Trigonometric Equations
158 Questions
2001
JEE Advanced
MCQ
IIT-JEE 2001 Screening
The number of distinct real roots of $\left| {\matrix{
{\sin x} & {\cos x} & {\cos x} \cr
{\cos x} & {\sin x} & {\cos x} \cr
{\cos x} & {\cos x} & {\sin x} \cr
} } \right|\,$
$\, = 0$ in the interval $ - {\pi \over 4} \le x \le {\pi \over 4}$ is
$\, = 0$ in the interval $ - {\pi \over 4} \le x \le {\pi \over 4}$ is
A.
0
B.
2
C.
1
D.
3
2000
JEE Advanced
MCQ
IIT-JEE 2000 Screening
Let $f\left( \theta \right) = \sin \theta \left( {\sin \theta + \sin 3\theta } \right)$. Then $f\left( \theta \right)$ is
A.
$ \ge 0\,\,$ only when $\theta \ge 0$
B.
$ \le 0$ for all real $\theta $
C.
$ \ge 0$ for all real $\theta $
D.
$ \le 0$ only when $\theta \le 0$
2000
JEE Advanced
Numerical
IIT-JEE 2000
In any triangle $ABC,$ prove that
$$\cot {A \over 2} + \cot {B \over 2} + \cot {C \over 2} = \cot {A \over 2}\cot {B \over 2}\cot {C \over 2}.$$
Correct Answer: Solve it.
1999
JEE Advanced
MCQ
IIT-JEE 1999
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 $a{x^2} + bx + c = 0\left( {a \ne 0} \right)$ then.
A.
$a + b = c$
B.
$a + c = b$
C.
$b + c = a$
D.
$b = c$
1999
JEE Advanced
MSQ
IIT-JEE 1999
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
${f_n}\left( \theta \right) = \left( {\tan {\theta \over 2}} \right)\,\left( {1 + \sec \theta } \right)\,\left( {1 + \sec 2\theta } \right)\,\left( {1 + \sec 4\theta } \right).....\left( {1 + \sec {2^n}\theta } \right).$ Then
A.
${f_2}\left( {{\pi \over {16}}} \right) = 1$
B.
${f_3}\left( {{\pi \over {32}}} \right) = 1$
C.
${f_4}\left( {{\pi \over {64}}} \right) = 1$
D.
${f_5}\left( {{\pi \over {128}}} \right) = 1$
1998
JEE Advanced
MCQ
IIT-JEE 1998
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
A.
0
B.
5
C.
6
D.
10
1998
JEE Advanced
MCQ
IIT-JEE 1998
Which of the following number(s) is /are rational?
A.
$\sin {\mkern 1mu} {15^ \circ }$
B.
$\cos {\mkern 1mu} {15^ \circ }$
C.
$\sin {\mkern 1mu} {15^ \circ }{\mkern 1mu} \cos {\mkern 1mu} {15^ \circ }$
D.
$\sin {\mkern 1mu} {15^ \circ }{\mkern 1mu} \cos {\mkern 1mu} {75^ \circ }$
1998
JEE Advanced
Numerical
IIT-JEE 1998
Prove that $\tan \,\alpha + 2\tan 2\alpha + 4\tan 4\alpha + 8\cot 8\alpha = \cot \alpha $
Correct Answer: Solve it.
1997
JEE Advanced
Numerical
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.
Correct Answer: Solve it.
1997
JEE Advanced
Numerical
IIT-JEE 1997
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 $x.$
Correct Answer: Solve it.
1997
JEE Advanced
Numerical
IIT-JEE 1997
The real roots of the equation $\,{\cos ^7}x + {\sin ^4}x = 1$ in the interval $\left( { - \pi ,\pi } \right)$ are ...., ...., and ______.
Correct Answer: $$ - {\pi \over 2},{\pi \over 2},0$$
1996
JEE Advanced
MCQ
IIT-JEE 1996
${\sec ^2}\theta = {{4xy} \over {{{\left( {x + y} \right)}^2}}}\,$ is true if and only if
A.
$x + y \ne 0\,$
B.
$x = y,\,x \ne 0$
C.
$x = y\,$
D.
$x \ne 0,\,y \ne 0$
1996
JEE Advanced
Numerical
IIT-JEE 1996
Find all values of $\theta $ in the interval $\left( { - {\pi \over 2},{\pi \over 2}} \right)$ satisfying the equation $\left( {1 - \tan \,\theta } \right)\left( {1 + \tan \,\theta } \right)\,\,{\sec ^2}\theta + \,\,{2^{{{\tan }^2}\theta }} = 0.$
Correct Answer: $$ \pm {\pi \over 3}$$
1996
JEE Advanced
Numerical
IIT-JEE 1996
General value of $\theta $ satisfying the equation ${\tan ^2}\theta + \sec \,2\,\theta = 1$ is _________.
Correct Answer: $$n\pi ,n\pi \pm {\pi \over 3}$$
1995
JEE Advanced
MCQ
IIT-JEE 1995
The minimum value of the expression $\sin \,\alpha + \sin \,\beta \, + \sin \,\gamma ,\,$ where $\alpha ,\,\beta ,\,\gamma $ are real numbers satisfying $\alpha + \beta + \gamma = \pi $ is
A.
positive
B.
zero
C.
negative
D.
-3
1995
JEE Advanced
MCQ
IIT-JEE 1995 Screening
The general values of $\theta $ satisfying the equation $2{\sin ^2}\theta - 3\sin \theta - 2 = 0$ is
A.
$n\pi + {\left( { - 1} \right)^n}\pi /6$
B.
$n\pi + {\left( { - 1} \right)^n}\pi /2 $
C.
$n\pi + {\left( { - 1} \right)^n}5\pi /6$
D.
$n\pi + {\left( { - 1} \right)^n}7\pi /6$
1995
JEE Advanced
MCQ
IIT-JEE 1995 Screening
$\,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) = $
A.
11
B.
12
C.
13
D.
14
1995
JEE Advanced
Numerical
IIT-JEE 1995
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 $x\, \in \,\left[ {0,2\pi } \right]$.
Correct Answer: $${{\pi \sqrt 2 } \over 4}$$
1994
JEE Advanced
MCQ
IIT-JEE 1994
If $\omega \,$ is an imaginary cube root of unity then the value of $\sin \left\{ {\left( {{\omega ^{10}} + {\omega ^{23}}} \right)\pi - {\pi \over 4}} \right\}$ is
A.
$ - {{\sqrt 3 } \over 2}\,$
B.
$ - {1 \over {\sqrt 2 }}$
C.
${1 \over {\sqrt 2 }}$
D.
${{\sqrt 3 } \over 2}$
1994
JEE Advanced
MCQ
IIT-JEE 1994
Let $0 < x < {\pi \over 4}$ then $\left( {\sec 2x - \tan 2x} \right)$ equals
A.
$\tan \left[ {x - {\pi \over 4}} \right]$
B.
$\tan \left[ {{\pi \over 4} - x} \right]$
C.
$\tan \left[ {x + {\pi \over 4}} \right]$
D.
${\tan ^2}\left[ {x + {\pi \over 4}} \right]$
1994
JEE Advanced
MCQ
IIT-JEE 1994
Let $n$ be a positive integer such that $\sin {\pi \over {2n}} + \cos {\pi \over {2n}} = {{\sqrt n } \over 2}.$ Then
A.
$6 \le n \le 8$
B.
$4 < n \le 8$
C.
$4 \le n \le 8$
D.
$4 < n < 8$
1994
JEE Advanced
MCQ
IIT-JEE 1994
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
A.
$\left( {{\pi \over 6},\,{{5\pi } \over 6}} \right)\,\,$
B.
$\left( { - 1,\,{{5\pi } \over 6}} \right)$
C.
$\left( { - 1,\,2} \right)\,\,\,$
D.
$\left( {{\pi \over 6},\,2} \right)$
1993
JEE Advanced
MCQ
IIT-JEE 1993
Number of solutions of the equation $\tan x + \sec x = 2\cos x\,$ lying in the interval $\left[ {0,2\pi } \right]$ is:
A.
0
B.
1
C.
2
D.
3
1993
JEE Advanced
Numerical
IIT-JEE 1993
Determine the smallest positive value of number $x$ (in degrees) for which
$$\tan \left( {x + {{100}^ \circ }} \right) = \tan \left( {x + {{50}^ \circ }} \right)\,\tan \left( x \right)\tan \left( {x - {{50}^ \circ }} \right).$$
Correct Answer: $${30^ \circ }$$
1993
JEE Advanced
Numerical
IIT-JEE 1993
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 ______.
Correct Answer: $${\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 8$}}$$
1993
JEE Advanced
Numerical
IIT-JEE 1993
If $A > 0,B > 0\,$ and $A + B = \pi /3,$ then the maximum value of tan A tan B is _______.
Correct Answer: $${\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 3$}}$$
1992
JEE Advanced
MCQ
IIT-JEE 1992
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}$${\rm I}$
(p) $\left( {{{13\pi } \over {48}},{{14\pi } \over {48}}} \right)$
(q) $\left( {{{14\pi } \over {48}},\,{{18\pi } \over {48}}} \right)$
(r) $\left( {{{18\pi } \over {48}},\,{{23\pi } \over {48}}} \right)$
(s) $\left( {0,\,{\pi \over 2}} \right)$
Options:-
A.
$\left( A \right) - r,\,\left( B \right) - q$
B.
