Class 11 Mathematics Trigonometric Functions MCQs Set 07

Question. Which of the following relation is correct
(a) \( \sin 1^\circ > \sin 1 \)
(b) \( \sin 1 > \sin 1^\circ \)
(c) \( \sin 1 = \sin 1^\circ \)
(d) \( \frac{\pi}{180} \sin 1 = \sin 1^\circ \)
Answer: (b)

Question. The radius of the circle whose arc of length 15 cm makes an angle of 3/4 radian at the centre is
(a) 10 cm
(b) 20 cm
(c) \( 11\frac{1}{4} \) cm
(d) \( 22\frac{1}{2} \) cm
Answer: (b)

Question. If \( \tan \theta = \frac{-4}{3} \), then \( \sin \theta = \)
(a) \( \frac{-4}{5} \) but not \( \frac{4}{5} \)
(b) \( \frac{-4}{5} \) or \( \frac{4}{5} \)
(c) \( \frac{4}{5} \) but not \( \frac{-4}{5} \)
(d) None of these
Answer: (b)

Question. If \( f(x) = \cos^2 x + \sec^2 x \), then
(a) \( f(x) < 1 \)
(b) \( f(x) = 1 \)
(c) \( 1 < f(x) < 2 \)
(d) \( f(x) \geq 2 \)
Answer: (d)

Question. If \( x = \sec \theta + \tan \theta \), then \( x + \frac{1}{x} = \)
(a) 1
(b) \( 2 \sec \theta \)
(c) 2
(d) \( 2 \tan \theta \)
Answer: (b)

Question. If A lies in the second quadrant and \( 3 \tan A + 4 = 0 \) then the value of \( 2 \cot A - 5 \cos A + \sin A \) is equal to
(a) \( \frac{-53}{10} \)
(b) \( \frac{-7}{10} \)
(c) \( \frac{7}{10} \)
(d) \( \frac{23}{10} \)
Answer: (d)

Question. \( \tan 1^\circ \tan 2^\circ \tan 3^\circ \tan 4^\circ \dots \tan 89^\circ = \)
(a) 1
(b) 0
(c) \( \infty \)
(d) 1/2
Answer: (a)

Question. The incorrect statement is
(a) \( \sin \theta = -\frac{1}{5} \)
(b) \( \cos \theta = 1 \)
(c) \( \sec \theta = \frac{1}{2} \)
(d) \( \tan \theta = 20 \)
Answer: (c)

Question. If \( \cos \theta - \sin \theta = \sqrt{2} \sin \theta \), then \( \cos \theta + \sin \theta \) is equal to
(a) \( \sqrt{2} \cos \theta \)
(b) \( \sqrt{2} \sin \theta \)
(c) \( 2 \cos \theta \)
(d) \( -\sqrt{2} \cos \theta \)
Answer: (a)

Question. If \( \sec \theta + \tan \theta = p \), then \( \tan \theta \) is equal to
(a) \( \frac{2p}{p^2 - 1} \)
(b) \( \frac{p^2 - 1}{2p} \)
(c) \( \frac{p^2 + 1}{2p} \)
(d) \( \frac{2p}{p^2 + 1} \)
Answer: (b)

Question. If \( \sin \theta - \cos \theta = 1 \), then \( \sin \theta \cos \theta = \)
(a) 0
(b) 1
(c) 2
(d) 1/2
Answer: (a)

Question. The value of \( \cos 1^\circ \cos 2^\circ \cos 3^\circ \dots \cos 179^\circ \) is
(a) \( \frac{1}{\sqrt{2}} \)
(b) 0
(c) 1
(d) None of these
Answer: (b)

Question. If \( \tan \theta = + \frac{1}{\sqrt{5}} \) and \( \theta \) lies in the 1st quadrant, then \( \cos \theta \) is
(a) \( \frac{1}{\sqrt{6}} \)
(b) \( -\frac{1}{\sqrt{6}} \)
(c) \( \sqrt{\frac{5}{6}} \)
(d) \( -\sqrt{\frac{5}{6}} \)
Answer: (c)

Question. If A lies in the third quadrant and \( 3 \tan A - 4 = 0 \), then \( 5 \sin 2A + 3 \sin A + 4 \cos A = \)
(a) 0
(b) \( -\frac{24}{5} \)
(c) \( \frac{24}{5} \)
(d) \( \frac{48}{5} \)
Answer: (a)

Question. \( (\sec^2 \theta - 1)(\csc^2 \theta - 1) = \)
(a) 0
(b) 1
(c) \( \sec \theta \cdot \csc \theta \)
(d) \( \sin^2 \theta - \cos^2 \theta \)
Answer: (b)

Question. If \( \tan \theta = \frac{20}{21} \), then \( \cos \theta \) will be
(a) \( \pm \frac{20}{41} \)
(b) \( \pm \frac{1}{21} \)
(c) \( \pm \frac{21}{29} \)
(d) \( \pm \frac{20}{21} \)
Answer: (c)

Question. If \( \csc A + \cot A = \frac{11}{2} \), then \( \tan A \) equal to
(a) \( \frac{21}{22} \)
(b) \( \frac{15}{16} \)
(c) \( \frac{44}{117} \)
(d) \( \frac{117}{43} \)
Answer: (c)

Question. If \( \sin \theta = \frac{24}{25} \) and \( \theta \) lies in the second quadrant, then \( \sec \theta + \tan \theta \) equal to
(a) -3
(b) -5
(c) -7
(d) -9
Answer: (c)

Question. If \( 5 \tan \theta = 4 \), then \( \frac{5 \sin \theta - 3 \cos \theta}{5 \sin \theta + 2 \cos \theta} \) equal to
(a) 0
(b) 1
(c) 1/6
(d) 6
Answer: (c)

Question. \( \frac{1 + \cos \theta}{\sin^2 \theta} \) equal to
(a) 0
(b) 1
(c) \( \frac{1}{1 - \cos \theta} \)
(d) \( \frac{1}{1 + \cos \theta} \)
Answer: (c)

Question. The expression \( \frac{1}{\tan A + \cot A} \) simplifies to
(a) \( \sec A \csc A \)
(b) \( \sin A \cos A \)
(c) \( \tan 2A \)
(d) \( \sin 2A \)
Answer: (b)

Question. If for real values of \( x \), \( \cos \theta = x + \frac{1}{x} \), then
(a) \( \theta \) is an acute angle
(b) \( \theta \) is a right angle
(c) \( \theta \) is an obtuse angle
(d) No value of \( \theta \) is possible
Answer: (d)

Question. If \( \sin x + \csc x = 2 \), then \( \sin^n x + \csc^n x \) is equal to
(a) 2
(b) \( 2^n \)
(c) \( 2^{n-1} \)
(d) \( 2^{n-2} \)
Answer: (a)

Question. One root of the equation \( \cos x - x + \frac{1}{2} = 0 \) lies in the interval
(a) \( [0, \frac{\pi}{2}] \)
(b) \( [-\frac{\pi}{2}, 0] \)
(c) \( [\frac{\pi}{2}, \pi] \)
(d) \( [\pi, \frac{3\pi}{2}] \)
Answer: (a)

Advance Level

Question. If \( \frac{2 \sin \alpha}{\{1 + \cos \alpha + \sin \alpha\}} = y \), then \( \frac{\{1 - \cos \alpha + \sin \alpha\}}{1 + \sin \alpha} = \)
(a) \( \frac{1}{y} \)
(b) \( y \)
(c) \( 1 - y \)
(d) \( 1 + y \)
Answer: (b)

Question. If \( \sin \theta + \sin^2 \theta + \sin^3 \theta = 1 \), then \( \cos^6 \theta - 4 \cos^4 \theta + 8 \cos^2 \theta = \)
(a) 4
(b) 2
(c) 1
(d) None of these
Answer: (a)

Question. If \( \theta \) and \( \phi \) are angles in the 1st quadrant such that \( \tan \theta = 1/7 \) and \( \sin \phi = 1/\sqrt{10} \). Then
(a) \( \theta + 2\phi = 90^\circ \)
(b) \( \theta + 2\phi = 60^\circ \)
(c) \( \theta + 2\phi = 30^\circ \)
(d) \( \theta + 2\phi = 45^\circ \)
Answer: (d)

Question. The value of \( \theta \) lying between 0 and \( \pi/2 \) and satisfying the equation \( \begin{vmatrix} 1 + \sin^2 \theta & \cos^2 \theta & 4 \sin 4\theta \\ \sin^2 \theta & 1 + \cos^2 \theta & 4 \sin 4\theta \\ \sin^2 \theta & \cos^2 \theta & 1 + 4 \sin 4\theta \end{vmatrix} = 0 \)
(a) \( \frac{7\pi}{24} \) or \( \frac{11\pi}{24} \)
(b) \( \frac{5\pi}{24} \)
(c) \( \frac{\pi}{24} \)
(d) None of these
Answer: (a)

Question. If \( \frac{3\pi}{4} < \alpha < \pi \), then \( \sqrt{\csc^2 \alpha + 2 \cot \alpha} \) is equal to
(a) \( 1 + \cot \alpha \)
(b) \( 1 - \cot \alpha \)
(c) \( -1 - \cot \alpha \)
(d) \( -1 + \cot \alpha \)
Answer: (c)

Question. If for all real values of \( x \), \( \frac{4x^2 + 1}{64x^2 - 96x \sin \alpha + 5} < \frac{1}{32} \), then \( \alpha \) lies in the interval
(a) \( (0, \frac{\pi}{3}) \)
(b) \( (\frac{\pi}{3}, \frac{2\pi}{3}) \)
(c) \( (\frac{2\pi}{3}, \pi) \)
(d) \( (\frac{4\pi}{3}, \frac{5\pi}{3}) \)
Answer: (c)

