Question 1 :
<br/>If $a \sin^{2}\theta+b\cos^{2}\theta=a\cos^{2}\phi+b\sin^{2}\phi=1$ and $a \tan\theta=b\tan\phi$, then choose the correct option.<br/>
Question 3 :
The function $f:\left [-\displaystyle \frac{1}{2},\:\displaystyle \frac{1}{2} \right ]\rightarrow \left [-\displaystyle \frac{\pi }{2},\:\displaystyle \frac{\pi}{2} \right ] $ defined by$ \sin^{-1}\left ( 3x-4x^{3} \right ) $ is
Question 4 :
Let $\displaystyle -\frac { \pi }{ 6 } <\theta <-\frac { \pi }{ 12 }$, Suppose$\displaystyle { \alpha }_{ 1 }$ and$\displaystyle { \beta }_{ 1 }$ are the roots of the equation$\displaystyle { x }^{ 2 }-2x\sec { \theta } +1=0$ and$\displaystyle { \alpha }_{ 2 }$ and $\displaystyle { \beta }_{ 2 }$ are the roots of the equation$\displaystyle { x }^{ 2 }+2x\tan { \theta } -1=0$. If$\displaystyle { \alpha }_{ 1 }>{ \beta }_{ 1 }$ and$\displaystyle { \alpha }_{ 2 }>{ \beta }_{ 2 }$, then$\displaystyle { \alpha }_{ 1 }+{ \beta }_{ 2 }$ equals to
Question 6 :
In a $\Delta ABC$, if $\cos A \cos B \cos C=\displaystyle\dfrac {\sqrt 3-1}{8}$ and $\sin A. \sin B. \sin C=\displaystyle \dfrac {3+\sqrt 3}{8}$, <br/><br/>then- On the basis of above information, answer the following questions:The value of $ \tan A \tan B + \tan B \tan C + \tan C \tan A$ is:
Question 8 :
If $\displaystyle\frac{\cos^{4}x }{\theta _{1}}+\displaystyle\frac{\sin^{4}x}{\theta _{2}}=\frac{1}{\theta _{1}+\theta _{2}},$ then $\displaystyle\frac{\theta _{2}}{\theta _{1}}$ equals<br>
Question 9 :
In a $\Delta ABC$, if $\cos A \cos B \cos C=\displaystyle\dfrac {\sqrt 3-1}{8}$ and $\sin A. \sin B. \sin C=\displaystyle \dfrac {3+\sqrt 3}{8}$, then- On the basis of above information, answer the following questions:The angles of $\Delta ABC$ are:<br/>
Question 10 :
Let $\displaystyle 0\leq \theta \leq \frac{\pi}{2}$ and $\displaystyle x=Xcos \: \theta +Ysin\: \theta ,y=Xsin\: \theta -Ycos\: \theta $ such that<br>$\displaystyle x^{2}+4xy+y^{2}=aX^{2}+bY^{2},$ where $\displaystyle a,b$ are constants. Then<br>
Question 11 :
Assertion: Statement 1:If $\displaystyle x+y+z= xyz,$ then at most one of the numbers can be negative.
Reason: Statement 2: In a triangle ABC, $\displaystyle \tan A+\tan B+\tan C= \tan A \tan B \tan C $ ,there can be at most one obtuse angle in a triangle.
Question 12 :
Let $x=(1+\sin A)(1-\sin B)(1+\sin C), y=(1-\sin A)(1-\sin B)(1-\sin C)$ and if $x=y$, then
Question 13 :
What is $\left(\dfrac{sec 18^{\circ}}{sec 144^{\circ}} + \dfrac{cosec 18^{\circ}}{cosec 144^{\circ}}\right)$ equals to?
Question 14 :
<br/>Given that $\sec\theta+\tan\theta=1$ then one root of the equation $(a-2b+c)x^{2}+(b-2c+a)x+(c-2a+b)= 0$ is<br/>
Question 15 :
If $ABCD$ is a cyclic quadrilateral such that $12$ $\tan A-5=0$ and 5 $\cos B+3=0$, then $\cos C\tan D=$<br/>
Question 16 :
If $cosec \theta -\sin \theta =m$ and $\sec \theta -\cos \theta =n$, eliminate $\theta $.<br><br>
Question 17 :
If $\displaystyle \frac { \sin ^{ 4 }{ x }  }{ 2 } +\frac { \cos ^{ 4 }{ x }  }{ 3 } =\frac { 1 }{ 5 } ,$ then:
Question 19 :
In $\triangle ABC$, the measure of $\angle B$ is $90^{\circ}, BC = 16$, and $AC = 20$. $\triangle DEF$ is similar to $\triangle ABC$, where vertices $D, E,$ and $F$ correspond to vertices. $A, B$, and $C$, respectively, and each side of $\triangle DEF$ is $\dfrac {1}{3}$ the length of the corresponding side of $\triangle ABC$. What is the value of $\sin F$?
Question 20 :
If$\displaystyle \cos \theta =\frac{3}{5},$ then the value of$\displaystyle \frac{\sin \theta -\tan \theta +1}{2\tan ^{2}\theta }$
Question 22 :
Assertion: For any real value of $\theta\neq (2n+1)\pi $ or $(2n + 1)\pi/2$, $n \in I$, the value of the expression $\displaystyle y=\frac{\cos ^2\theta -1}{\cos ^2\theta+\cos \theta}$ is $y \leq 0$ or $y \geq 2$ (either less than or equal to zero or greater than or equal to two)<br>Because
Reason: $\sec \theta \in (-\infty ,-1] \cup [1,\infty)$ for all real values of $\theta $.