$\left( A \right) - r,\,\left( B \right) - p$
C.
$\left( A \right) - s,\,\left( B \right) - r$
D.
$\left( A \right) - p,\,\left( B \right) - q$
1992
JEE Advanced
Numerical
IIT-JEE 1992
Show that the value of ${{\tan x} \over {\tan 3x}},$ wherever defined never lies between ${1 \over 3}$ and 3.
Correct Answer: Solve it.
1991
JEE Advanced
Numerical
IIT-JEE 1991
If $\exp \,\,\,\left\{ {\left( {\left( {{{\sin }^2}x + {{\sin }^4}x + {{\sin }^6}x + \,\,\,..............\infty } \right)\,In\,\,2} \right)} \right\}$ satiesfies the equation ${x^2} - 9x + 8 = 0,$ find the value of ${{\cos x} \over {\cos x + \sin x}},\,0 < x < {\pi \over 2}.$
Correct Answer: $${{\sqrt 3 - 1} \over 2}$$
1991
JEE Advanced
Numerical
IIT-JEE 1991
The value of
$\sin {\pi \over {14}}\sin {{3\pi } \over {14}}\sin {{5\pi } \over {14}}\sin {{7\pi } \over {14}}\sin {{9\pi } \over {14}}\sin {{11\pi } \over {14}}\sin {{13\pi } \over {14}}$ is equal to ______.
$\sin {\pi \over {14}}\sin {{3\pi } \over {14}}\sin {{5\pi } \over {14}}\sin {{7\pi } \over {14}}\sin {{9\pi } \over {14}}\sin {{11\pi } \over {14}}\sin {{13\pi } \over {14}}$ is equal to ______.
Correct Answer: $${\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle {64}$}}$$
1990
JEE Advanced
MCQ
IIT-JEE 1990
The equation $\left( {\cos p - 1} \right){x^2} + \left( {\cos p} \right)x + \sin p = 0\,$
In the variable x, has real roots. Then p can take any value in the interval
A.
$\left( {0,2\pi } \right)\,$
B.
$\left( { - \pi ,0} \right)\,\,\,$
C.
$\left[ { - {\pi \over 2},{\pi \over 2}} \right]\,$
D.
$\left( {0,\pi } \right)$
1990
JEE Advanced
Numerical
IIT-JEE 1990
$ABC$ is a triangle such that
$$\sin \left( {2A + B} \right) = \sin \left( {C - A} \right) = \, - \sin \left( {B + 2C} \right) = {1 \over 2}.$$
If $A,\,B$ and $C$ are in arithmetic progression, determine the values of $A,\,B$ and $C$.
Correct Answer: $${45^ \circ },\,\,\,{60^ \circ },\,\,\,{75^ \circ }$$
1989
JEE Advanced
MCQ
IIT-JEE 1989
The general solutions of $\,\sin x - 3\sin 2x + \sin 3x = \cos x - 3\cos 2x + \cos 3x$ is
A.
$n\pi + {\pi \over 8}$
B.
${{n\pi } \over 2} + {\pi \over 8}$
C.
${\left( { - 1} \right)^n}{{n\pi } \over 2} + {\pi \over 8}\,\,$
D.
$2n\pi + {\cos ^{ - 1}}{3 \over 2}$
1988
JEE Advanced
MCQ
IIT-JEE 1988
The value of the expression $\sqrt 3 \,\cos \,ec\,{20^0} - \sec \,{20^0}$ is equal to
A.
2
B.
$2\sin {20^0}/\sin {40^0}$
C.
4
D.
$4\sin {20^0}/\sin {40^0}$
1988
JEE Advanced
MSQ
IIT-JEE 1988
The values of $\theta $ lying between $\theta = \theta $ and $\theta = \pi /2$ and satisfying the equation
$\left| {\matrix{ {1 + {{\sin }^2}\theta } & {{{\cos }^2}\theta } & {4\sin 4\theta } \cr {{{\sin }^2}\theta } & {1 + {{\cos }^2}\theta } & {4\sin 4\theta } \cr {{{\sin }^2}\theta } & {{{\cos }^2}\theta } & {1 + 4\sin 4\theta } \cr } } \right| = 0$ are
A.
$7\pi /24$
B.
$5\pi /24$
C.
$11\pi /24$
D.
$\pi /24$
1987
JEE Advanced
MCQ
IIT-JEE 1987
The number of all possible triplets $\left( {{a_1},\,{a_2},\,{a_3}} \right)$ such that ${a_1} + {a_2}\,\,\cos \left( {2x} \right) + {a_3}{\sin ^2}\left( x \right) = 0\,$ for all $x$ is
A.
zero
B.
one
C.
three
D.
infinite
1987
JEE Advanced
Numerical
IIT-JEE 1987
The solution set of the system of equations $X + Y = {{2\pi } \over 3},$ $cox\,x + cos\,y = {3 \over 2},$ where x and y are real, is _____.
