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A Complete Resource Book in Mathematics for JEE Main

Dinesh Khattar

Chapter 25

Trigonometric Ratios and Identities - all with Video Answers

Educators

Chapter Questions

02:38

Problem 1

The value of $\cos$ a $\cos 2 a \cos 3 a \ldots . \cos 999 a$, where
$a=\frac{2 \pi}{1999}$, is
(A) $\frac{1}{2^{99}}$
(B) $\frac{1}{2^{999}}$
(C) $\frac{1}{2^{9999}}$
(D) $\frac{1}{2^{1999}}$

Aman Gupta
Aman Gupta
Numerade Educator
02:38

Problem 1

The value of $\cos$ a $\cos 2 a \cos 3 a \ldots . \cos 999 a$, where
$a=\frac{2 \pi}{1999}$, is
(A) $\frac{1}{2^{99}}$
(B) $\frac{1}{2^{999}}$
(C) $\frac{1}{2^{9999}}$
(D) $\frac{1}{2^{1999}}$

Aman Gupta
Aman Gupta
Numerade Educator
01:48

Problem 2

Let $a_{1}=\left(\tan \frac{\pi}{8}\right)^{\tan \frac{\pi}{8}}, a_{2}=\left(\tan \frac{\pi}{8}\right)^{\cot \frac{\pi}{8}}$,
$a_{3}=\left(\cot \frac{\pi}{8}\right)^{\tan \frac{\pi}{8}}, a_{4}=\left(\cot \frac{\pi}{8}\right)^{\cot \frac{\pi}{8}}$ Then,
(A) $a_{4}>a_{3}>a_{2}>a_{1}$
(B) $a_{3}>a_{4}>a_{2}>a_{1}$
(C) $a_{4}>a_{3}>a_{1}>a_{2}$
(D) $a_{3}>a_{1}>a_{2}>a_{4}$

Aman Gupta
Aman Gupta
Numerade Educator
01:48

Problem 2

Let $a_{1}=\left(\tan \frac{\pi}{8}\right)^{\tan \frac{\pi}{8}}, a_{2}=\left(\tan \frac{\pi}{8}\right)^{\cos \frac{\pi}{8}}$,
$a_{3}=\left(\cot \frac{\pi}{8}\right)^{\tan \frac{\pi}{8}}, a_{4}=\left(\cot \frac{\pi}{8}\right)^{\cos \frac{\pi}{8}}$ Then,
(A) $a_{4}>a_{3}>a_{2}>a_{1}$
(B) $a_{3}>a_{4}>a_{2}>a_{1}$
(C) $a_{4}>a_{3}>a_{1}>a_{2}$
(D) $a_{3}>a_{1}>a_{2}>a_{4}$

Aman Gupta
Aman Gupta
Numerade Educator
01:46

Problem 3

If $x \cos ^{2} 3 \theta+y \cos ^{4} \theta=16 \cos ^{6} \theta+9 \cos ^{2} \theta$ be an iden-
tity, then
(A) $x=-1, y=24$
(B) $x=1, y=24$
(C) $x=24, y=1$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
01:46

Problem 3

If $x \cos ^{2} 3 \theta+y \cos ^{4} \theta=16 \cos ^{6} \theta+9 \cos ^{2} \theta$ be an iden-
tity, then
(A) $x=-1, y=24$
(B) $x=1, y=24$
(C) $x=24, y=1$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
01:03

Problem 4

$|\tan \theta+\sec \theta|=|\tan \theta|+|\sec \theta|, 0 \leq \theta \leq 2 \pi$ is possible
only if
(A) $\theta \in[0, \pi]-\left\{\frac{\pi}{2}\right\}$
(B) $\theta \in[0, \pi]$
(C) $\theta \in\left[0, \frac{\pi}{2}\right)$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
01:03

Problem 4

$|\tan \theta+\sec \theta|=|\tan \theta|+|\sec \theta|, 0 \leq \theta \leq 2 \pi$ is possible
only if
(A) $\theta \in[0, \pi]-\left\{\frac{\pi}{2}\right\}$
(B) $\theta \in[0, \pi]$
(C) $\theta \in\left[0, \frac{\pi}{2}\right)$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
01:57

Problem 5

If $\sin \theta, \sin \phi$ and $\cos \theta$ are in G.P., then the roots of the equation $x^{2}+2 x \cot \phi+1=0$ are always
(A) real
(B) imaginary
(C) equal
(D) greater than 1

Aman Gupta
Aman Gupta
Numerade Educator
01:57

Problem 5

If $\sin \theta, \sin \phi$ and $\cos \theta$ are in G.P., then the roots of the equation $x^{2}+2 x \cot \phi+1=0$ are always
(A) real
(B) imaginary
(C) equal
(D) greater than 1

Aman Gupta
Aman Gupta
Numerade Educator
01:31

Problem 6

If $\cos 25^{\circ}+\sin 25^{\circ}=k$, then $\cos 50^{\circ}$ is equal to
(A) $k \sqrt{2-k^{2}}$
(B) $-\sqrt{2-k^{2}}$
(C) $\sqrt{2-k^{2}}$
(D) $-k \sqrt{2-k^{2}}$

Aman Gupta
Aman Gupta
Numerade Educator
01:31

Problem 6

If $\cos 25^{\circ}+\sin 25^{\circ}=k$, then $\cos 50^{\circ}$ is equal to
(A) $k \sqrt{2-k^{2}}$
(B) $-\sqrt{2-k^{2}}$
(C) $\sqrt{2-k^{2}}$
(D) $-k \sqrt{2-k^{2}}$

Aman Gupta
Aman Gupta
Numerade Educator
01:46

Problem 7

If $\frac{2 \sin \alpha}{1+\cos \alpha+\sin \alpha}=x$ then $\frac{1-\cos \alpha+\sin \alpha}{1+\sin \alpha}=\ldots$
(A) $\frac{1}{x}$
(B) $x$
(C) $1-x$
(D) $1+x$

Aman Gupta
Aman Gupta
Numerade Educator
01:46

Problem 7

If $\frac{2 \sin \alpha}{1+\cos \alpha+\sin \alpha}=x$ then $\frac{1-\cos \alpha+\sin \alpha}{1+\sin \alpha}=\ldots$
(A) $\frac{1}{x}$
(B) $x$
(C) $1-x$
(D) $1+x$

