Question 1 : Let A and B denote the statements<br/>A: $cos\,\theta_1 + cos \,\theta_2+cos \,\theta_3 =0$<br/>B: $sin\,\theta_1 + sin\,\theta_2 + sin\,\theta_3 = 0.$<br/><div>If $cos (\theta_1 - \theta_2) + cos (\theta_2 - \theta_3) + cos (\theta_3 - \theta_1) = \displaystyle \frac{-3}{2}$</div><div>Then which one of following is correct ?<br/></div>
Question 3 :
In a triangle $ABC$, $a- 2b +c = 0$. The value of $\cot \left(\dfrac{A}{2}\right) \cot\left(\dfrac{C}{2}\right)$ is
Question 4 :
If $ A+B+C = 180^o , $ then the value of $ ( \cot B + \cot C ) ( \cot C + \cot A ) ( \cot A + \cot B ) $ will be :
Question 5 :
If $\cot { x } -\tan { x } =2$, the generalised solution is (here, $n$ is integet)
Question 6 :
An extreme value of $4sin^2x+ 3 cos^2x- 24 sin \dfrac{x}{2}-24 cos\dfrac{x}{2}, $ where $0 \leq x \leq \dfrac{\pi}{2}$, is
Question 7 :
The general solution of the equation $\sin 2x + 2\sin x + 2\cos x + 1 = 0$ is
Question 8 :
The number of real roots of the equation $\displaystyle \cos ^{2}x+\sin ^{4}x=1$ in the interval $\displaystyle (-\pi ,\pi)$ are <br/>
Question 9 :
$\dfrac{{\cot x}}{{\cot x - \cot 3x}} + \dfrac{{\tan x}}{{\tan x - \tan 3x}} = $
Question 10 :
Solve for x, $\sqrt{13 -18\tan x}=6\tan x - 3$, where $-2\pi < x <2\pi$.<br>
Question 11 :
Solve the equation : $\cot x - 2\! \sin\! 2x = 1$.<br>
Question 12 : <p>The number of solutions of the equation $8{\tan ^2}\theta + 9 = 6\sec \theta $ in the interval $(\frac{-\pi}{2}, \frac{\pi}{2})$</p>
Question 13 :
The number of solution of the equation $\sin 5x \cos 3x=\sin 6x \cos 2x$ in the interval $[0,\pi]$ is
Question 14 :
Total number of solutions of $\sin x.\tan 4x= \cos x$ belonging to $\left ( 0, \pi \right )$ are:
Question 17 :
The general solution of $\tan x - \sin x = 1 - \tan x \sin x$
Question 18 :
Let a,b and c be the three sides of a triangle, then find the number of real roots of the equation $\displaystyle b^{2}x^{2}+(b^{2}+c^{2}-a^{2})x+c^{2}= 0 $<br/><br/>
Question 19 :
If $\displaystyle x\cos A=1$ and $\displaystyle y=\tan A,$ then $\displaystyle x^{2}-y^{2}$ is<br>
Question 20 :
If $\sin { A } +\cos { A } =1$, then $\sin { 2A }$ is equal to.
Question 22 :
If $\displaystyle \sin x+\sin ^{2}x=1$ then the value of $\displaystyle \cos ^{8}x+2\cos ^{6}x+\cos ^{4}x$ is
Question 23 :
Find all values of $\theta$ in the interval $(-\pi/2, \pi/2)$ satisfying the equation $(1 - tan \theta) (1 + tan \theta) sec^2 \theta + 2 ^{tan^2 \theta} = 0$
Question 24 :
If $ \sin { \theta } +\cos { \theta }=p$ and $ \tan { \theta +\cot { \theta =q } } $ then $q\left (p^{2}-1\right )=$<span><br></span>
Question 26 :
Solve $\displaystyle \cos ^{7}x+\sin ^{4}x=1$ in the interval $\displaystyle \left ( -\pi ,\pi \right )$<br>
Question 27 :
If $2\sin \alpha +3\cos \alpha =2$, then $3\sin \alpha -2\cos \alpha=$______?
