Class 11 Mathematics Trigonometric Equations MCQs Set 04

SINGLE CORRECT CHOICE

Question. The number of distinct real roots of the equation \( \sin^3 x + \sin^2 x + \sin x - \sin x.\sin 2x - \sin 2x - 2\cos x = 0 \) belonging to the interval \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (b) 1

Question. The number of distinct real roots of the equation \( \cos^8 x - \sin^8 x = 1, 0 \leq x \leq 2\pi \) is
(a) 4
(b) 8
(c) 3
(d) 6
Answer: (c) 3

Question. The number of distinct real roots of the equation \( \sqrt{\sin x} - \frac{1}{\sqrt{\sin x}} = \cos x \) is (where \( 0 \leq x \leq 2\pi \))
(a) 1
(b) 2
(c) 3
(d) more than 3
Answer: (a) 1

Question. The number of distinct real roots of the equation \( \sin \pi x = x^2 - x + \frac{5}{4} \) is
(a) 0
(b) 1
(c) 2
(d) 4
Answer: (b) 1

Question. If \( \alpha, \beta \) are two distinct solutions lying between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \) of the equaiton \( 2 \tan x + \sec x = 2 \) then \( \tan \alpha + \tan \beta = \)
(a) 0
(b) 1
(c) \( \frac{4}{3} \)
(d) \( \frac{8}{3} \)
Answer: (d) \( \frac{8}{3} \)

Question. In the range of \( 0 \leq x \leq 5\pi \), then equation \( (3 + \cos x)^2 = 4 - 2\sin^8 x \) has
(a) exactly one solution
(b) exactly three solutions
(c) exactly five solution
(d) infinite solutions
Answer: (b) exactly three solutions

Question. The number of solutions of the pair equations \( 2 \sin^2 \theta - \cos 2\theta = 0 \), \( 2 \cos^2 \theta - 3\sin \theta = 0 \), in the interval \( [0, 2\pi] \) is [IIT-2007]
(a) zero
(b) one
(c) two
(d) four
Answer: (c) two

Question. If \( m \) and \( n \ (n > m) \) are positive integers, the number of solutions of the equation \( n |\sin x| = m |\cos x| \) in \( [0, 2\pi] \) is
(a) m
(b) n
(c) mn
(d) 4
Answer: (d) 4

Question. If \( [\sin x] + \left[ \sqrt{2} \cos x \right] = -3, x \in [0, 2\pi] \) (\([.]\) denotes the greatest integer function), then \( x \) belongs to
(a) \( \left[ \pi, \frac{5\pi}{4} \right) \)
(b) \( \left[ \pi, \frac{5\pi}{4} \right] \)
(c) \( \left( \frac{5\pi}{4}, 2\pi \right] \)
(d) \( \left[ \frac{5\pi}{4}, 2\pi \right] \)
Answer: (a) \( \left[ \pi, \frac{5\pi}{4} \right) \)

Question. If \( f(x) = \max \{ \tan x, \cot x \} \). The number of roots of the equation \( f(x) = \frac{1}{2 + \sqrt{3}} \) in \( (0, 2\pi) \) is
(a) 0
(b) 2
(c) 4
(d) infinite
Answer: (a) 0

Question. Values of x and y satisfying the equation \( \sin^7 y = \left| x^3 - x^2 - 9x + 9 \right| + \left| x^3 - x^2 - 4x + 4 \right| + \sec^2 2y + \cos^4 y \) are
(a) \( x = 1, y = n\pi, n \in I \)
(b) \( x = 1, y = 2n\pi + \frac{\pi}{2}, n \in I \)
(c) \( x = 1, y = 2n\pi, n \in I \)
(d) None of the options
Answer: (b) \( x = 1, y = 2n\pi + \frac{\pi}{2}, n \in I \)

Question. If \( [y] = [\sin x] \) and \( y = \cos x \) are two given equations, then the number of solutions is (where \( [.] \) denotes the greatest integer function)
(a) 2
(b) 3
(c) 4
(d) infinitely many solutions
Answer: (d) infinitely many solutions

Question. The number of points inside the curve \( x^2 + y^2 \leq 4 \) satisfying \( \tan^4 x + \cot^4 x + 1 = 3\sin^2 y \) is.
(a) 4
(b) 6
(c) 8
(d) 10
Answer: (a) 4

ONE OR MORE THAN ONE CORRECT ANSWERS

Question. Consider the equation \( \sqrt{2} \tan^2 x - \sqrt{10} \tan x + \sqrt{2} = 0 \); \( 0 < x < \frac{\pi}{2} \) then the true statements among the following are
(a) The number of distinct real roots of the equation is 2
(b) The sum of all the real roots of the eqauation is \( \frac{\pi}{2} \)
(c) The absolute value of the difference of the real roots of the equation is \( \frac{\pi}{4} \)
(d) The equation has no real roots
Answer: (a) The number of distinct real roots of the equation is 2, (b) The sum of all the real roots of the eqauation is \( \frac{\pi}{2} \)

Question. The statement \( (3\sin^4 x - 2\cos^6 x + y - 2\sin^6 x + 3\cos^4 x)^2 = 9 \) is true for
(a) \( x = \frac{5\pi}{13} \text{ and } y = 2 \)
(b) \( x = \frac{7\pi}{16} \text{ and } y = -4 \)
(c) \( x = \frac{\pi}{7} \text{ and } y = 1 \)
(d) \( z = \frac{\pi}{5} \text{ and } y = -1 \)
Answer: (a) \( x = \frac{5\pi}{13} \text{ and } y = 2 \), (b) \( x = \frac{7\pi}{16} \text{ and } y = -4 \)

Question. The solution set of \( |\sin x| \leq |\cos 2x| \) contains
(a) \( \bigcup_{n \in I} \left[ n\pi - \frac{\pi}{6}, n\pi + \frac{\pi}{6} \right] \)
(b) \( \bigcup_{n \in I} \left\{ n\pi + \frac{\pi}{2} \right\} \)
(c) \( \bigcup_{n \in I} \left[ n\pi - \frac{\pi}{8}, n\pi + \frac{\pi}{8} \right] \)
(d) \( \bigcup_{n \in I} \left[ n\pi - \frac{\pi}{4}, n\pi + \frac{\pi}{4} \right] \)
Answer: (a) \( \bigcup_{n \in I} \left[ n\pi - \frac{\pi}{6}, n\pi + \frac{\pi}{6} \right] \), (b) \( \bigcup_{n \in I} \left\{ n\pi + \frac{\pi}{2} \right\} \), (c) \( \bigcup_{n \in I} \left[ n\pi - \frac{\pi}{8}, n\pi + \frac{\pi}{8} \right] \)

Question. If \( \sin \theta = \cos \phi \), then the possible values of \( \frac{1}{\pi} \left( \theta \pm \phi - \frac{\pi}{2} \right) \) are [IIT 2008]
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (a) 0, (c) 2

Question. For the smallest positive values of 'x' and 'y' the equation \( 2 (\sin x + \sin y) - 2\cos(x - y) = 3 \) has a solution then which of the following is/are true
(a) \( \sin \left( \frac{x+y}{2} \right) = 1 \)
(b) \( \cos \left( \frac{x-y}{2} \right) = \frac{1}{2} \)
(c) number of ordered pairs \( (x, y) \) is 2
(d) number of ordered pairs \( (x, y) \) is 3
Answer: (a) \( \sin \left( \frac{x+y}{2} \right) = 1 \), (b) \( \cos \left( \frac{x-y}{2} \right) = \frac{1}{2} \), (c) number of ordered pairs \( (x, y) \) is 2

Question. Which of the following inequalities hold true in any \( \Delta ABC \)?
(a) \( \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \leq \frac{1}{8} \)
(b) \( \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2} \leq \frac{3\sqrt{3}}{8} \)
(c) \( \sin^2 \frac{A}{2} + \sin^2 \frac{B}{2} + \sin^2 \frac{C}{2} < \frac{3}{4} \)
(d) \( \cos^2 \frac{A}{2} + \cos^2 \frac{B}{2} + \cos^2 \frac{C}{2} \leq \frac{9}{4} \)
Answer: (a) \( \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \leq \frac{1}{8} \), (b) \( \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2} \leq \frac{3\sqrt{3}}{8} \), (d) \( \cos^2 \frac{A}{2} + \cos^2 \frac{B}{2} + \cos^2 \frac{C}{2} \leq \frac{9}{4} \)

Question. If \( \frac{\cos(\alpha - 3\theta)}{\cos^3 \theta} = \frac{\sin(\alpha - 3\theta)}{\sin^3 \theta} = m \), then
(a) \( \cos 2\alpha = \frac{2m^4 - 9m^2 + 8}{m^2} \)
(b) \( \cos \alpha = \frac{2 - m^2}{m} \)
(c) \( \cos 2\alpha = \frac{2m^4 + 9m^2 + 8}{m^2} \)
(d) \( \cos \alpha = \frac{2 + m^2}{m} \)
Answer: (a) \( \cos 2\alpha = \frac{2m^4 - 9m^2 + 8}{m^2} \), (b) \( \cos \alpha = \frac{2 - m^2}{m} \)

Question. The equation \( \sin x = [1 + \sin x] + [1 - \cos x] \) has (where [x] is the greatest integer less than or equal to x )
(a) no solution in \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \)
(b) no solution in \( \left[ \frac{\pi}{2}, \pi \right] \)
(c) no solution in \( \left[ \pi, \frac{3\pi}{2} \right] \)
(d) no solution for \( x \in R \)
Answer: (a) no solution in \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \), (b) no solution in \( \left[ \frac{\pi}{2}, \pi \right] \), (c) no solution in \( \left[ \pi, \frac{3\pi}{2} \right] \), (d) no solution for \( x \in R \)

Question. The solution of the equation \( 9\cos^2 x + \cos^2 2x + 1 = 6\cos x \cos 2x + 6\cos x - 2\cos 2x \) is/are
(a) \( x = n\pi + \frac{\pi}{2}, n \in I \)
(b) \( x = n\pi + \cos^{-1} \left( \sqrt{\frac{2}{3}} \right), n \in I \)
(c) \( x = n\pi - \cos^{-1} \left( \sqrt{\frac{2}{3}} \right), n \in I \)
(d) None of the options
Answer: (a) \( x = n\pi + \frac{\pi}{2}, n \in I \), (b) \( x = n\pi + \cos^{-1} \left( \sqrt{\frac{2}{3}} \right), n \in I \), (c) \( x = n\pi - \cos^{-1} \left( \sqrt{\frac{2}{3}} \right), n \in I \)

Question. \( 2\sin^2 \left( \frac{\pi}{2} \cos^2 x \right) = 1 - \cos (\pi \sin 2x) \), If
(a) \( x = (2n + 1)\frac{\pi}{2}, n \in I \)
(b) \( \tan x = \frac{1}{2}, n \in I \)
(c) \( \tan x = -\frac{1}{2}, n \in I \)
(d) \( x = \frac{n\pi}{2}, n \in I \)
Answer: (a) \( x = (2n + 1)\frac{\pi}{2}, n \in I \), (b) \( \tan x = \frac{1}{2}, n \in I \), (c) \( \tan x = -\frac{1}{2}, n \in I \)

Question. If \( \sin \left( \frac{6}{5} x \right) = 0 \) and \( \cos \left( \frac{x}{5} \right) = 0 \) then
(a) \( x = (n - 5)\pi \)
(b) \( x = 6(n - 1)\pi \)
(c) \( x = 5\left( n - \frac{1}{2} \right)\pi \)
(d) \( x = 5\left( n + \frac{1}{2} \right)\pi \)
Answer: (c) \( x = 5\left( n - \frac{1}{2} \right)\pi \), (d) \( x = 5\left( n + \frac{1}{2} \right)\pi \)

COMPREHENSION PASSAGES

Passage:
A real number \( \theta \) is a root of the equation \( 7\cos^2 x + 4\cos x - 1 = 0 \).
If \( p\cos^2 2\theta + q\cos 2\theta + r = 0 \) where p,q,r are real constants then

