(b) \( n\pi + (-1)^n \frac{7\pi}{6} \)
(c) \( 2n\pi + \frac{7\pi}{6} \)
(d) \( 2n\pi + \frac{11\pi}{6} \)
Answer: (c) \( 2n\pi + \frac{7\pi}{6} \)

Question. If \( r > 0, -\pi \leq \theta \leq \pi \) and \( r, \theta \) satisfy \( r\sin \theta = 3 \) and \( r = 4(1 + \sin \theta) \) then the number of possible solutions of the pair \( (r, \theta) \) is
(a) 2
(b) 4
(c) 0
(d) infinite
Answer: (a) 2

Question. The number of solutions of the equation \( x^3 + x^2 + 4x + 2\sin x = 0 \) in \( 0 \leq x \leq 2\pi \) is
(a) zero
(b) one
(c) two
(d) four
Answer: (b) one

Question. The number of real solutions of \( \sin e^x \cdot \cos e^x = 2^{x-2} + 2^{-x-2} \) is
(a) zero
(b) one
(c) two
(d) infinite
Answer: (a) zero

Question. The least positive nonintegral of \( \sin \pi(x^2 + x) - \sin \pi x^2 = 0 \) is
(a) rational
(b) irrational of the form \( \sqrt{p} \)
(c) irrational of the form \( \frac{\sqrt{p} - 1}{4} \), where p is even integer
(d) irrational of the form \( \frac{\sqrt{p} + 1}{4} \), where p is an even integer
Answer: (c) irrational of the form \( \frac{\sqrt{p} - 1}{4} \), where p is even integer

Question. If the equation \( 2\cos x + \cos 2\lambda x = 3 \) has only one solution then \( \lambda \) is
(a) 1
(b) a rational number
(c) an irrational number
(d) None of the options
Answer: (c) an irrational number

Question. If \( 0 \leq x \leq 2\pi, 0 \leq y < 2\pi \) and \( \sin x + \sin y = 2 \) then the value of \( x + y \) is
(a) \( \pi \)
(b) \( \frac{\pi}{2} \)
(c) \( 3\pi \)
(d) None of the options
Answer: (a) \( \pi \)

Question. If \( -\pi < x \leq \pi, -\pi \leq y \leq \pi \) and \( \cos x + \cos y = 2 \) then the value \( \cos(x - y) \) is
(a) -1
(b) 0
(c) 1
(d) None of the options
Answer: (c) 1

Question. If \( 0 \leq x \leq 3\pi, 0 \leq y \leq 3\pi \) and \( \cos x \cdot \sin y = 1 \) then the possible number of values of the ordered pair \( (x, y) \) is
(a) 6
(b) 12
(c) 8
(d) 15
Answer: (a) 6

Question. If \( \theta \in [0, 5\pi] \) and \( r \in R \) such that \( 2\sin \theta = r^4 - 2r^2 + 3 \) then maximum number of values of the pair \( (r, \theta) \) is
(a) 8
(b) 10
(c) 6
(d) None of the options
Answer: (c) 6

Question. The number of values of \( x \) for which \( \sin 2x + \cos 4x = 2 \) is
(a) 0
(b) 1
(c) 2
(d) infinite
Answer: (a) 0

Question. If \( 2 \sin x + 1 \geq 0 \) and \( x \in [0, 2\pi] \) then the solution set for \( x \) is
(a) \( [0, \frac{7\pi}{6}] \)
(b) \( [0, \frac{7\pi}{6}] \cup [\frac{11\pi}{6}, 2\pi] \)
(c) \( [\frac{11\pi}{6}, 2\pi] \)
(d) None of the options
Answer: (b) \( [0, \frac{7\pi}{6}] \cup [\frac{11\pi}{6}, 2\pi] \)

Question. If \( 2\cos x < \sqrt{3} \) and \( x \in [-\pi, \pi] \) then the solution set for \( x \) is
(a) \( [-\pi, -\frac{\pi}{6}) \cup (\frac{\pi}{6}, \pi] \)
(b) \( (-\frac{\pi}{6}, \frac{\pi}{6}) \)
(c) \( [-\pi, -\frac{\pi}{6}] \cup [\frac{\pi}{6}, \pi] \)
(d) None of the options
Answer: (a) \( [-\pi, -\frac{\pi}{6}) \cup (\frac{\pi}{6}, \pi] \)

Question. If \( \cos x - \sin x \geq 1 \) and \( 0 \leq x \leq 2\pi \) then the solution set for \( x \) is
(a) \( [0, \frac{\pi}{4}] \cup [\frac{7\pi}{4}, 2\pi] \)
(b) \( [\frac{3\pi}{2}, \frac{7\pi}{4}] \cup \{0\} \)
(c) \( [\frac{3\pi}{2}, 2\pi] \cup \{0\} \)
(d) None of the options
Answer: (c) \( [\frac{3\pi}{2}, 2\pi] \cup \{0\} \)

