JEE Mathematics Trigonometrical Functions and Identities MCQs Set 02

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

Question. If \( \cos 2x + 2 \cos x = 1 \), then \( \sin^2 x(2 - \cos^2 x) \) is equal to
(a) 1
(b) -1
(c) \( -\sqrt{5} \)
(d) \( \sqrt{5} \)
Answer: (a) 1

Question. If \( 0 < \phi < \frac{\pi}{2} \), \( x = \sum_{n=0}^{\infty} \cos^{2n} \phi \), \( y = \sum_{n=0}^{\infty} \sin^{2n} \phi \) and \( z = \sum_{n=0}^{\infty} \cos^{2n} \phi \sin^{2n} \phi \), then
(a) \( xyz = xz + y \)
(b) \( xyz = xy + z \)
(c) \( xyz = x + y + z \)
(d) \( xyz = yz + x \)
Answer: (b) \( xyz = xy + z \)

Question. Let \( n \) be an odd integer. If \( \sin n\theta = \sum_{r=0}^{n} b_r \sin^r \theta \) for all real \( \theta \), then
(a) \( b_0 = 1, b_1 = 3 \)
(b) \( b_0 = 0, b_1 = n \)
(c) \( b_0 = -1, b_1 = n \)
(d) \( b_0 = 0, b_1 = n^2 - 3n - 3 \)
Answer: (b) \( b_0 = 0, b_1 = n \)

Question. If \( \cos 5\theta = a \cos^5 \theta + b \cos^3 \theta + c \cos \theta \), then \( c \) is equal to
(a) -5
(b) 1
(c) 5
(d) None of the options
Answer: (c) 5

Question. If \( \sin^3 x \cdot \sin 3x = \sum_{m=0}^{n} c_m \cos mx \) is an identity in \( x \), where \( c_m \)'s are constant, then the value of \( n \) is
(a) 4
(b) 6
(c) 9
(d) None of the options
Answer: (b) 6

Question. The value of \( \sin \frac{\pi}{14} \sin \frac{3\pi}{14} \sin \frac{5\pi}{14} \sin \frac{7\pi}{14} \sin \frac{9\pi}{14} \sin \frac{11\pi}{14} \sin \frac{13\pi}{14} \) is equal to
(a) 1
(b) \( \frac{1}{16} \)
(c) \( \frac{1}{64} \)
(d) None of the options
Answer: (c) \( \frac{1}{64} \)

Question. The numerical value of \( \sin \frac{\pi}{18} \sin \frac{3\pi}{18} \sin \frac{5\pi}{18} \sin \frac{7\pi}{18} \) is equal to
(a) 1
(b) \( \frac{1}{8} \)
(c) \( \frac{1}{4} \)
(d) \( \frac{1}{2} \)
Answer: (d) None of the options

Question. The value of \( \tan 63^\circ - \cot 63^\circ \) is equal to
(a) \( \frac{2}{\sqrt{5} + 1} \sqrt{10 - 2\sqrt{5}} \)
(b) \( \frac{2}{\sqrt{5} + 1} \sqrt{10 + 2\sqrt{5}} \)
(c) \( \frac{\sqrt{5} - 1}{4} \sqrt{10 - 2\sqrt{5}} \)
(d) None of the options
Answer: (a) \( \frac{2}{\sqrt{5} + 1} \sqrt{10 - 2\sqrt{5}} \)

Question. The value of \( \cos 9^\circ - \sin 9^\circ \) is
(a) \( \frac{-\sqrt{5 - \sqrt{5}}}{2} \)
(b) \( \frac{5 + \sqrt{5}}{4} \)
(c) \( \frac{1}{2}\sqrt{5 - \sqrt{5}} \)
(d) None of the options
Answer: (c) \( \frac{1}{2}\sqrt{5 - \sqrt{5}} \)