Aman Gupta
Aman Gupta
Numerade Educator
02:06

Problem 8

If $e^{-\pi 2}<\theta<\pi / 2$, then
(A) $\cos \log \theta<\log \cos \theta$
(B) $\cos \log \theta>\log \cos \theta$
(C) $\cos \log \theta \leq \log \cos \theta$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
02:06

Problem 8

If $e^{-\pi 2}<\theta<\pi / 2$, then
(A) $\cos \log \theta<\log \cos \theta$
(B) $\cos \log \theta>\log \cos \theta$
(C) $\cos \log \theta \leq \log \cos \theta$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
01:05

Problem 9

$\sin \theta=\frac{1}{2}\left(\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}\right)$ necessarily implies
(A) $x>y$
(B) $x<y$
(C) $x=y$
(D) both $x$ and $y$ are purely imaginary

Aman Gupta
Aman Gupta
Numerade Educator
01:05

Problem 9

$\sin \theta=\frac{1}{2}\left(\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}\right)$ necessarily implies
(A) $x>y$
(B) $x<y$
(C) $x=y$
(D) both $x$ and $y$ are purely imaginary

Aman Gupta
Aman Gupta
Numerade Educator
02:07

Problem 10

If $x y+y z+z x=1$, then $\sum \frac{x+y}{1-x y}=$
(A) $\frac{4}{x y z}$
(B) $\frac{1}{x y z}$
(C) $x y z$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
02:07

Problem 10

If $x y+y z+z x=1$, then $\sum \frac{x+y}{1-x y}=$
(A) $\frac{4}{x y z}$
(B) $\frac{1}{x y z}$
(C) $x y z$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
02:14

Problem 11

$\cos 12^{\circ} \cos 24^{\circ} \cos 36^{\circ} \cos 48^{\circ} \cos 72^{\circ} \cos 96^{\circ}$ equals
(A) $-\frac{1}{2^{6}}$
(B) $\frac{1}{2^{8}}$
(C) $\frac{1}{2^{7}}$
(D) $-\frac{1}{2^{7}}$

Aman Gupta
Aman Gupta
Numerade Educator
02:14

Problem 11

$\cos 12^{\circ} \cos 24^{\circ} \cos 36^{\circ} \cos 48^{\circ} \cos 72^{\circ} \cos 96^{\circ}$ equals
(A) $-\frac{1}{2^{6}}$
(B) $\frac{1}{2^{8}}$
(C) $\frac{1}{2^{7}}$
(D) $-\frac{1}{2^{7}}$

Aman Gupta
Aman Gupta
Numerade Educator
02:25

Problem 12

If $\alpha, \beta, \gamma \in\left(0, \frac{\pi}{2}\right)$, then $\frac{\sin (\alpha+\beta+\gamma)}{\sin \alpha+\sin \beta+\sin \gamma}$ is
$(\mathrm{A})<1$
$(\mathrm{B})>1$
$(\mathrm{C})=1$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
02:25

Problem 12

If $\alpha, \beta, \gamma \in\left(0, \frac{\pi}{2}\right)$, then $\frac{\sin (\alpha+\beta+\gamma)}{\sin \alpha+\sin \beta+\sin \gamma}$ is
$(\mathrm{A})<1$
(B) $>1$
$(\mathrm{C})=1$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
02:01

Problem 13

If $\left|\cos \theta\left\{\sin \theta+\sqrt{\sin ^{2} \theta+\sin ^{2} \alpha}\right\}\right| \leq k$, then the value
of $k$ is
(A) $\sqrt{1+\cos ^{2} \alpha}$
(B) $\sqrt{1+\sin ^{2} \alpha}$
(C) $\sqrt{2+\sin ^{2} \alpha}$
(D) $\sqrt{2+\cos ^{2} \alpha}$

Aman Gupta
Aman Gupta
Numerade Educator
02:01

Problem 13

If $\left|\cos \theta\left\{\sin \theta+\sqrt{\sin ^{2} \theta+\sin ^{2} \alpha}\right\}\right| \leq k$, then the value
of $k$ is
(A) $\sqrt{1+\cos ^{2} \alpha}$
(B) $\sqrt{1+\sin ^{2} \alpha}$
(C) $\sqrt{2+\sin ^{2} \alpha}$
(D) $\sqrt{2+\cos ^{2} \alpha}$

Aman Gupta
Aman Gupta
Numerade Educator
02:14

Problem 14

The maximum value of $\left(\cos \alpha_{1}\right)\left(\cos \alpha_{2}\right) \ldots_{\pi} \cdot(\cos a n)$
under the restrictions $0 \leq \alpha_{1}, \alpha_{2}, \ldots, \alpha_{n} \leq \frac{\pi}{2}$ and (cot $\left.\alpha_{1}\right)\left(\cot \alpha_{2}\right) \ldots\left(\cot \alpha_{n}\right)=1$ is
(A) $\frac{1}{2^{n / 2}}$
(B) $\frac{1}{2^{n}}$
(C) $\frac{1}{2 n}$
(D) 1

Aman Gupta
Aman Gupta
Numerade Educator
02:14

Problem 14

The maximum value of $\left(\cos \alpha_{1}\right)\left(\cos \alpha_{2}\right) \cdot \dot{\pi} .(\cos a n)$
under the restrictions $0 \leq \alpha_{1}, \alpha_{2}, \ldots, \alpha_{n} \leq \frac{\pi}{2}$ and (cot $\left.\alpha_{1}\right)\left(\cot \alpha_{2}\right) \ldots\left(\cot \alpha_{n}\right)=1$ is
(A) $\frac{1}{2^{n / 2}}$
(B) $\frac{1}{2^{n}}$
(C) $\frac{1}{2 n}$
(D) 1

Aman Gupta
Aman Gupta
Numerade Educator
02:29

Problem 15

The inequality $2^{\sin \theta}+2^{\cos \theta} \geq 2^{\left(1-\frac{1}{\sqrt{2}}\right)}$ holds for
(A) $0 \leq \theta<\pi$
(B) $\pi \leq \theta<2 \pi$
(C) for all real $\theta$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
02:29