Question 23 :
Find the relation obtained by eliminating$\displaystyle \theta $ from the equation$\displaystyle x=a\cos \theta +b\sin \theta $ and$\displaystyle y=a\sin \theta -b\cos \theta $
Question 24 :
In $\displaystyle A_{n}=\cos^{n}\theta+\sin^{n}\theta, n\in N$ and $\displaystyle \theta \in R$<br/><br/>If $\displaystyle A_{n-4}-A_{n-2}=\sin^{2}\theta\cos^{2}\theta A_{\lambda} $ , then $\displaystyle \lambda $ equals<br/>
Question 25 :
If the quadratic equation $ax^2+bx+c=0$ ($a > 0$) has $\sec^2\theta$ and $\text{cosec}^2\theta$ as its roots, then which of the following must hold good?<br>
Question 26 :
If $ \cos^{-1}\left ( 4x^{3}-3x \right )= 2\pi -3\cos^{-1}x $, then $ x $ lies in interval
Question 27 :
If $\displaystyle \frac{\sin x}{a}= \frac{\cos x}{b}= \frac{\tan x}{c}= k,$ then $\displaystyle bc+\frac{1}{ck}+\frac{ak}{1+bk} $ is equal to<br><br><br>
Question 28 :
If $5\cos { A } =4\sin { A } $, then $\tan { A=\_ \_ \_ } $
Question 29 :
If $0\leq x, y\leq 180^o$ and $\sin (x-y)=\cos(x+y)=\dfrac 12$, then the values of $x$ and $y$ are given by
Question 30 :
If $\sin\theta + \sin^{2}\theta = 1$, then $\cos^{2}\theta + \cos^{4}\theta = ......$
Question 33 :
In a triangle $ABC$, right angled at $C$, $a$, $b$ $c$ are the lengths of sides of triangle and hypotenuse respectively. Find the value of $\tan A+\tan B$.
Question 34 :
If $\sin x+\sin ^{2}x=1$,thenthe value of $\cos ^{12}x+3\cos ^{10}x+3\cos ^{8}x+\cos ^{6}x-2$ is equal to
Question 35 :
If $\sin (\alpha+\beta)=1$ and $\sin(\alpha -\beta)=1/2$ where $\alpha, \beta \epsilon [0, \pi /2]$ then
Question 36 :
$1)$ lf $\mathrm{x}$ lies in the lst quadrant and<br/>$\cos \mathrm{x}+\cos 3\mathrm{x}=\cos 2\mathrm{x}$ then $\mathrm{x}=30^{\mathrm{o}}$ or $45^{\mathrm{o}}$<br/>$2)\mathrm{x}\in(0,2\pi)$ and cosec $\mathrm{x}+2=0$ then $x=\displaystyle \frac{7\pi}{6},\frac{l1\pi}{6}$<br/>$3)\mathrm{x}\in[0,2\pi]$ and $(2 \cos \mathrm{x}- \mathrm{l}) (3+2\cos \mathrm{x})=0$ then $\displaystyle \mathrm{x}=\frac{\pi}{3}$ , $\displaystyle \frac{5\pi}{3}$ Which of the above statements are correct?<br/>
Question 38 :
The value of the expression $\displaystyle 1\, - \,\frac{{{{\sin }^2}y}}{{1\, + \cos \,y}}\, + \frac{{1\, + \cos \,y}}{{\sin \,y}}\, - \,\frac{{\sin \,y}}{{1\, - \cos \,y}}$ is equal to 
Question 39 :
In $\triangle ABC, \angle B = 90^{\circ}, BC = 7$ and $AC - AB = 1$, then $\cos C = .....$
Question 40 :
If $\sin A, \cos A$ and $\tan A$ are in G.P. then $\cot^6 A- \cot^2A$ is equal to
Question 41 :
The value of $ \cos y \cos\left(\dfrac{\pi}{2} -x\right) - \cos \left(\dfrac{\pi}{2}-y \right)\cos x + \sin y \cos\left(\dfrac{\pi}{2}-x\right)+ \cos x \sin\left(\dfrac{\pi}{2} -y\right)$ is zero if
Question 42 :
${\cos ^2}{48^ \circ } - {\sin ^2}{12^ \circ }$ is equal to -
Question 46 :
The number of ordered pairs $(\alpha, \beta)$, where $\alpha, \beta $ $\in$ $(-\pi, \pi)$ satisfying $\cos(\alpha -\beta)=1$ and $\cos(\alpha+\beta)=\dfrac {1}{e}$ is
Question 47 :
If $\sin x + \sin^{2}x=1,$ then the value of $\cos^{12} x + 3 \cos^{10} x + 3 cos^{8} x + cos ^{6} x -1$ is equal to :
Question 48 :
If $\cos9 \alpha= \sin \alpha$ and $9 \alpha < 90^{0}$, then the value of $\tan5 \alpha$ is<br/>
Question 50 :
If$\displaystyle \sin \Theta =\frac{3}{5} $ and$\displaystyle \Theta $ is acute then find the value of$\displaystyle \frac{\tan \Theta -2\cos \Theta }{3\sin \Theta +\sec \Theta }$
