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Mathematics
Trigonometric Functions & Equations
Previous Years Questions

If $$\tan 15^\circ + {1 \over {\tan 75^\circ }} + {1 \over {\tan 105^\circ }} + \tan 195^\circ = 2a$$, then the value of $$\left( {a + {1 \over a}} ... The set of all values of$$\lambda$$for which the equation$${\cos ^2}2x - 2{\sin ^4}x - 2{\cos ^2}x = \lambda $$has a real solution$$x$$, is... Let$$f(\theta ) = 3\left( {{{\sin }^4}\left( {{{3\pi } \over 2} - \theta } \right) + {{\sin }^4}(3\pi + \theta )} \right) - 2(1 - {\sin ^2}2\theta )...
The number of elements in the set $$S=\left\{x \in \mathbb{R}: 2 \cos \left(\frac{x^{2}+x}{6}\right)=4^{x}+4^{-x}\right\}$$ is :
Let $$S=\left\{\theta \in\left(0, \frac{\pi}{2}\right): \sum\limits_{m=1}^{9} \sec \left(\theta+(m-1) \frac{\pi}{6}\right) \sec \left(\theta+\frac{m \... Let$$S=\left\{\theta \in[0,2 \pi]: 8^{2 \sin ^{2} \theta}+8^{2 \cos ^{2} \theta}=16\right\} .$$Then$$n(s) + \sum\limits_{\theta \in S}^{} {\left( ...
$$2 \sin \left(\frac{\pi}{22}\right) \sin \left(\frac{3 \pi}{22}\right) \sin \left(\frac{5 \pi}{22}\right) \sin \left(\frac{7 \pi}{22}\right) \sin \le... The number of solutions of$$|\cos x|=\sin x$$, such that$$-4 \pi \leq x \leq 4 \pi$$is : If cot$$\alpha$$= 1 and sec$$\beta$$=$$ - {5 \over 3}$$, where$$\pi ...
Let for some real numbers $$\alpha$$ and $$\beta$$, $$a = \alpha - i\beta$$. If the system of equations $$4ix + (1 + i)y = 0$$ and $$8\left( {\cos {...$$\alpha = \sin 36^\circ $$is a root of which of the following equation? The value of$$\cos \left( {{{2\pi } \over 7}} \right) + \cos \left( {{{4\pi } \over 7}} \right) + \cos \left( {{{6\pi } \over 7}} \right)$$is equal ...$$16\sin (20^\circ )\sin (40^\circ )\sin (80^\circ )$$is equal to : The value of 2sin (12$$^\circ$$)$$-$$sin (72$$^\circ$$) is : The number of solutions of the equation$$\cos \left( {x + {\pi \over 3}} \right)\cos \left( {{\pi \over 3} - x} \right) = {1 \over 4}{\cos ^2}2x$$,... Let$$S = \left\{ {\theta \in [ - \pi ,\pi ] - \left\{ { \pm \,\,{\pi \over 2}} \right\}:\sin \theta \tan \theta + \tan \theta = \sin 2\theta } \r...
If n is the number of solutions of the equation $$2\cos x\left( {4\sin \left( {{\pi \over 4} + x} \right)\sin \left( {{\pi \over 4} - x} \right) - 1... The number of solutions of the equation$${32^{{{\tan }^2}x}} + {32^{{{\sec }^2}x}} = 81,\,0 \le x \le {\pi \over 4}$$is : The distance of the point (1,$$-$$2, 3) from the plane x$$-$$y + z = 5 measured parallel to a line, whose direction ratios are 2, 3,$$-$$6 is : The value of$$2\sin \left( {{\pi \over 8}} \right)\sin \left( {{{2\pi } \over 8}} \right)\sin \left( {{{3\pi } \over 8}} \right)\sin \left( {{{5\pi ...
The sum of solutions of the equation $${{\cos x} \over {1 + \sin x}} = \left| {\tan 2x} \right|$$, $$x \in \left( { - {\pi \over 2},{\pi \over 2}} \... If$$\tan \left( {{\pi \over 9}} \right),x,\tan \left( {{{7\pi } \over {18}}} \right)$$are in arithmetic progression and$$\tan \left( {{\pi \over ...