Problem 15

The inequality $2^{\sin \theta}+2^{\cos \theta} \geq 2^{\left(t-\frac{1}{\sqrt{2}}\right)}$ holds for
(A) $0 \leq \theta<\pi$
(B) $\pi \leq \theta<2 \pi$
(C) for all real $\theta$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
01:53

Problem 16

The expression $2^{\sin \theta}+2^{-\cos \theta}$ is minimum when $\theta$ is equal to
(A) $2 n \pi+\frac{\pi}{4}, n \in I$
(B) $2 n \pi+\frac{7 \pi}{4}, n \in I$
(C) $n \pi \pm \frac{\pi}{4}, n \in I$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
01:53

Problem 16

The expression $2^{\sin \theta}+2^{-\cos \theta}$ is minimum when $\theta$ is equal to
(A) $2 n \pi+\frac{\pi}{4}, n \in I$
(B) $2 n \pi+\frac{7 \pi}{4}, n \in I$
(C) $n \pi \pm \frac{\pi}{4}, n \in I$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
02:46

Problem 17

If $x_{n+1}=\sqrt{\frac{1+x_{n}}{2}}$, then $\cos \left(\frac{\sqrt{1-x_{0}^{2}}}{x_{1} x_{2} x_{3} \ldots \text { to } \infty}\right)\left(-1<x_{0}<1\right)$
is equal to
(A) $x_{0}$
(B) $1 / x_{0}$
(C) 1
(D) $-1$

Aman Gupta
Aman Gupta
Numerade Educator
02:46

Problem 17

If $x_{n+1}=\sqrt{\frac{1+x_{n}}{2}}$, then $\cos \left(\frac{\sqrt{1-x_{0}^{2}}}{x_{1} x_{2} x_{3} \ldots . \text { to } \infty}\right)\left(-1<x_{0}<1\right)$
is equal to
(A) $x_{0}$
(B) $1 / x_{0}$
(C) 1
(D) $-1$

Aman Gupta
Aman Gupta
Numerade Educator
01:31

Problem 18

If $0<\theta<\pi$, then
(A) $1+\cot \theta \leq \cot \frac{\theta}{2}$
(B) $1+\cot \theta \geq \cot \frac{\theta}{2}$
(C) $1+\cot \frac{\theta}{2} \geq \cot \theta$
(D) $1+\cot \frac{\theta}{2} \leq \cot \theta$

Aman Gupta
Aman Gupta
Numerade Educator
01:31

Problem 18

If $0<\theta<\pi$, then
(A) $1+\cot \theta \leq \cot \frac{\theta}{2}$
(B) $1+\cot \theta \geq \cot \frac{\theta}{2}$
(C) $1+\cot \frac{\theta}{2} \geq \cot \theta$
(D) $1+\cot \frac{\theta}{2} \leq \cot \theta$

Aman Gupta
Aman Gupta
Numerade Educator
02:45

Problem 19

If $\cos (\theta-\alpha)=a$ and $\sin (\theta-\beta)=b(0<\theta-\alpha, \theta-\beta<$
$\pi / 2$ ), then $\cos ^{2}(\alpha-\beta)+2 a b \sin (\alpha-\beta)$ is equal to
(A) $a^{2}-b^{2}$
(B) $a^{2}+b^{2}$
(C) $2 a^{2} b^{2}$
(D) $a^{2} b^{2}$

Aman Gupta
Aman Gupta
Numerade Educator
02:45

Problem 19

If $\cos (\theta-\alpha)=a$ and $\sin (\theta-\beta)=b(0<\theta-\alpha, \theta-\beta<$
$\pi / 2$ ), then $\cos ^{2}(\alpha-\beta)+2 a b \sin (\alpha-\beta)$ is equal to
(A) $a^{2}-b^{2}$
(B) $a^{2}+b^{2}$
(C) $2 a^{2} b^{2}$
(D) $a^{2} b^{2}$

Aman Gupta
Aman Gupta
Numerade Educator
02:05

Problem 20

If in the triangle $A B C, \tan \frac{A}{2}, \tan \frac{B}{2}$ and $\tan \frac{C}{2}$ are in
harmonic progression, then the least value of $\cot \frac{B}{2}$ is
(A) $\sqrt{2}$
(B) $\sqrt{3}$
(C) 2
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
02:05

Problem 20

If in the triangle $A B C, \tan \frac{A}{2}, \tan \frac{B}{2}$ and $\tan \frac{C}{2}$ are in
harmonic progression, then the least value of $\cot \frac{B}{2}$ is
(A) $\sqrt{2}$
(B) $\sqrt{3}$
(C) 2
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
02:24

Problem 21

If $x \sin a+y \sin 2 a+z \sin 3 a=\sin 4 a x \sin b+$
$y \sin 2 b+z \sin 3 b=\sin 4 b x \sin c+y \sin 2 c+z$
$\sin 3 c=\sin 4 c$, then the roots of the equation $t^{3}-\frac{z}{2} t^{2}-\frac{y+2}{4}+\frac{z-x}{8}=0 ; a, b, c \neq n \pi$, are
(A) $\sin a, \sin b, \sin c$
(B) $\cos a, \cos b, \cos c$
(C) $\sin 2 a, \sin 2 b, \sin 2 c$
(D) $\cos 2 a, \cos 2 b, \cos 2 c$

Aman Gupta
Aman Gupta
Numerade Educator
02:24

Problem 21

If $x \sin a+y \sin 2 a+z \sin 3 a=\sin 4 a x \sin b+$
$y \sin 2 b+z \sin 3 b=\sin 4 b x \sin c+y \sin 2 c+z$
$\sin 3 c=\sin 4 c$, then the roots of the equation $t^{3}-\frac{z}{2} t^{2}-\frac{y+2}{4}+\frac{z-x}{8}=0 ; a, b, c \neq n \pi$, are
(A) $\sin a, \sin b, \sin c$
(B) $\cos a, \cos b, \cos c$
(C) $\sin 2 a, \sin 2 b, \sin 2 c$
(D) $\cos 2 a, \cos 2 b, \cos 2 c$

Aman Gupta
Aman Gupta
Numerade Educator
01:10

Problem 22

If $a \sin x+b \cos (x+\theta)+b \cos (x-\theta)=d$, then the
minimum value of $|\cos \theta|$ is
(A) $\frac{1}{2|a|} \sqrt{d^{2}-a^{2}}$
(B) $\frac{1}{2|b|} \sqrt{d^{2}-a^{2}}$
(C) $\frac{1}{2|b|} \sqrt{a^{2}-d^{2}}$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
01:10

