Mathematics
Trigonometric Functions & Equations
Previous Years Questions

## Numerical

Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi}{4}If$\sin (\alpha+\beta)=\frac{1}{3}$and$\cos (\alpha-\beta)=\frac{2}{3}$, then the... Let f : R $$\to$$ R be a differentiable function with f(0) = 1 and satisfying the equation f(x + y) = f(x) f'(y) + f'(x) f(y) for all x, y$$\in$$ ... Consider the cube in the first octant with sides OP, OQ and OR of length 1, along the X-axis, Y-axis and Z-axis, respectively, where O(0, 0, 0) is the... The number of distinct solutions of the equation $${5 \over 4}{\cos ^2}\,2x + {\cos ^4}\,x + {\sin ^4}\,x + {\cos ^6}\,x + {\sin ^6}\,x\, = \,2$$$ i...
The positive integer value of $$n\, > \,3$$ satisfying the equation $${1 \over {\sin \left( {{\pi \over n}} \right)}} = {1 \over {\sin \left( {{{2... The number of all possible values of$$\theta $$where$$0 < \theta < \pi ,$$for which the system of equations$$$\left( {y + z} \right)\cos {... The number of values of $$\theta$$ in the interval, $$\left( { - {\pi \over 2},\,{\pi \over 2}} \right)$$ such that$$\,\theta \ne {{n\pi } \over... The maximum value of the expression$${1 \over {{{\sin }^2}\theta + 3\sin \theta \cos \theta + 5{{\cos }^2}\theta }}$$is Two parallel chords of a circle of radius 2 are at a distance$$\sqrt 3 + 1$$apart. If the chords subtend at the center , angles of$${\pi \over k}... ## MCQ (Single Correct Answer) Consider the following lists: .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-fa... Let f(x) = sin($$\pi$$ cos x) and g(x) = cos(2$$\pi$$ sin x) be two functions defined for x > 0. Define the following sets whose elements are wri... Let f(x) = sin($$\pi$$ cos x) and g(x) = cos(2$$\pi$$ sin x) be two functions defined for x > 0. Define the following sets whose elements are wri... The value of $$\sum\limits_{k = 1}^{13} {{1 \over {\sin \left( {{\pi \over 4} + {{\left( {k - 1} \right)\pi } \over 6}} \right)\sin \left( {{\pi \ov... Let$$S = \left\{ {x \in \left( { - \pi ,\pi } \right):x \ne 0, \pm {\pi \over 2}} \right\}.$$The sum of all distinct solutions of the equation$$\s... For $$x \in \left( {0,\pi } \right),$$ the equation $$\sin x + 2\sin 2x - \sin 3x = 3$$ has The number of points in $$\left( { - \infty \,\infty } \right),$$ for which $${x^2} - x\sin x - \cos x = 0,$$ is Let $$P = \{ \theta :\sin \theta - \cos \theta = \sqrt 2 \cos \theta \}$$ and $$Q = \{ \theta :\sin \theta + \cos \theta = \sqrt 2 \sin \theta \}... The number of solutions of the pair of equations$$$\,2{\sin ^2}\theta - \cos 2\theta = 0$2co{s^2}\theta - 3\sin \theta = 0$$in the interv... Let$$\theta \in \left( {0,{\pi \over 4}} \right)$$and$${t_1} = {\left( {\tan \theta } \right)^{\tan \theta }},\,\,\,\,{t_2} = \,\,{\left( {\tan \... The values of $$\theta \in \left( {0,2\pi } \right)$$ for which $$2\,{\sin ^2}\theta - 5\,\sin \theta + 2 > 0,$$ are $$\cos \left( {\alpha - \beta } \right) = 1$$ and $$\,\cos \left( {\alpha + \beta } \right) = 1/e$$ where $$\alpha ,\,\beta \in \left[ { - \pi ,\pi... Given both$$\theta $$and$$\phi $$are acute angles and$$\sin \,\theta = {1 \over 2},\,\cos \,\phi = {1 \over 3},$$then the value of$$\the... The number of integral values of $$k$$ for which the equation $$7\cos x + 5\sin x = 2k + 1$$ has a solution is The number of distinct real roots of $$\left| {\matrix{ {\sin x} & {\cos x} & {\cos x} \cr {\cos x} & {\sin x} & {\cos x} \cr... The maximum value of$$\left( {\cos {\alpha _1}} \right).\left( {\cos {\alpha _2}} \right).....\left( {\cos {\alpha _n}} \right),$$under the restric... If$$\alpha + \beta = \pi /2$$and$$\beta + \gamma = \alpha ,$$then$$\tan \,\alpha \,$$equals Let$$f\left( \theta \right) = \sin \theta \left( {\sin \theta + \sin 3\theta } \right)$$. Then$$f\left( \theta \right)$$is In a triangle$$PQR,\angle R = \pi /2$$. If$$\,\,\tan \left( {P/2} \right)$$and$$\tan \left( {Q/2} \right)$$are the roots of the equation$$a{x^2... Which of the following number(s) is /are rational? The number of values of $$x\,\,$$ in the interval $$\left[ {0,\,5\pi } \right]$$ satisfying the equation $$3\,{\sin ^2}x - 7\,\sin \,x + 2 = 0$$ is $${\sec ^2}\theta = {{4xy} \over {{{\left( {x + y} \right)}^2}}}\,$$ is true if and only if The minimum value of the expression $$\sin \,\alpha + \sin \,\beta \, + \sin \,\gamma ,\,$$ where $$\alpha ,\,\beta ,\,\gamma$$ are real numbers sa... $$\,3{\left( {\sin x - \cos x} \right)^4} + 6{\left( {\sin x + \cos x} \right)^2} + 4\left( {{{\sin }^6}x + {{\cos }^6}x} \right) =$$ The general values of $$\theta$$ satisfying the equation $$2{\sin ^2}\theta - 3\sin \theta - 2 = 0$$ is Let $$0 < x < {\pi \over 4}$$ then $$\left( {\sec 2x - \tan 2x} \right)$$ equals Let $$n$$ be a positive integer such that $$\sin {\pi \over {2n}} + \cos {\pi \over {2n}} = {{\sqrt n } \over 2}.