Class 11 Mathematics Trigonometric Equations MCQs Set 06

SINGLE ANSWER QUESTIONS

Question. If \( x = \sec \theta + \tan \theta, y = \operatorname{cosec} \theta - \cot \theta \) then \( y = \)
(a) \( \frac{1+x}{1-x} \)
(b) \( \frac{1-x}{1+x} \)
(c) \( \frac{x+1}{x-1} \)
(d) \( \frac{x-1}{x+1} \)
Answer: (d) \( \frac{x-1}{x+1} \)

Question. \( 0 \leq a \leq 3, 0 \leq b \leq 3 \) and the equation \( x^2 + 4 + 3\cos(ax+b) = 2x \) has at least one solution then the value of (a+b)
(a) \( \frac{\pi}{2} \)
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{3} \)
(d) \( \pi \)
Answer: (d) \( \pi \)

Question. If \( \frac{\sin^4 \theta}{a} + \frac{\cos^4 \theta}{b} = \frac{1}{a+b} \) then \( \frac{\sin^8 \theta}{a^3} + \frac{\cos^8 \theta}{b^3} = \)
(a) \( \frac{1}{(a+b)^3} \)
(b) \( \frac{1}{(a+b)^2} \)
(c) \( a+b \)
(d) \( \frac{1}{a+b} \)
Answer: (a) \( \frac{1}{(a+b)^3} \)

Question. If \( \sin^3 x \sin 3x = \sum_{m=0}^6 c_m \cos^m x \), where \( c_0, c_1, \dots, c_6 \) are constants, then
(a) \( c_0 + c_2 + c_4 + c_6 = 0 \)
(b) \( c_1 + c_3 + c_5 = 6 \)
(c) \( 2c_2 + 3c_6 = 0 \)
(d) \( c_4 + 2c_6 = 0 \)
Answer: (a) \( c_0 + c_2 + c_4 + c_6 = 0 \)
 

Question. The minimum value of \( 2\cos \theta + \frac{1}{\sin \theta} + \sqrt{2}\tan \theta \) in \( \left(0, \frac{\pi}{2}\right) \) is
(a) \( 2+\sqrt{2} \)
(b) \( 3\sqrt{2} \)
(c) \( 2\sqrt{3} \)
(d) \( 3+\sqrt{2} \)
Answer: (b) \( 3\sqrt{2} \)

Question. If \( \tan(\pi \cos \theta) = \cot(\pi \sin \theta), 0 < \theta < \frac{3\pi}{4} \) then \( \sin\left(\theta + \frac{\pi}{4}\right) = \)
(a) \( \frac{1}{\sqrt{2}} \)
(b) \( -\frac{1}{\sqrt{2}} \)
(c) \( \frac{1}{2\sqrt{2}} \)
(d) \( -\frac{1}{2\sqrt{2}} \)
Answer: (c) \( \frac{1}{2\sqrt{2}} \)

Question. If \( \tan \alpha = \frac{1}{\sqrt{x(x^2+x+1)}}, \tan \beta = \frac{\sqrt{x}}{\sqrt{x^2+x+1}} \) and \( \tan \gamma = \sqrt{x^{-3}+x^{-2}+x^{-1}} \) then \( \alpha + \beta = \)
(a) \( \gamma \)
(b) \( 2\gamma \)
(c) \( -\gamma \)
(d) \( \gamma/2 \)
Answer: (a) \( \gamma \)

Question. \( (\sin^8 75^\circ - \cos^8 75^\circ) = \)
(a) 1
(b) \( \frac{3\sqrt{3}}{8} \)
(c) \( \frac{3\sqrt{3}}{16} \)
(d) \( \frac{7\sqrt{3}}{16} \)
Answer: (d) \( \frac{7\sqrt{3}}{16} \)

Question. \( \sin 20^\circ(4 + \sec 20^\circ) = \)
(a) \( \frac{1}{2} \)
(b) \( \sqrt{2} \)
(c) \( \sqrt{3} \)
(d) 1
Answer: (c) \( \sqrt{3} \)

Question. \( 3\tan^6 10^\circ - 27\tan^4 10^\circ + 33\tan^2 10^\circ = \)
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (b) 1

Question. If \( 4n\alpha = \pi \), then the numerical value of \( \tan \alpha \tan 2\alpha \tan 3\alpha \dots \tan(2n-1)\alpha \) is equal to
(a) -1
(b) 0
(c) 1
(d) 2
Answer: (c) 1

Question. If \( \sec^4 \theta + \sec^2 \theta = 10 + \tan^4 \theta + \tan^2 \theta \) then \( \sin^2 \theta = \)
(a) \( \frac{2}{3} \)
(b) \( \frac{3}{4} \)
(c) \( \frac{4}{5} \)
(d) \( \frac{5}{6} \)
Answer: (c) \( \frac{4}{5} \)

Question. In \( \Delta ABC \), \( A = \frac{\pi}{4} \) and \( \tan B \tan C = P \) then all possible values of P is
(a) \( (-\infty, 3-2\sqrt{2}) \cup (3+2\sqrt{2}, \infty) \)
(b) \( (-\infty, 3-2\sqrt{2}] \cup [3+2\sqrt{2}, \infty) \)
(c) \( (3-2\sqrt{2}, 3+2\sqrt{2}) \)
(d) \( [3-2\sqrt{2}, 3+2\sqrt{2}] \)
Answer: (b) \( (-\infty, 3-2\sqrt{2}] \cup [3+2\sqrt{2}, \infty) \)

Question. \( \sin \frac{2\pi}{7} + \sin \frac{4\pi}{7} + \sin \frac{8\pi}{7} = \)
(a) \( \sqrt{7}/2 \)
(b) \( 7/2 \)
(c) \( -\sqrt{7}/2 \)
(d) \( -7/2 \)
Answer: (a) \( \sqrt{7}/2 \)

Question. \( \sin \frac{\pi}{7} \sin \frac{2\pi}{7} \sin \frac{4\pi}{7} = \)
(a) \( \frac{\sqrt{7}}{8} \)
(b) \( \frac{\sqrt{7}}{4} \)
(c) \( \frac{\sqrt{7}}{2} \)
(d) \( \sqrt{7} \)
Answer: (a) \( \frac{\sqrt{7}}{8} \)

Question. If \( \tan\left(x + \frac{\pi}{4}\right) = a \) then \( \sec^2 x = \)
(a) \( 1 + a^2 \)
(b) \( \frac{4a}{(a+1)^2} \)
(c) \( \frac{2(a^2+1)}{(a+1)^2} \)
(d) \( \left(\frac{a-1}{a+1}\right)^2 \)
Answer: (c) \( \frac{2(a^2+1)}{(a+1)^2} \)

Question. If \( \alpha, \beta, \gamma \) do not differ by a multiple of \( \pi \) and if \( \frac{\cos(\alpha+\theta)}{\sin(\beta+\gamma)} = \frac{\cos(\beta+\theta)}{\sin(\gamma+\alpha)} = \frac{\cos(\gamma+\theta)}{\sin(\alpha+\beta)} = k \). Then k equals
(a) \( \pm 2 \)
(b) \( \pm \frac{1}{2} \)
(c) 0
(d) \( \pm 1 \)
Answer: (d) \( \pm 1 \)

Question. If A, B, C are in A.P and \( B = \frac{\pi}{4} \) then \( \tan A \tan B \tan C = \)
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

Question. If \( \theta = \frac{\pi}{2^n+1} \) then \( \cos \theta \cos 2\theta \cos 2^2\theta \dots \cos 2^{n-1}\theta \) is equal to
(a) \( \frac{1}{2^n} \)
(b) \( \cos \theta \)
(c) 2
(d) \( 2^n \)
Answer: (a) \( \frac{1}{2^n} \)

Question. If ABC is a triangle and \( \tan \frac{A}{2}, \tan \frac{B}{2}, \tan \frac{C}{2} \) are in H.P then the minimum value of \( \cot \frac{B}{2} \) is equal to
(a) \( -\sqrt{3} \)
(b) \( \sqrt{3} \)
(c) 2
(d) -2
Answer: (b) \( \sqrt{3} \)

Question. \( \sum_{r=1}^{10} \cos^3 \frac{r\pi}{3} = \)
(a) \( -\frac{1}{8} \)
(b) \( -\frac{7}{8} \)
(c) \( -\frac{9}{8} \)
(d) \( \frac{1}{8} \)
Answer: (c) \( -\frac{9}{8} \)

Question. If \( \sin \frac{\pi}{18} \sin \frac{5\pi}{18} \sin \frac{7\pi}{18} = \)
(a) 8
(b) \( \frac{1}{8} \)
(c) 1/7
(d) 6
Answer: (b) \( \frac{1}{8} \)

Question. The value of \( \cos y \cos(\pi/2 - x) - \cos(\pi/2 - y)\cos x + \sin y \cos(\pi/2 - x) + \cos x\sin(\pi/2 - y) \) is zero if
(a) \( x = 0 \)
(b) \( y = 0 \)
(c) \( x = y \)
(d) \( x = n\pi - \pi/4 + y (n \in I) \)
Answer: (d) \( x = n\pi - \pi/4 + y (n \in I) \)

Question. If \( \tan \theta_1, \tan \theta_2, \tan \theta_3 \) and \( \tan \theta_4 \) are the roots of the equation \( x^4 - x^3\sin 2\beta + x^2\cos 2\beta - x\cos \beta - \sin \beta = 0 \) then \( \tan(\theta_1 + \theta_2 + \theta_3 + \theta_4) \) is equal to
(a) \( \sin \beta \)
(b) \( \cos \beta \)
(c) \( \tan \beta \)
(d) \( \cot \beta \)
Answer: (d) \( \cot \beta \)

Question. If \( \frac{\sin^3 \theta - \cos^3 \theta}{\sin \theta - \cos \theta} - \frac{\cos \theta}{\sqrt{1+\cot^2 \theta}} = -2\tan \theta \cot \theta = -1, \theta \in [0, 2\pi] \) then
(a) \( \theta \in (0, \pi/2) - \{\pi/4\} \)
(b) \( \theta \in \left(\frac{\pi}{2}, \pi\right) - \{3\pi/4\} \)
(c) \( \theta \in \left(\pi, \frac{3\pi}{2}\right) - \{5\pi/4\} \)
(d) \( \theta \in (0, \pi) - \{\pi/4, \pi/2\} \)
Answer: (d) \( \theta \in (0, \pi) - \{\pi/4, \pi/2\} \)

Question. If \( \frac{\tan(\alpha+\beta-\gamma)}{\tan(\alpha-\beta+\gamma)} = \frac{\tan \gamma}{\tan \beta} \) \( (\beta \neq \gamma) \) then \( \sin 2\alpha + \sin 2\beta + \sin 2\gamma = \)
(a) 0
(b) 1
(c) 2
(d) 1/2
Answer: (a) 0

Question. \( \cos 56^\circ + \cos 58^\circ - \cos 66^\circ - 4\cos 28^\circ \cos 29^\circ \sin 33^\circ = \)
(a) 0
(b) 1
(c) -1
(d) 2
Answer: (c) -1

Question. If \( \frac{x}{y} = \frac{\cos A}{\cos B} \) then \( \frac{x\tan A + y\tan B}{x+y} = \)
(a) \( \tan \frac{A+B}{2} \)
(b) \( \tan \frac{A-B}{2} \)
(c) \( \cot \frac{A+B}{2} \)
(d) \( \cot \frac{A-B}{2} \)
Answer: (a) \( \tan \frac{A+B}{2} \)

Question. If \( \sin x + \operatorname{cosec} x + \tan y + \cot y = 4 \) where \( x \) and \( y \in \left(0, \frac{\pi}{2}\right) \) \( \tan \frac{y}{2} \) is a root of the equation
(a) \( \alpha^2 + 2\alpha + 1 = 0 \)
(b) \( \alpha^2 + 2\alpha - 1 = 0 \)
(c) \( 2\alpha^2 - 2\alpha - 1 = 0 \)
(d) \( \alpha^2 - \alpha - 1 = 0 \)
Answer: (b) \( \alpha^2 + 2\alpha - 1 = 0 \)

Question. An integral value of a for which there is a solution of the equation \( a\cos x + \cot x + 1 = \operatorname{cosec} x \) is \( (\sin 2x \neq 0) \)
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3

Question. If \( y = (1+\tan A)(1-\tan B) \) where \( A - B = \frac{\pi}{4} \) then \( (y+1)^{y+1} = \)
(a) 27
(b) 9
(c) 81
(d) 4
Answer: (a) 27

Question. Let A, B, C be three angles such that \( A = \frac{\pi}{4} \) and \( \tan B \tan C = p \). Then, all possible values of p such that A, B, C are the angles of a triangle is
(a) \( \leq (\sqrt{2}+1)^2 \)
(b) \( \geq (\sqrt{2}+1)^2 \)
(c) \( > (\sqrt{2}+1)^2 \)
(d) \( < (\sqrt{2}+1)^2 \)
Answer: (b) \( \geq (\sqrt{2}+1)^2 \)

Question. If \( \tan \alpha = \frac{P}{q} \) where \( \alpha = 2\beta \), \( \alpha \) being an acute angle then \( \frac{1}{2}[p\operatorname{cosec} 2\beta + q\sec 2\beta] \) is equal to
(a) \( \sqrt{p^2+q^2} \)
(b) \( \sqrt{p^2-q^2} \)
(c) \( \sqrt{2p^2+q^2} \)
(d) \( 2\sqrt{p^2+q^2} \)
Answer: (a) \( \sqrt{p^2+q^2} \)

Question. If \( \frac{\cos x}{a} = \frac{\cos(x+\theta)}{b} = \frac{\cos(x+2\theta)}{c} = \frac{\cos(x+3\theta)}{d} \) then \( \frac{a+c}{b+d} \) is equal to
(a) a/d
(b) c/d
(c) b/c
(d) d/a
Answer: (c) b/c

Question. If \( \sqrt{2}\cos A = \cos B + \cos^3 B \) and \( \sqrt{2}\sin A = \sin B - \sin^3 B \) then \( \sin(A-B) = \)
(a) \( \pm 1 \)
(b) \( \pm \frac{1}{2} \)
(c) \( \pm \frac{1}{3} \)
(d) \( \pm \frac{1}{4} \)
Answer: (c) \( \pm \frac{1}{3} \)