Problem 22

If $a \sin x+b \cos (x+\theta)+b \cos (x-\theta)=d$, then the
minimum value of $|\cos \theta|$ is
(A) $\frac{1}{2|a|} \sqrt{d^{2}-a^{2}}$
(B) $\frac{1}{2|b|} \sqrt{d^{2}-a^{2}}$
(C) $\frac{1}{2|b|} \sqrt{a^{2}-d^{2}}$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
02:43

Problem 23

If $\sin \theta+\cos \theta=\frac{\sqrt{7}}{2}$ and $0<\theta<\pi / 6$, then $\tan \left(\frac{\theta}{2}\right)$
equals
(A) $\sqrt{7}-2$
(B) $\frac{1}{3}(\sqrt{7}-2)$
(C) $2-\sqrt{7}$
(D) $\frac{1}{3}(2-\sqrt{7})$

Aman Gupta
Aman Gupta
Numerade Educator
02:43

Problem 23

If $\sin \theta+\cos \theta=\frac{\sqrt{7}}{2}$ and $0<\theta<\pi / 6$, then $\tan \left(\frac{\theta}{2}\right)$
equals
(A) $\sqrt{7}-2$
(B) $\frac{1}{3}(\sqrt{7}-2)$
(C) $2-\sqrt{7}$
(D) $\frac{1}{3}(2-\sqrt{7})$

Aman Gupta
Aman Gupta
Numerade Educator
01:35

Problem 24

If $\sin (\theta+\alpha)=a$ and $\sin \left(\theta+\beta=b\left(0<\alpha, \beta, \theta<\frac{\pi}{2}\right)\right.$
then $\cos ^{2}(\alpha-\beta)-4 a b \cos (\alpha-\beta)$ is equal to
(A) $1-2 a^{2}-2 b^{2}$
(B) $1+2 a^{2}+2 b^{2}$
(C) $1-a^{2}-b^{2}$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
01:35

Problem 24

If $\sin (\theta+\alpha)=a$ and $\sin \left(\theta+\beta=b\left(0<\alpha, \beta, \theta<\frac{\pi}{2}\right)\right.$
then $\cos ^{2}(\alpha-\beta)-4 a b \cos (\alpha-\beta)$ is equal to
(A) $1-2 a^{2}-2 b^{2}$
(B) $1+2 a^{2}+2 b^{2}$
(C) $1-a^{2}-b^{2}$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
01:19

Problem 25

If $\sin x+\operatorname{cosec} x+\tan y+\cot y=4$, where $x$ and
$y \in\left[0, \frac{\pi}{2}\right]$, then $\tan \frac{y}{2}$ is a root of the equation
(A) $\alpha^{2}+2 \alpha+1=0$
(B) $\alpha^{2}+2 \alpha-1=0$
(C) $2 \alpha^{2}-2 \alpha-1=0$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
01:19

Problem 25

If $\sin x+\operatorname{cosec} x+\tan y+\cot y=4$, where $x$ and
$y \in\left[0, \frac{\pi}{2}\right]$, then $\tan \frac{y}{2}$ is a root of the equation
(A) $\alpha^{2}+2 \alpha+1=0$
(B) $\alpha^{2}+2 \alpha-1=0$
(C) $2 \alpha^{2}-2 \alpha-1=0$
(D) none of thes

Aman Gupta
Aman Gupta
Numerade Educator
02:37

Problem 26

The value of $2 \sin ^{2} \theta+4 \cos (\theta+\alpha) \sin \alpha \sin \theta+\cos 2$
$(\alpha+\theta)$ is
(A) $\cos \theta+\cos \alpha$
(B) independent of $\theta$
(C) independent of $\alpha$
(D) none of these

P Krishnamurthy
P Krishnamurthy
Numerade Educator
02:37

Problem 26

The value of $2 \sin ^{2} \theta+4 \cos (\theta+\alpha) \sin \alpha \sin \theta+\cos 2$
$(\alpha+\theta)$ is
(A) $\cos \theta+\cos \alpha$
(B) independent of $\theta$
(C) independent of $\alpha$
(D) none of these

P Krishnamurthy
P Krishnamurthy
Numerade Educator
03:24

Problem 27

The value of $\cos \theta \cdot \cos 2 \theta \cdot \cos 2^{2} \theta \ldots \cos 2^{n-1} \theta$ for
$\theta=\frac{\pi}{2^{n}+1}$ is
(A) 1
(B) $\frac{1}{2^{n}}$
(C) $2^{n}$
(D) none of these

P Krishnamurthy
P Krishnamurthy
Numerade Educator
03:24

Problem 27

The value of $\cos \theta \cdot \cos 2 \theta \cdot \cos 2^{2} \theta \ldots \cos 2^{n-1} \theta$ for
$\theta=\frac{\pi}{2^{n}+1}$ is
(A) 1
(B) $\frac{1}{2^{n}}$
(C) $2^{n}$
(D) none of these

P Krishnamurthy
P Krishnamurthy
Numerade Educator
03:20

Problem 28

The sum of the series $\sin \theta \cdot \sec 3 \theta+\sin 3 \theta \cdot \sec 3^{2} \theta+$
$\sin 3^{2} \theta \sec 3^{3} \theta+\ldots$ up to $n$ terms is
(A) $\frac{1}{2}\left(\tan 3^{n} \theta-\tan \theta\right)$
(B) $\left(\tan 3^{n} \theta-\tan \theta\right)$
(C) $\tan 3^{n} \theta-\tan 3^{n-1} \theta$
(D) none of these

P Krishnamurthy
P Krishnamurthy
Numerade Educator
03:20

Problem 28

The sum of the series $\sin \theta \cdot \sec 3 \theta+\sin 3 \theta \cdot \sec 3^{2} \theta+$
$\sin 3^{2} \theta \sec 3^{3} \theta+\ldots$ up to $n$ terms is
(A) $\frac{1}{2}\left(\tan 3^{n} \theta-\tan \theta\right)$
(B) $\left(\tan 3^{n} \theta-\tan \theta\right)$
(C) $\tan 3^{n} \theta-\tan 3^{n-1} \theta$
(D) none of these

P Krishnamurthy
P Krishnamurthy
Numerade Educator
02:51

Problem 29

If $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$ are in A.P. whose common difference is $\alpha$, then the value of $\sin \alpha\left[\sec x_{1} \sec x_{2}+\sec x_{2} \sec \right.$
$\left.x_{3}+\cdots+\sec x_{n-1} \sec x_{n}\right]$ is equal to
(A) $\frac{\sin n \alpha}{\cos x_{1} \cos x_{n}}$
(B) $\frac{\sin (n-1) \alpha}{\cos x_{1} \cos x_{n}}$
(C) $\frac{\sin (n+1) \alpha}{\cos x_{1} \cos x_{n}}$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
02:37

Problem 30

If $\cos 5 \theta=a \cos \theta+b \cos 3 \theta+c \cos 8 \theta+d$, then
(A) $a=20$
(B) $b=-20$
(C) $c=16$
(D) $d=5$

Aman Gupta
Aman Gupta
Numerade Educator
02:28

Problem 31

If $\frac{\sin \alpha}{\sin \beta}=\frac{\sqrt{3}}{2}$ and $\frac{\cos \alpha}{\cos \beta}=\frac{\sqrt{5}}{2}, 0<\alpha<\beta<\frac{\pi}{2}$, then
(A) $\tan \alpha=1$
(B) $\tan \alpha=\frac{\sqrt{3}}{\sqrt{5}}$
(C) $\tan \beta=\frac{\sqrt{3}}{\sqrt{5}}$
(D) $\tan \beta=1$

Aman Gupta
Aman Gupta
Numerade Educator
03:16

Problem 32

If $\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec} x=7$ and
$\sin 2 x=a-b \sqrt{7}$, then
(A) $a=8$
(B) $b=22$
(C) $a=22$
(D) $b=8$

Aman Gupta
Aman Gupta
Numerade Educator
02:22

Problem 33

Let $n$ be an odd integer. If $\sin n \theta=\sum_{r=0}^{n} b_{r} \sin ^{r} \theta$, for
every value of $\theta$, then
(A) $b_{0}=0$
(B) $b_{0}=n$
(C) $b_{1}=0$
(D) $b_{1}=n$

Aman Gupta
Aman Gupta
Numerade Educator
01:50

Problem 34

If $\alpha$ and $\beta$ be the solutions of $a \cos \theta+b \sin \theta=c$, ther
(A) $\sin \alpha+\sin \beta=\frac{2 b c}{a^{2}+b^{2}}$
(B) $\sin \alpha \sin \beta=\frac{c^{2}-a^{2}}{a^{2}+b^{2}}$
(C) $\sin \alpha+\sin \beta=\frac{2 a c}{b^{2}+c^{2}}$
(D) $\sin \alpha \cdot \sin \beta=\frac{a^{2}-b^{2}}{b^{2}+c^{2}}$

Aman Gupta
Aman Gupta
Numerade Educator
01:55

Problem 35

Let $n$ be a fixed positive integer such that $\sin \left(\frac{\pi}{2 n}\right)+\cos \left(\frac{\pi}{2 n}\right)=\frac{\sqrt{n}}{2}$, then
(A) $n=4$
(B) $n=5$
(C) $n=6$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
02:07

Problem 36

If $a \cos ^{2} 3 \alpha+b \cos ^{4} \alpha=16 \cos ^{6} \alpha+9 \cos ^{2} \alpha$ is iden-
tity, then
(A) $a=1$
(B) $a=24$
(C) $b=1$
(D) $b=24$

Aman Gupta
Aman Gupta
Numerade Educator
03:03

Problem 37

If $A$ and $B$ are acute angle such that $A+B$ and $A-B$ satisfy the equation $\tan ^{2} \theta-4 \tan \theta+1=0$, then
(A) $A=\frac{\pi}{4}$
(B) $B=\frac{\pi}{6}$
(C) $A=\frac{\pi}{6}$
(D) $B=\frac{\pi}{4}$

Aman Gupta
Aman Gupta
Numerade Educator
03:20

Problem 38

For $0<\phi<\pi / 2$, if $x=\sum_{n=0}^{\infty} \cos ^{2 n} \phi, y=\sum_{n=0}^{\infty} \sin ^{2 n} \phi$, and
$z=\sum_{n=0}^{x} \cos ^{2 n} \phi \sin ^{2 n} \phi$, then $x y z=$
(A) $x y+z$
(B) $x z+y$
(C) $x+y+z$
(D) $y z+x$

P Krishnamurthy
P Krishnamurthy
Numerade Educator
03:33

Problem 39

Let $f_{n}(\theta)=\tan \frac{\theta}{2}(1+\sec \theta)(1+\sec 2 \theta)(1+\sec 4 \theta) \ldots .$
$\left(1+\sec 2^{n} \theta\right)$, then
(A) $f_{2}\left(\frac{\pi}{16}\right)=1$
(B) $f_{3}\left(\frac{\pi}{32}\right)=1$
(C) $f_{4}\left(\frac{\pi}{64}\right)=1$
(D) $f_{5}\left(\frac{\pi}{128}\right)=1$

P Krishnamurthy
P Krishnamurthy
Numerade Educator
06:09

Problem 40

If $(a-b) \sin (\theta+\phi)=(a+b) \sin (\theta-\phi)$ and
$a \tan \frac{\theta}{2}-b \tan \frac{\phi}{2}=c$, then
(A) $b \tan \phi=a \tan \theta$
(B) $a \tan \phi=b \tan \theta$
(C) $\sin \phi=\frac{2 b c}{a^{2}-b^{2}-c^{2}}$
(D) $\sin \theta=\frac{2 a c}{a^{2}-b^{2}+c^{2}}$

P Krishnamurthy
P Krishnamurthy
Numerade Educator
06:12

Problem 41

If $\alpha, \beta$ and $\gamma$ are connected by the relation $2 \tan ^{2} \alpha$ $\tan ^{2} \beta \tan ^{2} \gamma+\tan ^{2} \alpha \tan ^{2} \beta+\tan ^{2} \beta \tan ^{2} \gamma+\tan ^{2} \gamma \tan ^{2} \alpha=$ 1 , then
(A) $\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=1$
(B) $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=2$
(C) $\cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma=1$
(D) $\cos (\alpha+\beta) \cos (\alpha-\beta)=-\cos ^{2} \gamma$

P Krishnamurthy
P Krishnamurthy
Numerade Educator
07:24

Problem 42

Column-I
I. If $8 \sin ^{3} \theta \sin 3 \theta=\sum_{k=0}^{n} \alpha_{k} \cos k \theta$ is an identity in $\theta$, where $\alpha_{k}$ 's are constants, then $n=$
II. If $\alpha \cos A=\beta \cos \left(A+\frac{2 \pi}{3}\right)=$ $\gamma \cos \left(A+\frac{4 \pi}{3}\right)$, then $\alpha \beta+\beta \gamma+\gamma \alpha=$
III. The number of all possible 5 -tuples $\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{3}\right)$ such that $a_{1}+a_{2} \sin x$ $+a_{3} \cos x+a_{4} \sin 2 x+a_{5} \cos 2 x=0$ holds for all $x$ is
IV. The value of $\prod_{p=1}^{10}\left(\sin \frac{2 p \pi}{11}-i \cos \frac{2 p \pi}{11}\right)$ is
Column-II
(A) 0
(B) 1
(C) 6
(D) $-1$

P Krishnamurthy
P Krishnamurthy
Numerade Educator
07:05

Problem 43

I. If $\sin \theta=3 \sin (\theta+2 \alpha)$, then the value
of $\tan (\theta+\alpha)+2 \tan \alpha$ is
II. If $p \sin \theta+q \cos \theta=a$ and $p \cos \theta-$
$q \sin \theta=b$ then $\frac{p+a}{q+b}+\frac{q-b}{p-a}+1$ is equal to
III. The value of the expression $\cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{10 \pi}{7}-\sin \frac{\pi}{14}$
$\sin \frac{3 \pi}{14} \sin \frac{5 \pi}{14}$ is
IV. If $\sec \theta+\tan \theta=1$, then one root of the equation $(a-2 b+c) x^{2}+(b-2 c+a) x$ $+(c-2 a+b)=0$ is
(A) 0
(B) 1
(C) $\sec \theta$
(D) $-\frac{1}{4}$

P Krishnamurthy
P Krishnamurthy
Numerade Educator
05:49

Problem 44

Column-I
I. Least ratio of the sides of regular $n$-sided polygon inside and outside the unit circle is
II. The tangents of two acute angles are 3 and $2 .$ The sine of twice their difference is
III. If $n=\frac{\pi}{4 \alpha}$, then $\tan \alpha \tan 2 \alpha \tan 3 \alpha \ldots$ $\tan (2 n-1) \alpha$ is equal to
IV. If $x=y \cos \frac{2 \pi}{3}=z \cos \frac{4 \pi}{3}$, then $x y+$ $y z+z x=$
$\mathrm{V}$. The ratio of the greatest value of $2-$ $\cos x+\sin ^{2} x$ to its least value is
If $\tan ^{-1} \frac{1}{1+2}+\tan ^{-1} \frac{1}{1+2.3}+\tan ^{-1} \frac{1}{1+3.4}$
$+\ldots+\tan ^{-1} \frac{1}{1+n(n+1)}$
$=\tan ^{-1} x$, then $x$ is equal to
(A) $\frac{n}{n+2}$
(B) $\frac{n}{n+1}$
(C) $\frac{n-1}{n+2}$
(D) none of these

P Krishnamurthy
P Krishnamurthy
Numerade Educator
07:27

Problem 45

Column-I
I. The value of $\frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{8 \pi}{15} \cos \frac{14 \pi}{15}$ is
II. If $A$ and $B$ be acute positive angles satisfying $3 \sin ^{2} A+2 \sin ^{2} B=1$ and $3 \sin$ $2 A-2 \sin 2 B=0$, then $\cos (A+2 B)=$
III. The number of integral values of $k$ for which the equation $7 \cos x+5 \sin x=$ $2 k+1$ has a solution is
IV. If $A=\tan 27 \theta-\tan \theta$ and $B=\frac{\sin \theta}{\cos 3 \theta}+\frac{\sin 3 \theta}{\cos 9 \theta}+\frac{\sin 9 \theta}{\cos 27 \theta}$ then $\frac{A}{B}=$
Column-II
(A) $\frac{2}{1}$
(B) $\frac{1}{16}$
(C) 0
(D) 8

P Krishnamurthy
P Krishnamurthy
Numerade Educator
08:37

Problem 46

Column-I
I. If $\alpha, \beta, \gamma$ and $\delta$ are four solutions of the equation $\tan \left(\theta+\frac{\pi}{4}\right)=3 \tan 3 \theta$,
no two of which have equal tangents, then the value of $\tan \alpha+\tan \beta+\tan \gamma$ $+\tan \delta_{15}$
II. If $\frac{\cos \left(\theta_{1}-\theta_{2}\right)}{\cos \left(\theta_{1}+\theta_{2}\right)}+\frac{\cos \left(\theta_{3}+\theta_{4}\right)}{\cos \left(\theta_{3}-\theta_{4}\right)}=0$, then $\tan \theta \tan \theta \tan \theta \tan \theta=$
III. If sec $(\alpha-\beta), \sec \alpha$ and $\sec (\alpha+\beta)$ are in A. P. (with $\beta \neq 0$ ), then $\cos \alpha \sec \frac{\beta}{2}=$
IV. If $\quad \alpha=\frac{2 \cos \beta-1}{2-\cos \beta}(0<\alpha<\beta<\pi)$,
then $\frac{\tan \alpha / 2}{\tan \beta / 2}$ is equal to
Column-II
(A) $\sqrt{2}$
(A) $\sqrt{2}$
(B) $\sqrt{3}$
(C) $-1$
(D) 0

P Krishnamurthy
P Krishnamurthy
Numerade Educator
06:45

Problem 47

Column-I
I. $\tan \alpha+2 \tan 2 \alpha+4 \tan 4 \alpha+8 \cot 8 \alpha$ $-\cot \alpha+1$ is equal to
II. $\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}$ is equal to
III. $\frac{1}{\cos 290^{\circ}}+\frac{1}{\sqrt{3} \sin 250^{\circ}}$ is equal to
IV. $\sin \frac{\pi}{18} \sin \frac{5 \pi}{18} \sin \frac{7 \pi}{18}$ is equal to
Column-II 1
(A) 1
(B) $\frac{1}{8}$
(C) 4
(D) $\frac{4}{\sqrt{3}}$

P Krishnamurthy
P Krishnamurthy
Numerade Educator
01:31

Problem 48

In a triangle $A B C, 2 c a \sin \frac{A-B+C}{2}$ is equal to:
(A) $a^{2}+b^{2}-c^{2}$
(B) $c^{2}+a^{2}-b^{2}$
(C) $b^{2}-c^{2}-a^{2}$
(D) $c^{2}-a^{2}-b^{2}$

Aman Gupta
Aman Gupta
Numerade Educator
01:25

Problem 49

$\sin ^{2} \theta=\frac{4 x y}{(x+y)^{2}}$ is true if and only if:
(A) $x+y \neq 0$
(B) $x=y, x \neq 0, y \neq 0$
(C) $x=y$
(D) $x \neq 0, y \neq 0$

Aman Gupta
Aman Gupta
Numerade Educator
01:03

Problem 50

The value of $\frac{1-\tan ^{2} 15^{\circ}}{1+\tan ^{2} 15^{\circ}}$ is:
(A) 1
(B) $\sqrt{3}$
(C) $\frac{\sqrt{3}}{2}$
(D) 2

Aman Gupta
Aman Gupta
Numerade Educator
01:03

Problem 51

If $\tan \theta=-\frac{4}{3}$, then $\sin \theta$ is:
(A) $-\frac{4}{5}$ but $\operatorname{not} \frac{4}{5}$
(B) $-\frac{4}{5}$ or $\frac{4}{5}$
(C) $\frac{4}{5}$ but not $-\frac{4}{5}$
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
01:55

Problem 52

If $\sin (\alpha+\beta)=1, \sin (\alpha-\beta)=\frac{1}{2}$, then $\tan (\alpha+2 \beta)$ tan $(2 \alpha+\beta)$ is equal to:
(A) 1
(B) $-1$
(C) zero
(D) none of these

Aman Gupta
Aman Gupta
Numerade Educator
01:13

Problem 53

If $y=\sin ^{2} \theta+\operatorname{cosec}^{2} \theta, \theta \neq 0$ then:
(A) $y=0$
(B) $y \leq 2$
(C) $y \geq-2$
(D) $y \geq 2$

Aman Gupta
Aman Gupta
Numerade Educator
01:09

Problem 54

In a triangle $A B C, a=4, b=3, \angle A=60^{\circ}$, then $c$ is the root of the equation:
(A) $c^{2}-3 c-7=0$
(B) $c^{2}+3 c+7=0$
(C) $c^{2}-3 c+7=0$
(D) $c^{2}+3 c-7=0$

Aman Gupta
Aman Gupta
Numerade Educator
01:40

Problem 55

In a $\triangle A B C, \tan \frac{A}{2}=\frac{5}{6}, \tan \frac{C}{2}=\frac{2}{5}$, then:
(A) $a, c, b$ are in $\mathrm{AP}$
(B) $a, b, c$ are in $\mathrm{AP}$
(C) $b, a, c$ are in $\mathrm{AP}$
(D) $a, b, c$ are in GP

Aman Gupta
Aman Gupta
Numerade Educator
01:25

Problem 56

The equation $a \sin x+b \cos x=c$ where $|c|>\sqrt{a^{2}+b^{2}}$ has:
(A) a unique solution
(B) infinite number of solutions
(C) no solution
(D) none of the above

Aman Gupta
Aman Gupta
Numerade Educator
01:12

Problem 57

If $\alpha$ is a root of $25 \cos ^{2} \theta+5 \cos \theta-12=0 \frac{\pi}{2}<\alpha<\pi$ then $\sin 2 \alpha$ is equal to :
(A) $\frac{24}{25}$
(B) $-\frac{24}{25}$
(C) $\frac{13}{18}$
(D) $-\frac{13}{18}$

Aman Gupta
Aman Gupta
Numerade Educator
01:05

Problem 58

If in a triangle $A B C a \cos ^{2}\left(\frac{C}{2}\right)+c \cos ^{2}\left(\frac{A}{2}\right)=\frac{3 b}{2}$ then the sides $a, b$ and $c$
(A) are in A.P.
(B) are in G.P.
(C) are in H.P.
(D) satisfy $a+b=c$

Aman Gupta
Aman Gupta
Numerade Educator
01:58

Problem 59

Let $\alpha, \beta$ be such that $\pi<\alpha-\beta<3 \pi$. If $\sin \alpha+\sin \beta=-\frac{21}{65}$ and $\cos \alpha+\cos \beta=-\frac{27}{65}$, then the value of $\cos \frac{\alpha-\beta}{2}$ is
(A) $-\frac{3}{\sqrt{130}}$
(B) $\frac{3}{\sqrt{130}}$
(C) $\frac{6}{65}$
(D) $-\frac{6}{65}$

Aman Gupta
Aman Gupta
Numerade Educator
02:06

Problem 60

If $u=\sqrt{a^{2} \cos ^{2} \theta+b^{2} \sin ^{2} \theta}+\sqrt{a^{2} \sin ^{2} \theta+b^{2} \cos ^{2} \theta}$, then the difference between the maximum and minimum values of $u^{2}$ is given by
(A) $2\left(a^{2}+b^{2}\right)$
(B) $2 \sqrt{a^{2}+b^{2}}$
(C) $(a+b)^{2}$
(D) $(a-b)^{2}$

Aman Gupta
Aman Gupta
Numerade Educator
01:20

Problem 61

The sides of a triangle are $\sin \alpha, \cos \alpha$ and $\sqrt{1+\sin \alpha \cos \alpha}$ for some $0<\alpha<\frac{\pi}{2}$. Then the greatest angle of the triangle is
(A) $60^{\circ}$
(B) $90^{\circ}$
(C) $120^{\circ}$
(D) $150^{\circ}$