$$ Then If $$\omega \,$$ is an imaginary cube root of unity then the value of $$\sin \left\{ {\left( {{\omega ^{10}} + {\omega ^{23}}} \right)\pi - {\pi \ov... Let$$2{\sin ^2}x + 3\sin x - 2 > 0$$and$${x^2} - x - 2 < 0$$($$x$$is measured in radians). Then$$x$$lies in the interval Number of solutions of the equation$$\tan x + \sec x = 2\cos x\,$$lying in the interval$$\left[ {0,2\pi } \right]$$is: In this questions there are entries in columns 1 and 2. Each entry in column 1 is related to exactly one entry in column 2. Write the correct letter f... The equation$$\left( {\cos p - 1} \right){x^2} + \left( {\cos p} \right)x + \sin p = 0\,$$In the variable x, has real roots. Then p can take any va... The general solutions of$$\,\sin x - 3\sin 2x + \sin 3x = \cos x - 3\cos 2x + \cos 3x$$is The value of the expression$$\sqrt 3 \,\cos \,ec\,{20^0} - \sec \,{20^0}$$is equal to The number of all possible triplets$$\left( {{a_1},\,{a_2},\,{a_3}} \right)$$such that$${a_1} + {a_2}\,\,\cos \left( {2x} \right) + {a_3}{\sin ^2}\... The expression $$2\left[ {{{\sin }^6}\left( {{\pi \over 2} + \alpha } \right) + {{\sin }^6}\left( {5\pi - \alpha } \right)} \right]$$ is equal to $$\left( {1 + \cos {\pi \over 8}} \right)\left( {1 + \cos {{3\pi } \over 8}} \right)\left( {1 + \cos {{5\pi } \over 8}} \right)\left( {1 + \cos {{7\p... The general solution of the trigonometric equation sin x+cos x=1 is given by: Given$$A = {\sin ^2}\theta + {\cos ^4}\theta $$then for all real values of$$\theta $$The equation$$\,2{\cos ^2}{x \over 2}{\sin ^2}x = {x^2} + {x^{ - 2}};\,0 < x \le {\pi \over 2}$$has If$$\tan \theta = - {4 \over 3},then\sin \theta \,is\,$$If$$\alpha + \beta + \gamma = 2\pi ,$$then ## MCQ (More than One Correct Answer) For non-negative integers n, let$$f(n) = {{\sum\limits_{k = 0}^n {\sin \left( {{{k + 1} \over {n + 2}}\pi } \right)} \sin \left( {{{k + 2} \over {n + ... In a $$\Delta$$PQR = 30$$^\circ$$ and the sides PQ and QR have lengths 10$$\sqrt 3$$ and 10, respectively. Then, which of the following statement(s... Let $$\alpha$$ and $$\beta$$ be non zero real numbers such that $$2(\cos \beta - \cos \alpha ) + \cos \alpha \cos \beta = 1$$. Then which of the f... Let $$f\left( x \right) = x\sin \,\pi x,\,x > 0.$$ Then for all natural numbers $$n,\,f'\left( x \right)$$ vanishes at Let $$\theta ,\,\varphi \, \in \,\left[ {0,2\pi } \right]$$ be such that $$2\cos \theta \left( {1 - \sin \,\varphi } \right) = {\sin ^2}\theta \,\,\l... For$$0 < \theta < {\pi \over 2},$$the solution (s) of$$$\sum\limits_{m = 1}^6 {\cos ec\,\left( {\theta + {{\left( {m - 1} \right)\pi } \o...
If $${{{{\sin }^4}x} \over 2} + {{{{\cos }^4}x} \over 3} = {1 \over 5},$$ then
For a positive integer $$\,n$$, let $${f_n}\left( \theta \right) = \left( {\tan {\theta \over 2}} \right)\,\left( {1 + \sec \theta } \right)\,\le... The values of$$\theta $$lying between$$\theta = \theta $$and$$\theta = \pi /2$$and satisfying the equation$$\left| {\matrix{ {1 + {{\sin ...

## Subjective

Find the range of values of $$\,t$$ for which $$2\,\sin \,t = {{1 - 2x + 5{x^2}} \over {3{x^2} - 2x - 1}},\,\,\,\,\,t\, \in \,\left[ { - {\pi \over ... In any triangle$$ABC,$$prove that$$$\cot {A \over 2} + \cot {B \over 2} + \cot {C \over 2} = \cot {A \over 2}\cot {B \over 2}\cot {C \over 2}.$$Prove that$$\tan \,\alpha + 2\tan 2\alpha + 4\tan 4\alpha + 8\cot 8\alpha = \cot \alpha $$Prove that the values of the function$${{\sin x\cos 3x} \over {\sin 3x\cos x}}$$do not lie between$${1 \over 3}$$and 3 for any real$$x.$$Prove that$$\sum\limits_{k = 1}^{n - 1} {\left( {n - k} \right)\,\cos \,{{2k\pi } \over n} = - {n \over 2},} $$where$$n \ge 3$$is an integer. Find all values of$$\theta $$in the interval$$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$satisfying the equation$$\left( {1 - \tan \,\th... Find the smallest positive number $$p$$ for which the equation $$\cos \left( {p\,\sin x} \right) = \sin \left( {p\cos x} \right)$$ has a solution $$x\... Determine the smallest positive value of number$$x$$(in degrees) for which$$$\tan \left( {x + {{100}^ \circ }} \right) = \tan \left( {x + {{50}^ ...
Show that the value of $${{\tan x} \over {\tan 3x}},$$ wherever defined never lies between $${1 \over 3}$$ and 3.
If $$\exp \,\,\,\left\{ {\left( {\left( {{{\sin }^2}x + {{\sin }^4}x + {{\sin }^6}x + \,\,\,..............\infty } \right)\,In\,\,2} \right)} \right\}...$$ABC$$is a triangle such that$$$\sin \left( {2A + B} \right) = \sin \left( {C - A} \right) = \, - \sin \left( {B + 2C} \right) = {1 \over 2}.$$I... Find the values of$$x \in \left( { - \pi , + \pi } \right)$$which satisfy the equation$${g^{(1 + \left| {\cos x} \right| + \left| {{{\cos }^2}x} \r... Find all solutions of $$4{\cos ^2}\,x\sin x - 2{\sin ^2}x = 3\sin x$$ Show that $$16\cos \left( {{{2\pi } \over {15}}} \right)\cos \left( {{{4\pi } \over {15}}} \right)\cos \left( {{{8\pi } \over {15}}} \right)\cos \lef... Without using tables, prove that$$\left( {\sin \,{{12}^ \circ }} \right)\left( {\sin \,{{48}^ \circ }} \right)\left( {\sin \,{{54}^ \circ }} \right)... Given $$\alpha + \beta - \gamma = \pi ,$$ prove that $$\,{\sin ^2}\alpha + {\sin ^2}\beta - {\sin ^2}\gamma = 2\sin \alpha {\mkern 1mu} \sin \b... Given$$A = \left\{ {x:{\pi \over 6} \le x \le {\pi \over 3}} \right\}$$and$$f\left( x \right) = \cos x - x\left( {1 + x} \right);$$find$$f\lef... For all $$\theta$$ in $$\left[ {0,\,\pi /2} \right]$$ show that, $$\cos \left( {\sin \theta } \right) \ge \,\sin \,\left( {\cos \theta } \right).$$ (a) Draw the graph of $$y = {1 \over {\sqrt 2 }}\left( {cinx + \cos x} \right)$$ from $$x = - {\pi \over 2}$$ to $$x = {\pi \over 2}$$. (b) If $$\c... If$$\tan \alpha = {m \over {m + 1}}\,$$and$$\tan \beta = {2 \over {2m + 1}},$$find the possible values of$$\left( {\alpha + \beta } \right).$...

## Fill in the Blanks

The real roots of the equation $$\,{\cos ^7}x + {\sin ^4}x = 1$$ in the interval $$\left( { - \pi ,\pi } \right)$$ are ...., ...., and ______.
General value of $$\theta$$ satisfying the equation $${\tan ^2}\theta + \sec \,2\,\theta = 1$$ is _________.
If $$K = \sin \left( {\pi /18} \right)\sin \left( {5\pi /18} \right)\sin \left( {7\pi /18} \right),$$ then the numerical value of K is ______.
If $$A > 0,B > 0\,$$ and $$A + B = \pi /3,$$ then the maximum value of tan A tan B is _______.
The value of $$\sin {\pi \over {14}}\sin {{3\pi } \over {14}}\sin {{5\pi } \over {14}}\sin {{7\pi } \over {14}}\sin {{9\pi } \over {14}}\sin {{11\pi... The solution set of the system of equations$$X + Y = {{2\pi } \over 3},cox\,x + cos\,y = {3 \over 2},$$where x and y are real, is _____. The sides of a triangle inscribed in a given circle subtend angles$$\alpha $$,$$\beta $$and$$\gamma $$at the centre. The minimum value of the ari... The set of all$$x$$in the interval$$\left[ {0,\,\pi } \right]$$for which$$2\,{\sin ^2}x - 3\sin x + 1 \ge 0,$$is _____. Suppose$${\sin ^3}\,x\sin 3x = \sum\limits_{m = 0}^n {{C_m}\cos \,mx} $$is an identity in x, where C0, C1 ,....Cn are constants, and$${C_n} \ne 0\$...

## True or False

There exists a value of $$\theta$$ between 0 and $$2\pi$$ that satisfies the equation $$\,\,{\sin ^4}\theta - 2{\sin ^2}\theta - 1 = 0.$$
If $$\tan \,A = \left( {1 - \cos B} \right)/\sin B,$$ then $$tan2A = tan\,B$$.
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