Question. If \( |\tan x| \leq 1 \) and \( x \in [-\pi, \pi] \) then the solution set for \( x \) is
(a) \( [-\pi, -\frac{3\pi}{4}] \cup [-\frac{\pi}{4}, \frac{\pi}{4}] \cup [\frac{3\pi}{4}, \pi] \)
(b) \( [-\frac{3\pi}{4}, \frac{\pi}{4}] \cup [\frac{3\pi}{4}, \pi] \)
(c) \( [-\frac{\pi}{4}, \frac{\pi}{4}] \)
(d) None of the options
Answer: (a) \( [-\pi, -\frac{3\pi}{4}] \cup [-\frac{\pi}{4}, \frac{\pi}{4}] \cup [\frac{3\pi}{4}, \pi] \)

Question. If \( 4\sin^2x - 8 \sin x + 3 \leq 0, 0 \leq x \leq 2\pi \), then the solution set for \( x \) is
(a) \( [0, \frac{\pi}{6}] \)
(b) \( [0, \frac{5\pi}{6}] \)
(c) \( [\frac{5\pi}{6}, 2\pi] \)
(d) \( [\frac{\pi}{6}, \frac{5\pi}{6}] \)
Answer: (d) \( [\frac{\pi}{6}, \frac{5\pi}{6}] \)

Question. The set of values of \( x \) for which \( \sin x \cdot \cos^3 x > \cos x \cdot \sin^3 x, 0 \leq x \leq 2\pi \), is
(a) \( (0, \pi) \)
(b) \( (0, \frac{\pi}{4}) \)
(c) \( (\frac{\pi}{4}, \pi) \)
(d) None of the options
Answer: (b) \( (0, \frac{\pi}{4}) \)

Question. The number of values of \( x \in [0, 4\pi] \) satisfying \( | \sqrt{3} \cos x - \sin x | \geq 2 \) is
(a) 2
(b) 0
(c) 4
(d) 8
Answer: (c) 4

Type 2: Choose the correct options. One or more options may be correct.

Question. If \( 0 \leq x \leq 2\pi \) and \( | \cos x | \leq \sin x \) then
(a) the set of value of \( x \) is \( [\frac{\pi}{4}, \frac{\pi}{2}] \)
(b) the number of solutions that are integral of \( \frac{\pi}{4} \) is three
(c) the sum of the largest and the smallest solution is \( \frac{3\pi}{4} \)
(d) \( x \in [\frac{\pi}{4}, \frac{\pi}{2}] \cup [\frac{\pi}{2}, \frac{3\pi}{4}] \)
Answer: (b) the number of solutions that are integral of \( \frac{\pi}{4} \) is three, (d) \( x \in [\frac{\pi}{4}, \frac{\pi}{2}] \cup [\frac{\pi}{2}, \frac{3\pi}{4}] \)

Question. \( 3^{\sin 2x + 2\cos^2 x} + 3^{1 - \sin 2x + 2\sin^2 x} = 28 \) is satisfied by
(a) those values of \( x \) for which \( \tan x - 1 \)
(b) those values of \( x \) for which \( \tan x = -\frac{1}{2} \)
(c) those values of \( x \) for which \( \cos x = 0 \)
(d) those values of \( x \) for which \( \tan x = 1 \)
Answer: (a) those values of \( x \) for which \( \tan x - 1 \), (c) those values of \( x \) for which \( \cos x = 0 \)

Question. Let [x] = the greatest integer less than or equal to x. The equation \( \sin x = [1 + \sin x] + [1 - \cos x] \) has
(a) no solution in \( [-\frac{\pi}{2}, \frac{\pi}{2}] \)
(b) no solution in \( [\frac{\pi}{2}, \pi] \)
(c) no solution in \( [\pi, \frac{3\pi}{2}] \)
(d) none of the options solution for \( x \in R \)
Answer: (a) no solution in \( [-\frac{\pi}{2}, \frac{\pi}{2}] \), (b) no solution in \( [\frac{\pi}{2}, \pi] \), (c) no solution in \( [\pi, \frac{3\pi}{2}] \), (d) none of the options solution for \( x \in R \)

Question. If \( \sin \theta = a \) for exactly one values of \( \theta \in [0, \frac{7\pi}{3}] \) then the value of \( a \) is
(a) \( \frac{\sqrt{3}}{2} \)
(b) 1
(c) 0
(d) -1
Answer: (b) 1, (d) -1