Aman Gupta
Aman Gupta
Numerade Educator
01:54

Problem 62

In a triangle $P Q R, \angle R=\frac{\pi}{2} .$ If $\tan \left(\frac{P}{2}\right)$ and $\tan \left(\frac{Q}{2}\right)$ are the roots of $a x^{2}+b x+c=0, a \neq 0$ then
(A) $a=b+c$
(B) $c=a+b$
(C) $b=c$
(D) $b=a+c$

Aman Gupta
Aman Gupta
Numerade Educator
01:36

Problem 63

In a triangle $A B C, \angle C=\frac{\pi}{2}$ If $r$ is the inradius and $R$ is the circumradius of the the triangle $A B C$, then $2(r+$ $R$ ) equals
(A) $b+c$
(B) $a+b$
(C) $a+b+c$
(D) $\overline{c+a}$

Aman Gupta
Aman Gupta
Numerade Educator
01:23

Problem 64

If the roots of the quadratic equation $x^{2}+p x+q=0$ are $\tan 30^{\circ}$ and $\tan 15^{\circ}$, respectively then the value of $2+$ $q-p$ is
(A) 2
(B) 3
(C) 0
(D) 1

Aman Gupta
Aman Gupta
Numerade Educator
01:17

Problem 65

The number of values of $x$ in the interval $[0,3 \pi]$ satisfying the equation $2 \sin ^{2} x+5 \sin x-3=0$ is
(A) 4
(B) 6
(C) 1
(D) 2

Aman Gupta
Aman Gupta
Numerade Educator
01:04

Problem 66

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $x$. The maximum area enclosed by the park is
(A) $\frac{3}{2} x^{2}$
(B) $\sqrt{\frac{x^{3}}{8}}$
(C) $\frac{1}{2} x^{2}$
(D) $\pi x^{2}$

Aman Gupta
Aman Gupta
Numerade Educator
01:35

Problem 67

If $0<x<$ and $\cos x+\sin x=\frac{1}{2}$, then $\tan x$ is
(A) $\frac{(1-\sqrt{7})}{4}$
(B) $\frac{(4-\sqrt{7})}{3}$
(C) $-\frac{(4+\sqrt{7})}{3}$
(D) $\frac{(1+\sqrt{7})}{4}$

Aman Gupta
Aman Gupta
Numerade Educator
01:48

Problem 68

Let $\mathrm{A}$ and $\mathrm{B}$ denote the statements
A: $\cos \alpha+\cos \beta+\cos \lambda=0$
B: $\sin \alpha+\sin \beta+\sin \lambda=0$
If $\cos (\beta-\lambda)+\cos (\beta-\alpha)+\cos (\alpha-\beta)=-\frac{3}{2}$, then
(A) A is true and B is false
(B) A is false and $\mathrm{B}$ is true
(C) both $\mathrm{A}$ and $\mathrm{B}$ are true
(D) both A and B are false

Aman Gupta
Aman Gupta
Numerade Educator
02:02

Problem 69

Let $\cos (\alpha+\beta)=\frac{4}{5}$ and $\sin (\alpha-\beta)=\frac{5}{13}$, where $0 \leq \alpha, \beta \leq \frac{\pi}{4}$,
then $\tan 2 \alpha=$
(A) $\frac{56}{33}$
(B) $\frac{19}{12}$
(C) $\frac{20}{7}$
(D) $\frac{25}{16}$

Aman Gupta
Aman Gupta
Numerade Educator
01:08

Problem 70

For a regular polygon, let $r$ and $R$ be the respective radii of the inscribed and the circumscribed circles. A false statement among the following is [2010]
(A) There is a regular polygon with $\frac{r}{R}=\frac{1}{\sqrt{2}}$
(B) There is a regular polygon with $\frac{r}{R}=\frac{2}{3}$
(C) There is a regular polygon with $\frac{r}{R}=\frac{\sqrt{3}}{2}$
(D) There is a regular polygon with $\frac{r}{R}=\frac{1}{2}$

Aman Gupta
Aman Gupta
Numerade Educator
01:22

Problem 71

If $A=\sin ^{2} x+\cos ^{4} x$ then, for all real values of $x$,
(A) $\frac{13}{16} \leq A \leq 1$
(B) $1 \leq A \leq 2$
(C) $\frac{3}{4} \leq A \leq \frac{13}{16}$
(D) $\frac{3}{4} \leq A \leq 1$

Aman Gupta
Aman Gupta
Numerade Educator
02:20

Problem 72

In a $\Delta P Q R$, if $3 \sin P+4 \cos Q=6$ and $4 \sin Q+3 \cos$ $P=1$, then the angle $R$ is equal to
(A) $\frac{5 \pi}{6}$
(B) $\frac{\pi}{6}$
(C) $\frac{\pi}{4}$
(D) $\frac{3 \pi}{4}$

Aman Gupta
Aman Gupta
Numerade Educator
02:24

Problem 73

$A B C D$ is a trapezium such that $A B$ and $C D$ are parallel and $B C \perp C D .$ If angle $A D B=B, B C=P$ and $C D=q$, then $A B$ is equal to
(A) $\frac{p^{2}+q^{2} \cos \theta}{p \cos \theta+q \sin \theta}$
(B) $\frac{p^{2}+q^{2}}{p^{2} \cos \theta+q^{2} \sin \theta}$
(C) $\frac{\left(p^{2}+q^{2}\right) \sin \theta}{(p \cos \theta+q \sin \theta)^{2}}$
(D) $\frac{\left(p^{2}+q^{2}\right) \sin \theta}{p \cos \theta+q \sin \theta}$

Aman Gupta
Aman Gupta
Numerade Educator
01:42

Problem 74

The expression $\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}$ can be written as $[\mathbf{2 0 1 3}]$
(A) $\sec A \operatorname{cosec} A+1$
(B) $\tan A+\cot A$
(C) $\sec A+\operatorname{cosec} A$
(D) $\sin A \cos A+1$

Aman Gupta
Aman Gupta
Numerade Educator
01:49

Problem 75

Let $f_{k}(x)=\frac{1}{k}\left(\sin ^{k} x+\cos ^{4} x\right)$ where $x \in R$ and $k \geq 1$ then, the value of $f_{4}(x)-f_{6}(x)$ equals
(A) $\frac{1}{6}$
(B) $\frac{1}{3}$
(C) $\frac{1}{4}$
(D) $\frac{1}{12}$

Aman Gupta
Aman Gupta
Numerade Educator