Trigonometric Functions & Equations | Mathematics | JEE Advanced Previous Year Questions

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

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

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 from column 2 against the entry number in column 1 in your answer book.

$${{\sin \,3\alpha } \over {\cos 2\alpha }}$$ is

Column $${\rm I}$$

(A) positive

(B) negative

Column $${\rm I}$$$${\rm I}$$

(p) $$\left( {{{13\pi } \over {48}},{{14\pi } \over {48}}} \right)$$

(q) $$\left( {{{14\pi } \over {48}},\,{{18\pi } \over {48}}} \right)$$

(r) $$\left( {{{18\pi } \over {48}},\,{{23\pi } \over {48}}} \right)$$

(s) $$\left( {0,\,{\pi \over 2}} \right)$$

Options:-

IIT-JEE 1992

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 value in the interval

IIT-JEE 1990

The general solutions of $$\,\sin x - 3\sin 2x + \sin 3x = \cos x - 3\cos 2x + \cos 3x$$ is

IIT-JEE 1989

The value of the expression $$\sqrt 3 \,\cos \,ec\,{20^0} - \sec \,{20^0}$$ is equal to

IIT-JEE 1988

The number of all possible triplets $$\left( {{a_1},\,{a_2},\,{a_3}} \right)$$ such that $${a_1} + {a_2}\,\,\cos \left( {2x} \right) + {a_3}{\sin ^2}\left( x \right) = 0\,$$ for all $$x$$ is

IIT-JEE 1987

The expression $$2\left[ {{{\sin }^6}\left( {{\pi \over 2} + \alpha } \right) + {{\sin }^6}\left( {5\pi - \alpha } \right)} \right]$$ is equal to

IIT-JEE 1986

$$\left( {1 + \cos {\pi \over 8}} \right)\left( {1 + \cos {{3\pi } \over 8}} \right)\left( {1 + \cos {{5\pi } \over 8}} \right)\left( {1 + \cos {{7\pi } \over 8}} \right)$$ is equal to

IIT-JEE 1984

The general solution of the trigonometric equation sin x+cos x=1 is given by:

IIT-JEE 1981

The equation $$\,2{\cos ^2}{x \over 2}{\sin ^2}x = {x^2} + {x^{ - 2}};\,0 < x \le {\pi \over 2}$$ has

Given $$A = {\sin ^2}\theta + {\cos ^4}\theta $$ then for all real values of $$\theta $$

If $$\alpha + \beta + \gamma = 2\pi ,$$ then

IIT-JEE 1979

If $$\tan \theta = - {4 \over 3},then\sin \theta \,is\,$$

IIT-JEE 1979

MCQ (More than One Correct Answer)

Let $P Q R S$ be a quadrilateral in a plane, where

$Q R=1, \angle P Q R=\angle Q R S=70^{\circ}, \angle P Q S=15^{\circ}$ and $\angle P R S=40^{\circ}$.

If $\angle R P S=\theta^{\circ}, P Q=\alpha$ and $P S=\beta$, then the interval(s) that contain(s) the value of

$4 \alpha \beta \sin \theta^{\circ}$ is/are

JEE Advanced 2022 Paper 2 Online

For non-negative integers n, let

$$f(n) = {{\sum\limits_{k = 0}^n {\sin \left( {{{k + 1} \over {n + 2}}\pi } \right)} \sin \left( {{{k + 2} \over {n + 2}}\pi } \right)} \over {\sum\limits_{k = 0}^n {{{\sin }^2}\left( {{{k + 1} \over {n + 2}}\pi } \right)} }}$$

Assuming cos^$$-1$$ x takes values in [0, $$\pi $$], which of the following options is/are correct?

JEE Advanced 2019 Paper 2 Offline

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) is(are) TRUE?

JEE Advanced 2018 Paper 1 Offline

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 following is/are true?

JEE Advanced 2017 Paper 2 Offline

Let $$f\left( x \right) = x\sin \,\pi x,\,x > 0.$$ Then for all natural numbers $$n,\,f'\left( x \right)$$ vanishes at

JEE Advanced 2013 Paper 1 Offline

Let $$\theta ,\,\varphi \, \in \,\left[ {0,2\pi } \right]$$ be such that
$$2\cos \theta \left( {1 - \sin \,\varphi } \right) = {\sin ^2}\theta \,\,\left( {\tan {\theta \over 2} + \cot {\theta \over 2}} \right)\cos \varphi - 1,\,\tan \left( {2\pi - \theta } \right) > 0$$ and $$ - 1 < \sin \theta \, < - {{\sqrt 3 } \over 2},$$

then $$\varphi $$ cannot satisfy

IIT-JEE 2012 Paper 1 Offline

For $$0 < \theta < {\pi \over 2},$$ the solution (s) of $$$\sum\limits_{m = 1}^6 {\cos ec\,\left( {\theta + {{\left( {m - 1} \right)\pi } \over 4}} \right)\,\cos ec\,\left( {\theta + {{m\pi } \over 4}} \right) = 4\sqrt 2 } $$$ is (are)

IIT-JEE 2009 Paper 2 Offline

If $${{{{\sin }^4}x} \over 2} + {{{{\cos }^4}x} \over 3} = {1 \over 5},$$ then

IIT-JEE 2009 Paper 1 Offline

For a positive integer $$\,n$$, let
$${f_n}\left( \theta \right) = \left( {\tan {\theta \over 2}} \right)\,\left( {1 + \sec \theta } \right)\,\left( {1 + \sec 2\theta } \right)\,\left( {1 + \sec 4\theta } \right).....\left( {1 + \sec {2^n}\theta } \right).$$ Then

IIT-JEE 1999

The values of $$\theta $$ lying between $$\theta = \theta $$ and $$\theta = \pi /2$$ and satisfying the equation

$$\left| {\matrix{ {1 + {{\sin }^2}\theta } & {{{\cos }^2}\theta } & {4\sin 4\theta } \cr {{{\sin }^2}\theta } & {1 + {{\cos }^2}\theta } & {4\sin 4\theta } \cr {{{\sin }^2}\theta } & {{{\cos }^2}\theta } & {1 + 4\sin 4\theta } \cr } } \right| = 0$$ are

IIT-JEE 1988

Subjective

Find the range of values of $$\,t$$ for which $$$2\,\sin \,t = {{1 - 2x + 5{x^2}} \over {3{x^2} - 2x - 1}},\,\,\,\,\,t\, \in \,\left[ { - {\pi \over 2},\,{\pi \over 2}} \right].$$$

IIT-JEE 2005

In any triangle $$ABC,$$ prove that $$$\cot {A \over 2} + \cot {B \over 2} + \cot {C \over 2} = \cot {A \over 2}\cot {B \over 2}\cot {C \over 2}.$$$

IIT-JEE 2000

Prove that $$\tan \,\alpha + 2\tan 2\alpha + 4\tan 4\alpha + 8\cot 8\alpha = \cot \alpha $$

IIT-JEE 1998

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 $$x.$$

IIT-JEE 1997

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.

IIT-JEE 1997

Find all values of $$\theta $$ in the interval $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$ satisfying the equation $$\left( {1 - \tan \,\theta } \right)\left( {1 + \tan \,\theta } \right)\,\,{\sec ^2}\theta + \,\,{2^{{{\tan }^2}\theta }} = 0.$$

IIT-JEE 1996

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 $$x\, \in \,\left[ {0,2\pi } \right]$$.

IIT-JEE 1995

Determine the smallest positive value of number $$x$$ (in degrees) for which $$$\tan \left( {x + {{100}^ \circ }} \right) = \tan \left( {x + {{50}^ \circ }} \right)\,\tan \left( x \right)\tan \left( {x - {{50}^ \circ }} \right).$$$

Show that the value of $${{\tan x} \over {\tan 3x}},$$ wherever defined never lies between $${1 \over 3}$$ and 3.

IIT-JEE 1992

If $$\exp \,\,\,\left\{ {\left( {\left( {{{\sin }^2}x + {{\sin }^4}x + {{\sin }^6}x + \,\,\,..............\infty } \right)\,In\,\,2} \right)} \right\}$$ satiesfies the equation $${x^2} - 9x + 8 = 0,$$ find the value of $${{\cos x} \over {\cos x + \sin x}},\,0 < x < {\pi \over 2}.$$

IIT-JEE 1991

$$ABC$$ is a triangle such that $$$\sin \left( {2A + B} \right) = \sin \left( {C - A} \right) = \, - \sin \left( {B + 2C} \right) = {1 \over 2}.$$$

If $$A,\,B$$ and $$C$$ are in arithmetic progression, determine the values of $$A,\,B$$ and $$C$$.

IIT-JEE 1990

Find the values of $$x \in \left( { - \pi , + \pi } \right)$$ which satisfy the equation $${g^{(1 + \left| {\cos x} \right| + \left| {{{\cos }^2}x} \right| + \left| {{{\cos }^3}x} \right| + ...)}} = {4^3}$$

IIT-JEE 1984

Find all solutions of $$4{\cos ^2}\,x\sin x - 2{\sin ^2}x = 3\sin x$$

IIT-JEE 1983

Show that $$$16\cos \left( {{{2\pi } \over {15}}} \right)\cos \left( {{{4\pi } \over {15}}} \right)\cos \left( {{{8\pi } \over {15}}} \right)\cos \left( {{{16\pi } \over {15}}} \right) = 1$$$

IIT-JEE 1983

Without using tables, prove that $$\left( {\sin \,{{12}^ \circ }} \right)\left( {\sin \,{{48}^ \circ }} \right)\left( {\sin \,{{54}^ \circ }} \right) = {1 \over 8}.$$

IIT-JEE 1982

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 $$f\left( A \right).$$

Given $$\alpha + \beta - \gamma = \pi ,$$ prove that
$$\,{\sin ^2}\alpha + {\sin ^2}\beta - {\sin ^2}\gamma = 2\sin \alpha {\mkern 1mu} \sin \beta {\mkern 1mu} \cos y$$

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 $$\cos \left( {\alpha + \beta } \right) = {4 \over 5},\,\,\sin \,\left( {\alpha - \beta } \right) = \,{5 \over {13}},$$ and $$\alpha ,\,\beta $$ lies between 0 and $${\pi \over 4}$$, find tan2$$\alpha $$.

IIT-JEE 1979

If $$\tan \alpha = {m \over {m + 1}}\,$$ and $$\tan \beta = {2 \over {2m + 1}},$$ find the possible values of $$\left( {\alpha + \beta } \right).$$

IIT-JEE 1978

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 ______.

IIT-JEE 1997

General value of $$\theta $$ satisfying the equation $${\tan ^2}\theta + \sec \,2\,\theta = 1$$ is _________.

IIT-JEE 1996

If $$A > 0,B > 0\,$$ and $$A + B = \pi /3,$$ then the maximum value of tan A tan B 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 ______.