Question. The value of \( 2 \tan \frac{\pi}{10} + 3 \sec \frac{\pi}{10} - 4 \cos \frac{\pi}{10} \) is
(a) 0
(b) \( \sqrt{5} \)
(c) 1
(d) None of the options
Answer: (a) 0

Question. The value of \( \tan 20^\circ + 2 \tan 50^\circ - \tan 70^\circ \) is
(a) 1
(b) 0
(c) \( \tan 50^\circ \)
(d) None of the options
Answer: (b) 0

Question. If \( \alpha, \beta, \gamma \) and \( \delta \) be four angles of a cyclic quadrilateral then the value of \( \cos \alpha + \cos \beta + \cos \gamma + \cos \delta \) is
(a) 1
(b) 0
(c) -1
(d) None of the options
Answer: (b) 0

Question. If \( 4n\alpha = \pi \) then \( \cot \alpha \cdot \cot 2\alpha \cdot \cot 3\alpha \dots \cot(2n - 1)\alpha \) is equal to
(a) 1
(b) -1
(c) \( \infty \)
(d) None of the options
Answer: (a) 1

Question. The value of \( \cos 12^\circ \cdot \cos 24^\circ \cdot \cos 36^\circ \cdot \cos 48^\circ \cdot \cos 72^\circ \cdot \cos 84^\circ \) is
(a) \( \frac{1}{64} \)
(b) \( \frac{1}{32} \)
(c) \( \frac{1}{16} \)
(d) \( \frac{1}{128} \)
Answer: (a) \( \frac{1}{64} \)

Question. The value of \( \cos \frac{\pi}{11} + \cos \frac{3\pi}{11} + \cos \frac{5\pi}{11} + \cos \frac{7\pi}{11} + \cos \frac{9\pi}{11} \) is
(a) 0
(b) 1
(c) \( \frac{1}{2} \)
(d) None of the options
Answer: (c) \( \frac{1}{2} \)

Question. \( \sum_{r=1}^{n-1} \cos^2 \frac{r\pi}{n} \) is equal to
(a) \( \frac{n}{2} \)
(b) \( \frac{n-1}{2} \)
(c) \( \frac{n}{2} - 1 \)
(d) None of the options
Answer: (c) \( \frac{n}{2} - 1 \)

Question. The value of \( \sin \frac{\pi}{n} + \sin \frac{3\pi}{n} + \sin \frac{5\pi}{n} + \dots \) to \( n \) terms is equal to
(a) 1
(b) 0
(c) \( \frac{n}{2} \)
(d) None of the options
Answer: (b) 0

Question. The sum of the real roots of \( \cos^6 x + \sin^4 x = 1 \) in the interval \( -\pi < x \leq \pi \) is equal to
(a) 0
(b) \( \pi \)
(c) \( -\pi \)
(d) None of the options
Answer: (a) 0

Question. The number of real solutions of the equation \( \cos^7 x + \sin^4 x = 1 \) in the interval \( [-\pi, \pi] \) is
(a) 2
(b) 3
(c) 5
(d) None of the options
Answer: (b) 3

Question. If the solutions for \( \theta \) from the equation \( \sin^2 \theta - 2 \sin \theta + \lambda = 0 \) lie in \( \bigcup_{n \in \mathbb{Z}} \left( 2n\pi - \frac{\pi}{6}, 2n\pi + \frac{\pi}{6} + \pi \right) \), then the set of possible values of \( \lambda \) is
(a) \( \left(\frac{5}{4}, 1\right] \)
(b) \( (-\infty, 1) \)
(c) \( \left(-\frac{5}{4}, +\infty\right] \)
(d) \( [1] \)
Answer: (a) \( \left(\frac{5}{4}, 1\right] \)

Question. If ABCD is a convex quadrilateral such that \( 4 \sec A + 5 = 0 \) then the quadratic equation whose roots are \( \tan A \) and \( \csc A \) is
(a) \( 12x^2 - 29x + 15 = 0 \)
(b) \( 12x^2 - 11x - 15 = 0 \)
(c) \( 12x^2 + 11x - 15 = 0 \)
(d) None of the options
Answer: (b) \( 12x^2 - 11x - 15 = 0 \)

Question. If ABCD is a cyclic quadrilateral such that \( 12 \tan A - 5 = 0 \) and \( 5 \cos B + 3 = 0 \) then the quadratic equation whose roots are \( \cos C \), \( \tan D \) is
(a) \( 39x^2 - 16x - 48 = 0 \)
(b) \( 39x^2 + 88x + 48 = 0 \)
(c) \( 39x^2 - 88x + 48 = 0 \)
(d) None of the options
Answer: (a) \( 39x^2 - 16x - 48 = 0 \)

Question. The number of real solution of the equation \( \sin(e^x) = 2^x + 2^{-x} \) is
(a) 1
(b) 0
(c) 2
(d) infinite
Answer: (b) 0

Question. The equation \( (\cos p - 1)x^2 + (\cos p)x + \sin p = 0 \) in \( x \) has real roots. Then the set of values of \( p \) is
(a) \( [0, 2\pi] \)
(b) \( [-\pi, 0] \)
(c) \( [-\frac{\pi}{2}, \frac{\pi}{2}] \)
(d) \( [0, \pi] \)
Answer: (d) \( [0, \pi] \)

Question. If \( e^{\sin x} - e^{-\sin x} - 4 = 0 \) then the number of real values of \( x \) is
(a) 0
(b) 2
(c) 1
(d) infinite
Answer: (a) 0

Question. If \( \sin \alpha = p \), where \( |p| \leq 1 \) then the quadratic equation whose roots are \( \tan \frac{\alpha}{2} \) and \( \cot \frac{\alpha}{2} \) is
(a) \( px^2 + 2x + p = 0 \)
(b) \( px^2 - x + p = 0 \)
(c) \( px^2 - 2x + p = 0 \)
(d) None of the options
Answer: (c) \( px^2 - 2x + p = 0 \)

Question. If \( \sec \alpha \) and \( \csc \alpha \) are the roots of \( x^2 - px + q = 0 \) then
(a) \( p^2 = q(q - 2) \)
(b) \( p^2 = q(q + 2) \)
(c) \( p^2 + q^2 = 2q \)
(d) None of the options
Answer: (b) \( p^2 = q(q + 2) \)

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

Question. If \( x = \alpha, \beta \) satisfies both the equations \( \cos^2 x + a \cos x + b = 0 \) and \( \sin^2 x + p \sin x + q = 0 \) then relation between \( a, b, p \) and \( q \) is
(a) \( 1 + b + a^2 = p^2 - q - 1 \)
(b) \( a^2 + b^2 = p^2 + q^2 \)
(c) \( 2(b + q) = a^2 + p^2 - 2 \)
(d) None of the options
Answer: (d) None of the options

Question. If \( 0 \leq a \leq 3 \), \( 0 < b \leq 3 \) and the equation \( x^2 + 4 + 3 \cos(ax + b) = 2x \) has at least one solution then the value of \( a + b \) is
(a) 0
(b) \( \frac{\pi}{2} \)
(c) \( \pi \)
(d) None of the options
Answer: (c) \( \pi \)

Question. The equation \( \cos \theta = x + \frac{p}{x} \) for all \( x \in \mathbb{R} \) has a real solution for \( \theta \). Then
(a) \( p = \frac{1}{2} \)
(b) \( p \leq \frac{1}{4} \)
(c) \( p \geq \frac{1}{4} \)
(d) None of the options
Answer: (b) \( p \leq \frac{1}{4} \)

Question. If \( f(x) = \frac{\sin 3x}{\sin x} \), where \( x \neq n\pi \), then the range of values of \( f(x) \) for real values of \( x \) is
(a) \( [-1, 3] \)
(b) \( (-\infty, -1) \)
(c) \( (3, +\infty) \)
(d) \( [-1, 3) \)
Answer: (d) \( [-1, 3) \)

Question. The set of values of \( k \in \mathbb{R} \) such that the equation \( \cos 2\theta + \cos \theta + k = 0 \) admits of a solution for \( \theta \) is
(a) \( [0, 9/8] \)
(b) \( [0, +\infty] \)
(c) \( [-2, 0] \)
(d) None of the options
Answer: (a) \( [0, 9/8] \)

Question. The set of values of \( \lambda \in \mathbb{R} \) such that \( \tan^2 \theta + \sec \theta = \lambda \) holds for some \( \theta \) is
(a) \( (-\infty, 1) \)
(b) \( (-\infty, -1] \)
(c) \( \phi \)
(d) \( [-1, +\infty) \)
Answer: (d) \( [-1, +\infty) \)

Question. If \( \tan A + \tan B + \tan C = \tan A \cdot \tan B \cdot \tan C \) then
(a) A, B, C must be angles of a triangle
(b) the sum of any two of A, B, C is equal to the third
(c) \( A + B + C \) must be an integral multiple of \( \pi \)
(d) None of the options
Answer: (c) \( A + B + C \) must be an integral multiple of \( \pi \)

Choose the correct options. One or more options may be correct

Question. If \( 7 \cos x - 4 \sin x = \lambda \cos(x + \alpha) < \frac{\pi}{2} \), be true for all \( x \in \mathbb{R} \) then
(a) \( \lambda = 25 \)
(b) \( \alpha = \sin^{-1} \frac{24}{25} \)
(c) \( \lambda = -25 \)
(d) \( \alpha = \cos^{-1} \frac{7}{25} \)
Answer: (a) \( \lambda = 25 \)
(b) \( \alpha = \sin^{-1} \frac{24}{25} \)
(d) \( \alpha = \cos^{-1} \frac{7}{25} \)

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

Question. If \( \tan \theta = a \neq 0, \tan 2\theta = b \neq 0 \) and \( \tan \theta + \tan 2\theta = \tan 3\theta \) then
(a) \( a = b \)
(b) \( ab = 1 \)
(c) \( a + b = 0 \)
(d) \( b = 2a \)
Answer: (c) \( a + b = 0 \)

Question. \( \cos^3 x \cdot \sin 2x = \sum_{m=1}^{n} a_m \sin mx \) is an identity in \( x \). Then
(a) \( a_3 = \frac{3}{8}, a_2 = 0 \)
(b) \( n = 6, a_1 = \frac{1}{2} \)
(c) \( n = 5, a_1 = \frac{1}{4} \)
(d) \( \sum a_m = \frac{3}{4} \)
Answer: (a) \( a_3 = \frac{3}{8}, a_2 = 0 \)
(c) \( n = 5, a_1 = \frac{1}{4} \)
(d) \( \sum a_m = \frac{3}{4} \)

Question. If \( 1 + \cos(x - y) = 0 \) then
(a) \( \cos x - \cos y = 0 \)
(b) \( \cos x + \cos y = 0 \)
(c) \( \sin x + \sin y = 0 \)
(d) \( \cos x + \sin y = 1 \)
Answer: (b) \( \cos x + \cos y = 0 \)
(c) \( \sin x + \sin y = 0 \)

Question. If \( A \geq 0, B > 0, A + B = \frac{\pi}{3} \) and \( y = \tan A \cdot \tan B \) then
(a) the maximum value of \( y \) is 3
(b) the minimum value of \( y \) is \( \frac{1}{3} \)
(c) the maximum value of \( y \) is \( \frac{1}{3} \)
(d) the minimum value of \( y \) is 0
Answer: (c) the maximum value of \( y \) is \( \frac{1}{3} \)
(d) the minimum value of \( y \) is 0

Question. \( \cos \frac{2\pi}{7} + \cos \frac{4\pi}{7} + \cos \frac{6\pi}{7} \) is equal to
(a) an integer
(b) a positive rational number
(c) a negative rational number
(d) an irrational number
Answer: (c) a negative rational number

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