Question. p can be
(a) 16
(b) 49
(c) 64
(d) 81
Answer: (b) 49

Question. q can be
(a) 49
(b) 54
(c) 38
(d) 20
Answer: (c) 38

Question. r can be
(a) -7
(b) -16
(c) 14
(d) -5
Answer: (a) -7

Passage:
If curve of \( y = f(x) \) and \( y = g(x) \) intersects at n different points \( x = x_1, x_2, x_3 \dots \) then equation \( f(x) = g(x) \) is said to have n solutions

Question. Number of solutions of \( |\cos x| = 2[x] \) is (where [x] is integral part of x)
(a) 0
(b) 1
(c) 2
(d) infinite
Answer: (a) 0

Question. The number of solutions of \( \sin \pi x = |\log_e |x|| \) is
(a) 0
(b) 6
(c) 4
(d) 8
Answer: (b) 6

Question. Number of solutions of the equation \( \frac{\sin^5 x - \cos^5 x}{\cos x - \sin x} = \frac{1}{\sin x \cos x} \) (\( \sin x \neq \cos x \)) is
(a) 0
(b) 1
(c) 2
(d) infinite
Answer: (a) 0

Passage-3:
Consider the system of equations
\( \sin x \cos 2y = (a^2 - 1)^2 + 1, \cos x \sin 2y = a + 1 \).

Question. The number of values of a for which the system has a solution is
(a) 1
(b) 2
(c) 3
(d) Infinite
Answer: (a) 1

Question. The number of values of \( x \in [0, 2\pi] \) when the system has solution for permissible values of 'a' is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Passage-4:
Consider the cubic equation
\( x^3 - (1 + \cos \theta + \sin \theta)x^2 + (\cos \theta \sin \theta + \cos \theta + \sin \theta)x - \sin \theta \cos \theta = 0 \)
whose roots are \( x_1, x_2, \) and \( x_3 \).

Question. The value of \( x_1^2 + x_2^2 + x_3^2 \) equals
(a) 1
(b) 2
(c) \( 2\cos \theta \)
(d) \( \sin \theta (\sin \theta + \cos \theta) \)
Answer: (b) 2

Question. Number of values of \( \theta \) in \( [0, 2\pi] \) for which at least two roots are equal
(a) 3
(b) 4
(c) 5
(d) 6
Answer: (c) 5

Question. Greatest possible difference between two of the roots if \( \theta \in [0, 2\pi] \) is
(a) 2
(b) 1
(c) \( \sqrt{2} \)
(d) \( 2\sqrt{2} \)
Answer: (a) 2

ASSERTION AND REASONING

(a) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.
(b) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
(c) Statement-1 is true, statement-2 is false.
(d) Statement-1 is false, statement-2 is true.

Question. Statement 1 : The equations \( \cos 2\theta = 1 \) and \( \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = 1 \) do not have the same solution set.
Statement 2 : The functions \( f(\theta) = \cos 2\theta \) and \( g(\theta) = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} \) are not equal.
(a) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.
(b) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
(c) Statement-1 is true, statement-2 is false.
(d) Statement-1 is false, statement-2 is true.
Answer: (d) Statement-1 is false, statement-2 is true.

Question. Statement 1: The equation \( \tan x = x \) has infinitely many roots.
Statement 2: The straight line \( y = x \) intersects the graph of \( y = \tan x \) at infinitely many points.
(a) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.
(b) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
(c) Statement-1 is true, statement-2 is false.
(d) Statement-1 is false, statement-2 is true.
Answer: (a) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.

Question. Consider two functions \( f(x) = \frac{1 + e^{\cot^2 x}}{1 - \cos 2x} \) and \( g(x) = \sqrt{2|\sin x| - 1} + \frac{1 - \cos 2x}{1 + \sin^4 x} \)
Statement-1:The solutions of the equation f (x) = g (x) is given by
\( x = (2n + 1)\frac{\pi}{2} \ \forall \ n \in I \).
Statement-2:If f (x) ≥ k and g (x) ≤ k (where \( k \in R \)) then solutions of the equation f (x) = g (x) is the solution corresponding to the equation f (x) = k.
(a) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.
(b) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
(c) Statement-1 is true, statement-2 is false.
(d) Statement-1 is false, statement-2 is true.
Answer: (c) Statement-1 is true, statement-2 is false.