MCQ (Single Correct Answer)
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$$]...
The probability distribution of random variable X is given by :
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Let z1 and z2 be two complex numbers such that $$\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} }}...