JEE Mathematics Solutions of Equations and Inequations MCQs Set 01

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

Question. The number of distinct solutions of \( \sin 5\theta \cdot \cos 3\theta = \sin 9\theta \cdot \cos 7\theta \) in \( [0, \pi/2] \) is
(a) 4
(b) 5
(c) 8
(d) 9
Answer: (d) 9

Question. The number of solutions of \( \sin^2 \theta + 3\cos \theta = \sqrt{3}, 0 \leq \theta \leq 3\pi \), is
(a) 4
(b) 2
(c) 0
(d) None of the options
Answer: (d) None of the options

Question. The number of distinct solutions of \( \sec \theta + \tan \theta = \sqrt{3}, 0 \leq \theta \leq 3\pi \), is
(a) 3
(b) 5
(c) 4
(d) 0
Answer: (d) 0

Question. The number of solution of the equations \( 5\sec \theta - 13 = 12\tan \theta \) in \( [0, 2\pi] \) is
(a) 2
(b) 1
(c) 4
(d) 0
Answer: (a) 2

Question. The equations \( p\cos x - q\sin x = r \) admits of a solution for \( x \) only if
(a) \( r < \max \{p, q\} \)
(b) \( -\sqrt{p^2 + q^2} < r < \sqrt{p^2 + q^2} \)
(c) \( r^2 = p^2 + q^2 \)
(d) None of the options
Answer: (d) None of the options

Question. The equations \( k\cos x - 2\sin x = k + 1 \) is solvable only if \( k \) belongs to the interval
(a) \( [4, +\infty) \)
(b) \( [-4, 4] \)
(c) \( (-\infty, 4] \)
(d) None of the options
Answer: (c) \( (-\infty, 4] \)

Question. The number of solutions of \( \cos \theta + \sqrt{3}\sin \theta = 5, 0 \leq \theta \leq 5\pi \), is
(a) 4
(b) 0
(c) 5
(d) None of the options
Answer: (b) 0

Question. If the equation \( a_1 + a_2\cos 2x + a_3\sin^2 x = 1 \) is satisfied by every real every of \( x \) then the number of possible values of the triplet \( (a_1, a_2, a_3) \) is
(a) 0
(b) 1
(c) 3
(d) infinite
Answer: (d) infinite

Question. The number of solutions of the equations \( \tan x + \sec x = 2\cos x \) lying in the interval \( [0, 2\pi] \) is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (c) 2

Question. The number of values of \( x \) in \( [0, 5\pi] \) satisfying the equation \( 3\cos 2x - 10\cos x + 7 = 0 \) is
(a) 5
(b) 6
(c) 8
(d) 10
Answer: (c) 8

Question. The smallest positive integral value of \( p \) for which the equation \( \cos(p\sin x) = \sin(p\cos x) \) in \( x \) has solution in \( [0, 2\pi] \) is
(a) 2
(b) 1
(c) 3
(d) None of the options
Answer: (a) 2

Question. If \( \sin^2\theta - 2\sin \theta - 1 = 0 \) is to be satisfied for exactly 4 distinct values of \( \theta \in [0, n\pi], n \in N \), then the least value of \( n \) is
(a) 2
(b) 6
(c) 4
(d) 1
Answer: (c) 4

Question. If \( 2\tan^2 x - 5\sec x \) is equal to 1 for exactly 7 distinct values of \( x \in \left[ 0, \frac{n\pi}{2} \right], n \in N \), then the greatest value of \( n \) is
(a) 6
(b) 12
(c) 13
(d) 15
Answer: (d) 15

Question. The number of values of \( x \in [0, 2\pi] \) that satisfy \( \cot x - \csc x = 2\sin x \) is
(a) 3
(b) 2
(c) 1
(d) 0
Answer: (d) 0

Question. If \( \sin \alpha, 1 \) and \( \cos 2\alpha \) are in GP then \( \alpha \) is equal to
(a) \( n\pi + (-1)^n \frac{\pi}{2}, n \in Z \)
(b) \( n\pi + (-1)^{n-1} \frac{\pi}{2}, n \in Z \)
(c) \( 2n\pi, n \in Z \)
(d) None of the options
Answer: (b) \( n\pi + (-1)^{n-1} \frac{\pi}{2}, n \in Z \)

Question. If \( \frac{1}{6}\sin \theta, \cos \theta \) and \( \tan \theta \) are in GP the general value if \( \theta \) is
(a) \( 2n\pi \pm \frac{\pi}{3}, n \in Z \)
(b) \( 2n\pi \pm \frac{\pi}{6}, n \in Z \)
(c) \( n\pi + (-1)^n \frac{\pi}{3}, n \in Z \)
(d) \( n\pi + \frac{\pi}{3}, n \in Z \)
Answer: (a) \( 2n\pi \pm \frac{\pi}{3}, n \in Z \)

Question. The general values of \( x \) for which \( \cos 2x, \frac{1}{2} \) and \( \sin 2x \) are in AP are given by
(a) \( n\pi, n\pi + \frac{\pi}{2} \)
(b) \( n\pi, n\pi + \frac{\pi}{4} \)
(c) \( n\pi + \frac{\pi}{4} \)
(d) \( n\pi \)
Answer: (b) \( n\pi, n\pi + \frac{\pi}{4} \)

Question. The most general values of \( \theta \) satisfying \( \tan \theta + \tan \left( \frac{3\pi}{4} + \theta \right) = 2 \) are given by
(a) \( n\pi \pm \frac{\pi}{3}, n \in Z \)
(b) \( 2n\pi + \frac{\pi}{3}, n \in Z \)
(c) \( 2n\pi \pm \frac{\pi}{3}, n \in Z \)
(d) \( n\pi + (-1)^n \frac{\pi}{3}, n \in Z \)
Answer: (a) \( n\pi \pm \frac{\pi}{3}, n \in Z \)