Question 28 :
The value of $\tan 1^0 \tan 2^0 \tan 3^0....\tan 89^0$ is<br/>
Question 29 :
The real roots of the equation $cos^7 x + sin^4 x = 1$ in the interval $(-\pi, \pi)$ are ..... and ...........
Question 30 :
Principal solutions of the equation $\sin 2x+\cos2x=0$, where $\pi < x < 2\pi$ are
Question 32 :
If $ f\left ( x \right )= \sin x, \:g\left ( x \right )= 1+x^{3} $, then $ \left ( fog \right )\:\left ( x \right ) $ equals
Question 33 :
Which of the following can be the solution $\sqrt {\sin (1-x)}=\sqrt {\cos x}$ for $x\in [0, 2\pi]$
Question 34 :
The value of $\tan 27 \tan 31 + \tan 32 \tan 31 +\tan 31 \tan 27 $ is
Question 35 :
<span>If $x \, \varepsilon \ $</span><span>$\displaystyle [0,2\pi]$, then t</span>he equation $|sin x | = sin x +3$ has
Question 36 :
What is sin A cos A tan A + cos A sin A cot A<span> equal to ?</span>
Question 37 :
If $\cos\alpha + \cos\beta =0= \sin\alpha + \sin\beta,$ then $\cos2\alpha + \cos2\beta$ is equal to-
Question 38 :
The possible values of $x$, which satisfy the trigonometric equation $\tan^{-1}\left (\dfrac {x - 1}{x - 2}\right ) + \tan^{-1}\left (\dfrac {x + 1}{x + 2}\right ) =\dfrac {\pi}{4}$ are
Question 39 :
The interval for which $2\tan ^{ -1 }{ x } +\sin ^{ -1 }{ \dfrac { 2 }{ 1+x } } $ is independent of $x$ is
Question 42 :
When n is an odd natural number other than 1, then the value of x is
Question 43 :
The value of $ \displaystyle \cos ^{4}\frac{\pi }{4}-\cos ^{4}\frac{\pi }{6}+\sin ^{4}\frac{\pi }{6}+\sin ^{4}\frac{4\pi }{3} $ is
Question 47 :
What is the simplified value of ${ \left( \cfrac { \sec { A } }{ \cot { A } +\tan { A } } \right) }^{ 2 }$
Question 49 :
Solve the equation $\displaystyle 2^{\cos x} 3^{\sin y} = z^2 - 2z + 7$.<br/>Then find the value of $x+y+z$
Question 50 :
Assume that $\theta$ is rational multiple of $\pi$ such that $\displaystyle \cos \theta $ is a distinct rational. The number of values of $\displaystyle \cos \theta $ is<br>
Question 51 :
If $\alpha, \beta$ are solutions of $\sin^{2}\mathrm{x}+ a\sin x +\mathrm{b}=0$ and $\cos^{2}\mathrm{x}+ c\cos x +\mathrm{d}=0$ then $\sin(\alpha+\beta)$<br/>equals<br/>
Question 52 :
The most general solutions of the equation $\sec x-1=\left ( \sqrt{2}-1 \right )\tan x$ are given by
Question 54 :
Solve for $x, (- \pi \leq x \leq \pi)$, the equation $2 (cos x + cos 2x) + sin 2x (1 + 2 cos x) = 2 sin x$
Question 55 :
If $\cos x= \tan y, \cot y = \tan z$ and $\cot z = \tan x$; then $\sin x = $
Question 56 :
If $\alpha, \beta$ are acute angles such that $(\alpha+\beta)$ and $(\alpha-\beta)$ satisfy the equation $\tan^{2} \theta -4 \tan \theta +1=0,$ then:<br/>
Question 57 :
If $\sin\alpha =p, |p|\leq 1$ then the quadratic equation whose roots are $\displaystyle \tan\frac{\alpha }{2}$ and $\displaystyle \cot \frac{\alpha}{2}$ is:
Question 59 :
The solution of the equation $\displaystyle \sin ^{4}x+\cos ^{7}x= 1$ is given by $\left ( \forall \: k \: \epsilon\: l \right )$
Question 61 :
The general solution of the equation ${\sin ^{100}}x - {\cos ^{100}}x = 1$ is-
Question 62 :
If $\sin^{-1}\left ( \tan\dfrac{\pi}{4} \right )-\sin^{-1}\sqrt{\dfrac{3}{x}}=\dfrac{\pi}{6}$, then $x$ is a root of the equation:<br/>
Question 64 :
If $2 \sin \alpha \cdot \cos \beta \cdot \sin \gamma = \sin \beta \cdot \sin (\alpha + \gamma )$, then $\tan \alpha ,\tan \beta, \tan \gamma$ are in
Question 66 :
The number of solutions of $\displaystyle \left [ \sin x+\cos x \right ]=3+\left [ -\sin x \right ]+\left [ -\cos x \right ]$(where $\displaystyle \left [ \right ]$ denotes the greatest interger function),$\displaystyle x\epsilon \left [ 0,2\pi \right ]$ is<br>
Question 67 :
The value of $x$ in $(0, \pi/2)$ satisfying $\dfrac{\sqrt3 - 1}{\sin x} + \dfrac{\sqrt3 + 1}{\cos x} = 4\sqrt2$ is
Question 68 :
If $x\in \left[ 0,\dfrac { \pi }{ 2 } \right] $, the number of solutions of the equation, $\sin { 7x } +\sin { 4x } +\sin { x=0 }$ is
Question 69 :
If $\displaystyle a<c $ in $\displaystyle \Delta ABC,$ then sum to infinite terms of the series $\displaystyle n\frac{a}{c}\sin B +\frac{n(n+1)}{2!} \frac{a^{2}}{c^{2}}\sin 2B+.....\infty$ is equal to<br>
Question 70 :
The number of solutions of the equation $\sin { x } =\cos { 3x } $ in $\left[ 0,\pi \right] $ is
Question 71 :
For which values of $a$ does the equation $4 sin(x+\dfrac{\pi }{ 3})cos(x-\dfrac{\pi }{ 6})=a^{2}+\sqrt{3}sin2x-cos2x$ have solutions? Find the solutions for $a=0$, if exist.
Question 76 :
If $sin^2A=x$then sin 3A. sin A is polynomial in x, whose degree is equal to <br/>
Question 77 :
If $\tan \alpha, \tan \beta$ satisfy (1) and $\cos \gamma, \cos \delta$ satisfy (2) then $\tan \alpha \tan \beta + \cos \gamma + \cos \delta$ can be equal to
Question 78 :
The value of $\dfrac {1}{\tan \alpha} + \dfrac {1}{\tan \beta} + \dfrac {1}{\tan \gamma} + \dfrac {1}{\tan \delta }$ is<br>
Question 79 :
$\displaystyle \frac{\sin 3 \alpha}{\cos 2 \alpha} < 0$, if $\alpha$ lies in
Question 80 :
In a $\triangle ABC,$ the angles $A$ and $B$ are two different values of $\theta$ satisfying $\sqrt { 3 } \cos { \theta } +\sin { \theta } =k,\left| k \right| <2.$ The triangle:
Question 81 :
The equation $\displaystyle 4\sin (x + \frac{\pi}{3}) \cos (x - \frac{\pi}{6}) = a^2 + \sqrt{3} \sin 2 x - \cos 2x$ has a solution if the value of a is
Question 82 :
Consider the cubic equation ${ x }^{ 3 }-\left( 1+\cos { \theta } +\sin { \theta } \right) { x }^{ 2 }+\left( \cos { \theta } \sin { \theta } +\cos { \theta } +\sin { \theta } \right) x-\sin { \theta } \cos { \theta } =0$<br>whose roots are ${ x }_{ 1 },{ x }_{ 2 },{ x }_{ 3 }$<br>The greatest possible difference between two of the roots if $\displaystyle \theta \epsilon \left [ 0,2\pi \right ]$ is<br><br>