If $$\sin \theta + \cos \theta = {1 \over 2}$$, then 16(sin(2$$\theta$$) + cos(4$$\theta$$) + sin(6$$\theta$$)) is equal to :
The value of $$\cot {\pi \over {24}}$$ is :
The sum of all values of x in [0, 2$$\pi$$], for which sin x + sin 2x + sin 3x + sin 4x = 0, is equal to :
If 15sin4$$\alpha$$ + 10cos4$$\alpha$$ = 6, for some $$\alpha$$$$\in$$R, then the value of 27sec6$$\alpha$$ + 8cosec6$$\alpha$$ is equal to :...
The number of solutions of the equation x + 2tanx = $${\pi \over 2}$$ in the interval [0, 2$$\pi$$] is :
The number of roots of the equation, (81)sin2x + (81)cos2x = 30 in the interval [ 0, $$\pi$$ ] is equal to :...
If for x $$\in$$ $$\left( {0,{\pi \over 2}} \right)$$, log10sinx + log10cosx = $$-$$1 and log10(sinx + cosx) = $${1 \over 2}$$(log10 n $$-$$ 1), n &g...
If 0 < x, y < $$\pi$$ and cosx + cosy $$-$$ cos(x + y) = $${3 \over 2}$$, then sinx + cosy is equal to :
All possible values of $$\theta$$ $$\in$$ [0, 2$$\pi$$] for which sin 2$$\theta$$ + tan 2$$\theta$$ > 0 lie in :
If $${e^{\left( {{{\cos }^2}x + {{\cos }^4}x + {{\cos }^6}x + ...\infty } \right){{\log }_e}2}}$$ satisfies the equation t2 - 9t + 8 = 0, then the val...
If L = sin2$$\left( {{\pi \over {16}}} \right)$$ - sin2$$\left( {{\pi \over {8}}} \right)$$ and M = cos2$$\left( {{\pi \over {16}}} \right)$$ - sin...
If the equation cos4 $$\theta$$ + sin4 $$\theta$$ + $$\lambda$$ = 0 has real solutions for $$\theta$$, then $$\lambda$$ lies in the interval :...
If $$x = \sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)}^n}{{\tan }^{2n}}\theta }$$ and $$y = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\thet... The value of$${\cos ^3}\left( {{\pi \over 8}} \right){\cos}\left( {{3\pi \over 8}} \right)$$+$${\sin ^3}\left( {{\pi \over 8}} \right){\si...
If [x] denotes the greatest integer $$\le$$ x, then the system of linear equations [sin $$\theta$$]x + [–cos$$\theta$$]y = 0, [cot$$\theta$$]x + ...
Let S be the set of all $$\alpha$$ $$\in$$ R such that the equation, cos2x + $$\alpha$$sinx = 2$$\alpha$$– 7 has a solution. Then S is equal to :
The number of solutions of the equation 1 + sin4 x = cos23x, $$x \in \left[ { - {{5\pi } \over 2},{{5\pi } \over 2}} \right]$$ is :...
The equation y = sinx sin (x + 2) – sin2 (x + 1) represents a straight line lying in :
The value of sin 10º sin30º sin50º sin70º is :-
Let S = {$$\theta$$ $$\in$$ [–2$$\pi$$, 2$$\pi$$] : 2cos2$$\theta$$ + 3sin$$\theta$$ = 0}. Then the sum of the elements of S is
The value of cos210° – cos10°cos50° + cos250° is
If cos($$\alpha$$ + $$\beta$$) = 3/5 ,sin ( $$\alpha$$ - $$\beta$$) = 5/13 and 0 < $$\alpha , \beta$$ < $$\pi \over 4$$, then tan(2$$\alpha ... The maximum value of 3cos$$\theta $$+ 5sin$$\left( {\theta - {\pi \over 6}} \right)$$for any real value of$$\theta $$is : The value of$$\cos {\pi \over {{2^2}}}.\cos {\pi \over {{2^3}}}\,.....\cos {\pi \over {{2^{10}}}}.\sin {\pi \over {{2^{10}}}}$$is - The sum of all values of$$\theta  \in \left( {0,{\pi \over 2}} \right)$$satisfying sin2 2$$\theta $$+ cos4 2$$\theta $$=$${3 \over 4}$$... If 0$$ \le $$x <$${\pi \over 2}$$, then the number of values of x for which sin x$$-$$sin 2x + sin 3x = 0, is : For any$$\theta \in \left( {{\pi \over 4},{\pi \over 2}} \right)$$, the expression$$3{(\cos \theta - \sin \theta )^4} + 6{(\sin \theta + ...
If sum of all the solutions of the equation $$8\cos x.\left( {\cos \left( {{\pi \over 6} + x} \right).\cos \left( {{\pi \over 6} - x} \right) - {1 \... PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR. If the angles of elevation of the top of the tower at P, Q ... The number of solutions of sin3x = cos 2x, in the interval$$\left( {{\pi \over 2},\pi } \right)$$is : Let a vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P be a point on the ground such that AP = 2AB. If$$\angl...
If $$5\left( {{{\tan }^2}x - {{\cos }^2}x} \right) = 2\cos 2x + 9$$, then the value of $$\cos 4x$$ is
Let f(x) = sin4x + cos4 x. Then f is an increasing function in the interval :
The number of x $$\in$$ [0,   2$$\pi$$ ] for which $$\left| {\sqrt {2{{\sin }^4}x + 18{{\cos }^2}x} - \sqrt {2{{\cos }^4}x + 18{{\sin }^2}x... If m and M are the minimum and the maximum values of 4 +$${1 \over 2}$$sin2 2x$$-$$2cos4 x, x$$ \in $$R, then M$$-$$m is equal to :... If$$0 \le x < 2\pi $$, then the number of real values of$$x$$, which satisfy the equation$$\,\cos x + \cos 2x + \cos 3x + \cos 4x = 0$$is: Let$$fk\left( x \right) = {1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)$$where$$x \in R$$and$$k \ge \,.$$Then$${f_4}\left( x \right) ...
The expression $${{\tan {\rm A}} \over {1 - \cot {\rm A}}} + {{\cot {\rm A}} \over {1 - \tan {\rm A}}}$$ can be written as:
$$ABCD$$ is a trapezium such that $$AB$$ and $$CD$$ are parallel and $$BC \bot CD.$$ If $$\angle ADB = \theta ,\,BC = p$$ and $$CD = q,$$ then AB is e...
In a $$\Delta PQR,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu}$$ If $$3{\mkern 1mu} \sin {\mkern 1mu} P + 4{\mkern 1mu} \cos {\mkern 1mu} Q = 6$$ and $$4... If$$A = {\sin ^2}x + {\cos ^4}x,$$then for all real$$x$$: Let$$\cos \left( {\alpha + \beta } \right) = {4 \over 5}$$and$$\sin \,\,\,\left( {\alpha - \beta } \right) = {5 \over {13}},$$where$$0 \le \alp...
Let A and B denote the statements A: $$\cos \alpha + \cos \beta + \cos \gamma = 0$$ B: $$\sin \alpha + \sin \beta + \sin \gamma = 0$$ If $$\cos... If$$0 < x < \pi $$and$$\cos x + \sin x = {1 \over 2},$$then$$\tan x$$is The number of values of$$x$$in the interval$$\left[ {0,3\pi } \right]\,$$satisfying the equation$$2{\sin ^2}x + 5\sin x - 3 = 0$$is Let$$\alpha ,\,\beta $$be such that$$\pi < \alpha - \beta < 3\pi $$. If$$sin{\mkern 1mu} \alpha + \sin \beta = - {{21} \over {65}}$$a... If$$u = \sqrt {{a^2}{{\cos }^2}\theta + {b^2}{{\sin }^2}\theta } + \sqrt {{a^2}{{\sin }^2}\theta + {b^2}{{\cos }^2}\theta } $$then the differen... A line makes the same angle$$\theta $$, with each of the$$x$$and$$z$$axis. If the angle$$\beta \,$$, which it makes with y-axis, is such that... The period of$${\sin ^2}\theta $$is The number of solution of$$\tan \,x + \sec \,x = 2\cos \,x$$in$$\left[ {0,\,2\,\pi } \right]$$is Which one is not periodic ## Numerical If m and n respectively are the numbers of positive and negative values of$$\theta$$in the interval$$[-\pi,\pi]$$that satisfy the equation$$\cos ...
Let $$\mathrm{S = \{ \theta \in [0,2\pi ):\tan (\pi \cos \theta ) + \tan (\pi \sin \theta ) = 0\}}$$. Then $$\sum\limits_{\theta \in S} {{{\sin }^2}... Let$$S=\left\{\theta \in(0,2 \pi): 7 \cos ^{2} \theta-3 \sin ^{2} \theta-2 \cos ^{2} 2 \theta=2\right\}$$. Then, the sum of roots of all the equation... Let$$S=\left[-\pi, \frac{\pi}{2}\right)-\left\{-\frac{\pi}{2},-\frac{\pi}{4},-\frac{3 \pi}{4}, \frac{\pi}{4}\right\}$$. Then the number of elements i... If the sum of solutions of the system of equations$$2 \sin ^{2} \theta-\cos 2 \theta=0$$and$$2 \cos ^{2} \theta+3 \sin \theta=0$$in the interval ... Let$${S_1} = \{ x \in [0,12\pi ]:{\sin ^5}x + {\cos ^5}x = 1\} $$and$${S_2} = \{ x \in [0,8\pi ]:{\sin ^7}x + {\cos ^7}x = 1\} $$Then$$n({S_1}) -...
The number of solutions of the equation $$\sin x = {\cos ^2}x$$ in the interval (0, 10) is _________.
The number of elements in the set $$S = \{ \theta \in [ - 4\pi ,4\pi ]:3{\cos ^2}2\theta + 6\cos 2\theta - 10{\cos ^2}\theta + 5 = 0\}$$ is _____...
The number of solutions of the equation $$2\theta - {\cos ^2}\theta + \sqrt 2 = 0$$ in R is equal to ___________.
If $${\sin ^2}(10^\circ )\sin (20^\circ )\sin (40^\circ )\sin (50^\circ )\sin (70^\circ ) = \alpha - {1 \over {16}}\sin (10^\circ )$$, then $$16 + {\... The number of values of x in the interval$$\left( {{\pi \over 4},{{7\pi } \over 4}} \right)$$for which$$14\cos e{c^2}x - 2{\sin ^2}x = 21 - 4{\cos...
Let S be the sum of all solutions (in radians) of the equation $${\sin ^4}\theta + {\cos ^4}\theta - \sin \theta \cos \theta = 0$$ in [0, 4$$\pi$$]...
The probability distribution of random variable X is given by : .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-sty...
Let z1 and z2 be two complex numbers such that $$\arg ({z_1} - {z_2}) = {\pi \over 4}$$ and z1, z2 satisfy the equation | z $$-$$ 3 | = Re(z). Then t...
Let S = {1, 2, 3, 4, 5, 6, 9}. Then the number of elements in the set T = {A $$\subseteq$$ S : A $$\ne$$ $$\phi$$ and the sum of all the elements of...
The number of solutions of the equation $$|\cot x| = \cot x + {1 \over {\sin x}}$$ in the interval [ 0, 2$$\pi$$ ] is
If $$\sqrt 3 ({\cos ^2}x) = (\sqrt 3 - 1)\cos x + 1$$, the number of solutions of the given equation when $$x \in \left[ {0,{\pi \over 2}} \right]$$...
The number of integral values of 'k' for which the equation $$3\sin x + 4\cos x = k + 1$$ has a solution, k$$\in$$R is ___________.
If $${{\sqrt 2 \sin \alpha } \over {\sqrt {1 + \cos 2\alpha } }} = {1 \over 7}$$ and \sqrt {{{1 - \cos 2\beta } \over 2}} = {1 \over {\sqrt {10} }}...
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