Question. If \( \tan \theta = \sqrt{\frac{3}{2}} \), then the sum of the infinite series \( 1 + 2(1 - \cos \theta) + 3(1 - \cos \theta)^2 + 4(1 - \cos \theta)^3 + \dots \infty \) is
(a) \( \frac{2}{3} \)
(b) \( \frac{3}{4} \)
(c) \( \frac{5}{2\sqrt{2}} \)
(d) \( \frac{5}{2} \)
Answer: (d)

Question. Let \( A_0 A_1 A_2 A_3 A_4 A_5 \) be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments \( A_0 A_1 \), \( A_0 A_2 \) and \( A_0 A_4 \) is
(a) \( \frac{3}{4} \)
(b) \( 3\sqrt{3} \)
(c) 3
(d) \( \frac{3\sqrt{3}}{2} \)
Answer: (c)

Trigonometrical Ratios of Allied Angles

Question. If \( x = \frac{\sin 45^\circ \cos 60^\circ \tan^2 60^\circ \csc 30^\circ}{\sec 45^\circ \cot^2 30^\circ} \), then \( x = \)
(a) 2
(b) 4
(c) 8
(d) 16
Answer: (c)

Question. \( \cos A + \sin(270^\circ - A) - \sin(270^\circ + A) + \cos(180^\circ + A) = \)
(a) -1
(b) 0
(c) 1
(d) None of these
Answer: (b)

Question. \( \sin(\pi + \theta) \sin(\pi - \theta) \csc^2 \theta = \)
(a) 1
(b) -1
(c) \( \sin \theta \)
(d) \( -\sin \theta \)
Answer: (b)

Question. The value of \( \sin 600^\circ \cos 330^\circ + \cos 120^\circ \sin 150^\circ \) is
(a) -1
(b) 1
(c) \( \frac{1}{\sqrt{2}} \)
(d) \( \frac{\sqrt{3}}{2} \)
Answer: (a)

Question. If \( A = 130^\circ \) and \( x = \sin A + \cos A \), then
(a) \( x > 0 \)
(b) \( x < 0 \)
(c) \( x = 0 \)
(d) \( x \leq 0 \)
Answer: (a)

Question. \( \tan \theta \sin(\frac{\pi}{2} + \theta) \cos(\frac{\pi}{2} - \theta) = \)
(a) 1
(b) 0
(c) \( \frac{1}{\sqrt{2}} \)
(d) None of these
Answer: (d)

Question. \( \sin^2 5^\circ + \sin^2 10^\circ + \sin^2 15^\circ + \dots + \sin^2 85^\circ + \sin^2 90^\circ = \)
(a) 7
(b) 8
(c) 9
(d) \( 9\frac{1}{2} \)
Answer: (d)

Question. Values of \( \theta \) \( (0 < \theta < 360^\circ) \) satisfying \( \csc \theta + 2 = 0 \) are
(a) \( 210^\circ, 300^\circ \)
(b) \( 240^\circ, 300^\circ \)
(c) \( 210^\circ, 240^\circ \)
(d) \( 210^\circ, 330^\circ \)
Answer: (d)

Question. The value of \( \tan(-945^\circ) \) is
(a) -1
(b) -2
(c) -3
(d) -4
Answer: (a)

Question. The value of \( \frac{\cot 54^\circ}{\tan 36^\circ} + \frac{\tan 20^\circ}{\cot 70^\circ} \) is
(a) 2
(b) 3
(c) 1
(d) 0
Answer: (a)

Question. \( \tan 9^\circ - \tan 27^\circ - \tan 63^\circ + \tan 81^\circ = \)
(a) 1/2
(b) 2
(c) 4
(d) 8
Answer: (c)

Question. \( \cos 1^\circ + \cos 2^\circ + \cos 3^\circ + \dots + \cos 180^\circ = \)
(a) 0
(b) 1
(c) -1
(d) 2
Answer: (c)

Question. If \( \tan(A - B) = 1 \), \( \sec(A + B) = \frac{2}{\sqrt{3}} \), then the smallest positive value of B is
(a) \( \frac{25\pi}{24} \)
(b) \( \frac{19\pi}{24} \)
(c) \( \frac{13\pi}{24} \)
(d) \( \frac{11\pi}{24} \)
Answer: (b)

Question. If \( x = \sin 130^\circ \cos 80^\circ \), \( y = \sin 80^\circ \cos 130^\circ \), \( z = 1 + xy \), which one of the following is true
(a) \( x > 0, y > 0, z > 0 \)
(b) \( x > 0, y < 0, 0 < z < 1 \)
(c) \( x > 0, y < 0, z > 1 \)
(d) \( x < 0, y < 0, 0 < z < 1 \)
Answer: (b)

Question. If \( \alpha = 22^\circ 30' \), then \( (1 + \cos \alpha)(1 + \cos 3\alpha)(1 + \cos 5\alpha)(1 + \cos 7\alpha) \) equals
(a) 1/8
(b) 1/4
(c) \( \frac{1 + \sqrt{2}}{2\sqrt{2}} \)
(d) \( \frac{\sqrt{2} - 1}{\sqrt{2} + 1} \)
Answer: (a)