Correct Answer: $$\phi $$
1987
JEE Advanced
Numerical
IIT-JEE 1987
The set of all $x$ in the interval $\left[ {0,\,\pi } \right]$ for which $2\,{\sin ^2}x - 3$ $\sin x + 1 \ge 0,$ is _____.
Correct Answer: $$\,\left[ {0,{\pi \over 6}} \right]U\left\{ {{\pi \over 2}} \right\}U\left[ {{{s\pi } \over 6},\pi } \right]$$
1987
JEE Advanced
Numerical
IIT-JEE 1987
The sides of a triangle inscribed in a given circle subtend angles $\alpha $, $\beta $ and $\gamma $ at the centre. The minimum value of the arithmetic mean of $cos\left[ {\alpha + {\pi \over 2}} \right],\,\cos \left[ {\beta + {\pi \over 2}} \right]$ and $cos\left[ {\gamma + {\pi \over 2}} \right]$ is equal to _______.
Correct Answer: $${{ - \sqrt 3 } \over 2}$$
1986
JEE Advanced
MCQ
IIT-JEE 1986
The expression $2\left[ {{{\sin }^6}\left( {{\pi \over 2} + \alpha } \right) + {{\sin }^6}\left( {5\pi - \alpha } \right)} \right]$ is equal to
A.
0
B.
1
C.
3
D.
$\sin \,4\,\alpha + \cos \,6\,\alpha \,\,\,\,$
1984
JEE Advanced
MCQ
IIT-JEE 1984
$\left( {1 + \cos {\pi \over 8}} \right)\left( {1 + \cos {{3\pi } \over 8}} \right)\left( {1 + \cos {{5\pi } \over 8}} \right)\left( {1 + \cos {{7\pi } \over 8}} \right)$ is equal to
A.
${1 \over 2}$
B.
$\cos {\pi \over 8}$
C.
${1 \over 8}$
D.
${{1 + \sqrt 2 } \over {2\sqrt 2 }}$
1984
JEE Advanced
Numerical
IIT-JEE 1984
Find the values of $x \in \left( { - \pi , + \pi } \right)$ which satisfy the equation ${g^{(1 + \left| {\cos x} \right| + \left| {{{\cos }^2}x} \right| + \left| {{{\cos }^3}x} \right| + ...)}} = {4^3}$
Correct Answer: $$ \pm {\pi \over 3}, \pm {{2\pi } \over 3}$$
1984
JEE Advanced
MCQ
IIT-JEE 1984
There exists a value of $\theta $ between 0 and $2\pi $ that satisfies the equation $\,\,{\sin ^4}\theta - 2{\sin ^2}\theta - 1 = 0.$
A.
TRUE
B.
FALSE
1983
JEE Advanced
Numerical
IIT-JEE 1983
Show that $$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$$
Correct Answer: Solve by yourself
1983
JEE Advanced
Numerical
IIT-JEE 1983
Find all solutions of $4{\cos ^2}\,x\sin x - 2{\sin ^2}x = 3\sin x$
Correct Answer: Solve by yourself
1983
JEE Advanced
MCQ
IIT-JEE 1983
If $\tan \,A = \left( {1 - \cos B} \right)/\sin B,$ then $tan2A = tan\,B$.
A.
TRUE
B.
FALSE
1982
JEE Advanced
Numerical
IIT-JEE 1982
Without using tables, prove that $\left( {\sin \,{{12}^ \circ }} \right)\left( {\sin \,{{48}^ \circ }} \right)\left( {\sin \,{{54}^ \circ }} \right) = {1 \over 8}.$
Correct Answer: Solve it.
1981
JEE Advanced
MCQ
IIT-JEE 1981
The general solution of the trigonometric equation sin x+cos x=1 is given by:
A.
$2n\pi ;\,n = 0,\, \pm 1,\, \pm 2....$
B.
$x = 2n\pi + \pi /2;\,n = 0,\, \pm 1,\, \pm 2....$
C.
$x = n\pi + {\left( { - 1} \right)^n}\,\,\,\,\,\,\,{\pi \over 4} - {\pi \over 4}$ ; $n = 0,\, \pm 1,\, \pm 2..$
D.
none of these
1981
JEE Advanced
Numerical
IIT-JEE 1981
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 ${C_n} \ne 0$ , then the value of n is _____.
Correct Answer: 6
1980
JEE Advanced
MCQ
IIT-JEE 1980
The equation $\,2{\cos ^2}{x \over 2}{\sin ^2}x = {x^2} + {x^{ - 2}};\,0 < x \le {\pi \over 2}$ has
A.
no real solution
B.
one real solution
C.
more than one solution
D.
none of these