Question. The most general solutions of the equation \( \sec x = - (\sqrt{2} - 1) \tan x \) are given by
(a) \( n\pi + \frac{\pi}{8} \)
(b) \( 2n\pi, 2n\pi - \frac{\pi}{4} \)
(c) \( 2n\pi \)
(d) None of the options
Answer: (b) \( 2n\pi, 2n\pi - \frac{\pi}{4} \)

Question. The most general solutions of the equation \( \sec^2 x = \sqrt{2} (1 - \tan^2 x) \) are given by
(a) \( n\pi + \frac{\pi}{8} \)
(b) \( n\pi \pm \frac{\pi}{8} \)
(c) \( n\pi \pm \frac{\pi}{8} \)
(d) None of the options
Answer: (c) \( n\pi \pm \frac{\pi}{8} \)

Question. The most general values of \( x \) for which \( \sin x + \cos x = \min \{1, a^2 - 4a + 6\} \) are given by
(a) \( 2n\pi \)
(b) \( 2n\pi - \frac{\pi}{2} \)
(c) \( n\pi + (-1)^n \frac{\pi}{4} - \frac{\pi}{4} \)
(d) None of the options
Answer: (c) \( n\pi + (-1)^n \frac{\pi}{4} - \frac{\pi}{4} \)

Question. If \( \max \{5\sin \theta + 3\sin(\theta - \alpha)\} = 7 \) then the set of possible values of \( \alpha \) is \( \theta \in R \)
(a) \( \{x | x = 2n\pi \pm \frac{\pi}{3}; n \in Z\} \)
(b) \( \{x | x = 2n\pi \pm \frac{2\pi}{3}; n \in Z\} \)
(c) \( \left[ \frac{\pi}{3}, \frac{2\pi}{3} \right] \)
(d) None of the options
Answer: (a) \( \{x | x = 2n\pi \pm \frac{\pi}{3}; n \in Z\} \)

Question. Let \( \alpha, \beta \) be any two positive values of \( x \) for which \( 2\cos x, |\cos x| \) and \( 1-3\cos^2x \) are in GP. The minimum values of \( |\alpha - \beta| \) is
(a) \( \frac{\pi}{3} \)
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{2} \)
(d) None of the options
Answer: (d) None of the options

Question. The sum of all the solution of the equation \( \cos x \cdot \cos \left( \frac{\pi}{3} + x \right) \cdot \cos \left( \frac{\pi}{3} - x \right) = \frac{1}{4}, x \in [0, 6\pi] \) is
(a) \( 15\pi \)
(b) \( 30\pi \)
(c) \( \frac{110\pi}{3} \)
(d) None of the options
Answer: (b) \( 30\pi \)

Question. The values of \( x \in [-2\pi, 2\pi] \) such that \( \frac{\sin x + i\cos x}{1 + i}, i = \sqrt{-1} \), is purely imaginary, are given by
(a) \( n\pi - \frac{\pi}{4} \)
(b) \( n\pi + \frac{\pi}{4} \)
(c) \( n\pi \)
(d) None of the options
Answer: (a) \( n\pi - \frac{\pi}{4} \)

Question. If \( x \in \left[ -\frac{5\pi}{2}, \frac{5\pi}{2} \right] \), the greatest positive solution of \( 1 + \sin^4 x = \cos^2 3x \) is
(a) \( \pi \)
(b) \( 2\pi \)
(c) \( \frac{5\pi}{2} \)
(d) None of the options
Answer: (b) \( 2\pi \)

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

Question. If \( \alpha \in [-2\pi, 2\pi] \) and \( \cos \frac{\alpha}{2} + \sin \frac{\alpha}{2} = \sqrt{2}(\cos 36^\circ - \sin 18^\circ) \) then a value of \( \alpha \) is
(a) \( \frac{7\pi}{6} \)
(b) \( \frac{\pi}{6} \)
(c) \( -\frac{5\pi}{6} \)
(d) \( -\frac{\pi}{6} \)
Answer: (a) \( \frac{7\pi}{6} \), (d) \( -\frac{\pi}{6} \)

Question. If \( \cos x = \sqrt{1 - \sin 2x}, 0 \leq x \leq \pi \), then a value of \( x \) is
(a) \( \pi \)
(b) 0
(c) \( \tan^{-1} 2 \)
(d) None of the options
Answer: (b) 0, (c) \( \tan^{-1} 2 \)

Question. If \( \sin 3\theta = \cos 2\theta \) then \( \theta \) is equal to
(a) \( (4n + 1) \frac{\pi}{2}, (4n + 1) \frac{\pi}{10} \); when \( n \) is an even integer only
(b) \( (4n + 1) \frac{\pi}{2}, (4n + 1) \frac{\pi}{10} \); when \( n \) is an any integer
(c) \( (4n + 1) \frac{\pi}{2}, (4n + 1) \frac{\pi}{10} \); when \( n \) is an odd integer only
(d) None of the options
Answer: (b) \( (4n + 1) \frac{\pi}{2}, (4n + 1) \frac{\pi}{10} \); when \( n \) is an any integer

Question. \( \sin \theta + \sqrt{3} \cos \theta = 6x - x^2 - 11, 0 \leq \theta \leq 4\pi, x \in R \), holds for
(a) no value of \( x \) and \( \theta \)
(b) one value of \( x \) and two values of \( \theta \)
(c) two values of \( x \) and two values of \( \theta \)
(d) two pairs of values of \( (x, \theta) \)
Answer: (b) one value of \( x \) and two values of \( \theta \), (d) two pairs of values of \( (x, \theta) \)