Question 83 :
For $X\in [-2\pi ,3\pi ] \:and\:y \in R$ the number of ordered pair satisfying equation $(\sqrt{3}\sin x-\cos x-3y^2+6y-5)=0$ will be<br/>
Question 84 :
The most general solution of $\displaystyle 2^{1+\left | \cos x \right |+\cos^{2}x+\left | \cos x\right |^{3}+\cdots \infty}=4$ are given by
Question 85 :
$cos \alpha + cos \beta + cos \gamma$ can be equal to
Question 86 :
The number of solutions of the equation, $\displaystyle \frac{\tan 3x}{\tan x}=2$ is
Question 87 :
<br/>If $\tan\alpha,\tan\beta$ are the roots of the equation $x^{2}+px+q=0(p\neq 0)$ then<br/>$\sin^{2}(\alpha+\beta)+p\sin(\alpha+\beta)\cos(\alpha+\beta)+q\cos^{2}(\alpha+\beta)=$<br/>
Question 88 :
Solve : $\cos(\pi .\: 3^x)-2\cos^2(\pi .\: 3^x)+2\cos(4\pi .\: 3^x)-\cos(7\pi .\: 3^x)$<br> $\sin (\pi .\: 3^x)+2\sin ^2(\pi .\: 3^x)-2\sin (4\pi .\: 3^x)+2\sin(\pi .\: 3^{x+1})-\sin (7\pi .\: 3^x)$<br>
Question 89 :
In a triangle $ABC$, $\angle B < \angle C$ and the values of $B$ & $C$ satisfy the equation $2 tan x-k (1+tan^2x)=0$ where $(0 < k < 1)$. Then the measure of $\angle A$ is:
Question 90 :
Solve for x : $\sin3\alpha = 4\! \sin\! \alpha \! \sin(x + \alpha ) \:\sin(x - \alpha )$ where $\alpha$ is a constant $\neq n\pi , \:n\in I$.<br>
Question 91 :
The most general value of $\theta$ satisfying $3-2 \cos\theta -4 \sin\theta -\cos2\theta+\sin 2\theta=0$:
Question 92 :
If $(cosec^{2}\theta -4)x^{2}+(cot\theta +\sqrt{3})x+cos^{2}\frac{3\pi}{2}=0$ holds true for all real x, then the most general values of $\theta$ can be given by $(n\in z)$
Question 93 :
If $\sin^{2}x-2\sin x - 1=0$has exactly four different solution in $x\in [0, n\pi]$, then value / values of n is / are $(n \in N)$
Question 94 :
If $4\sin ^{4}x+\cos ^{4}x=1$,then x is equal to $\left ( n\in Z \right )$
Question 96 :
For the smallest positive values of $x$ and $y$, the equation $2 ( \sin x + \sin y) - 2 \cos (x - y) = 3$ has a solution, then which of the following is/are true?
Question 97 :
If $\cos { \alpha } +\cos { \beta } +\cos { \gamma } =\sin { \alpha } +\sin { \beta } +\sin { \gamma } =0$, then the value of $\cos { 3\alpha } +\cos { 3\beta } +\cos { 3\gamma } $ is
Question 98 :
If $f(x)=\sin x+\sin x\cos {\cfrac{x}{2}}+\sin x\cos ^{ 2 }{ \cfrac { x }{ 2 } } +....$, and $g(x)=f(x).\sin {\cfrac{x}{4}}-\cos {\cfrac{x}{4}}$, then-
Question 99 :
The number of solutions of the equation $\left | \sin x \right |= \left | \cos 3x \right |$ in $\left [ -2\pi , 2\pi \right ]$ is:
Question 100 :
The value of $\sin \left (\dfrac {\pi}{18}\right ) \sin \left (\dfrac {5\pi}{18}\right )\sin \left (\dfrac {7\pi}{18}\right )$ is:
