Trigonometric Functions

Q.

What is the domain and range of $y=\tan x$?

Q.

$\frac{\tan \left(80°\right)-\tan \left(10°\right)}{\tan \left(70°\right)}$ is equal to:

1. $0$

2. $1$

3. $2$

4. $3$

Q.

If $y\left(\alpha \right)=\sqrt{2\left(\frac{\tan \alpha +cot\alpha }{1+{\tan }^2\alpha }\right)+\frac{1}{{\sin }^2\alpha }}$ where $\alpha \in \left(\frac{3\mathrm{\pi }}{4},\mathrm{\pi }\right)$ then $\frac{dy}{d\alpha }$ at $\alpha =\frac{5\mathrm{\pi }}{6}$ is

1. $-\frac{1}{4}$

2. $\frac{4}{3}$

3. $4$

4. $-4$

Q. If $2\cos 3B+3\cos 4A=3$ and $2\sin 3B-3\sin 4A=0$, where $2A$ and $2B$ are positive acute angles. If $2A+3B=\frac{\pi }{J}$, then the value of $J$ is

1. $1$
2. $2$
3. $3$
4. $4$

Q. Write the following functions in the simplest form :
${\tan }^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right),0<x<\pi .$

Q. The value of ${\cos }^{-1}\sqrt{\frac{2}{3}}-{\cos }^{-1}\frac{\sqrt{6}+1}{2\sqrt{3}}$ is equal to

1. $\frac{\pi }{3}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{2}$
4. $\frac{\pi }{6}$

Q. The value of $\underset{-2021\pi }{\overset{2021\pi }{\int }}\left\{x^{2021}(1+\tan (\frac{\pi }{3}-x))(1+\tan (x-\frac{\pi }{12}))\right\}dx$ is

1. $\frac{\pi }{2021}$
2. $\frac{\pi }{2022}$
3. $0$
4. $\frac{(2022{)}^2}{2}$

Q. In a triangle $ABC,\cos A\cos B+\sin A\sin B\sin C=1,$ then $a:b:c$ is:

1. $1:1:\sqrt{2}$
2. $1:\sqrt{2}:1$
3. $\sqrt{2}:1:1$
4. $\sqrt{2}:\sqrt{2}:1$

Q. The equation ${\tan }^{-1}x-{\cot }^{-1}x={\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)$ has

1. no solution
2. unique solution
3. infinite number of solutions
4. two solutions

Q. cos(cot^–1x) is equal to, where x ≥ 0

cos(cot^–1x) का मान है, जहाँ x ≥ 0

1. $\sqrt{1+x^2}$
2. x
3. ${\left(1+x^2\right)}^{--\frac{3}{2}}$
4. $x{\left(1+x^2\right)}^{--\frac{1}{2}}$

Q. Number of value(s) of x satisfying sin^–1x + cos^–1x + sec^–1x + cosec^–1x = π is

sin^–1x + cos^–1x + sec^–1x + cosec^–1x = π को संतुष्ट करने वाले x के मान/मानों की संख्या है/हैं

1. 1
2. 2
3. Infinitely many
   
   अपरिमित रूप से अनेक
4. 0

Q. The maximum value of sin^–1x + tan^–1x is

sin^–1x + tan^–1x का अधिकतम मान है

1. $\frac{\pi }{4}$
2. $\frac{\pi }{2}$
3. $\frac{3\pi }{4}$
4. π

Q. The value of $\cos [2{\cos }^{-1}\frac{1}{5}+{\sin }^{-1}\frac{1}{5}]$ is

1. $\frac{2\sqrt{6}}{5}$
2. $-\frac{2\sqrt{6}}{5}$
3. $\frac{1}{5}$
4. $-\frac{1}{5}$

Q. If $\frac{\sqrt{2}\sin \alpha }{\sqrt{1+\cos 2\alpha }}=\frac{1}{7}$ and $\sqrt{\frac{1-\cos 2\beta }{2}}=\frac{1}{\sqrt{10}},\alpha ,\beta \in (0,\frac{\pi }{2})$, then $\tan (\alpha +2\beta )$ is equal to

Q. If $\alpha ={\cos }^{-1}\left(\frac{3}{5}\right),\beta ={\tan }^{-1}\left(\frac{1}{3}\right),$ where $0<\alpha ,\beta <\frac{\pi }{2},$ then $\alpha -\beta$ is equal to:

1. ${\sin }^{-1}\left(\frac{9}{5\sqrt{10}}\right)$
2. ${\cos }^{-1}\left(\frac{9}{5\sqrt{10}}\right)$
3. ${\tan }^{-1}\left(\frac{9}{5\sqrt{10}}\right)$
4. ${\tan }^{-1}\left(\frac{9}{14}\right)$

Q. If tan $\alpha =2$ and $\alpha \in (\pi ,\frac{3\pi }{2})$ then the value of the expression $\frac{\cos \alpha }{{\sin }^3\alpha +{\cos }^3\alpha }$ is equal to ..................

1. $\frac{5}{9}$
2. 1
3. $\frac{7}{9}$
4. $\frac{5}{7}$

Q. If $\cos \left(\alpha +\beta \right)$ $=\frac{4}{5}$ $and$ $\sin (\alpha -\beta )$ $=\frac{5}{13}$ $\tan 2\alpha =?$

Q. If $a=5,b=4$ and $\cos (A-B)=\frac{31}{32},$ then $\tan \frac{C}{2}=$

1. $\frac{\sqrt{7}}{3}$
2. $\frac{3}{\sqrt{7}}$
3. $\frac{1}{\sqrt{7}}$
4. $\frac{2}{\sqrt{7}}$

Q. The sum of the series ${\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{2}{9}+{\tan }^{-1}\frac{4}{33}+\cdots \text{upto}\infty$ terms is

1. $\frac{\pi }{4}$
2. $\frac{\pi }{2}$
3. $\pi$
4. $\frac{2\pi }{3}$

Q. Differentiate
$\frac{\sin x-x\cos x}{x\sin x+\cos x}$

Q. The value of the integral $\int \frac{x^2}{(x^2+1)(x^2+4)}dx$
(where $C$ is integration constant)

1. $\frac{1}{3}{\tan }^{-1}x+\frac{2}{3}{\tan }^{-1}\left(\frac{x}{2}\right)+C$
2. $\frac{1}{3}{\tan }^{-1}x+\frac{2}{3}{\tan }^{-1}\left(\frac{x}{3}\right)+C$
3. $-\frac{1}{3}{\tan }^{-1}x+\frac{2}{3}{\tan }^{-1}\left(\frac{x}{2}\right)+C$
4. $-\frac{1}{3}{\tan }^{-1}x+\frac{4}{3}{\tan }^{-1}\left(\frac{x}{2}\right)+C$

Q. If $\cos (\alpha +\beta )=\frac{3}{5}$ and $\sin (\alpha -\beta )=\frac{4}{13}$ then $\tan 2\alpha =?$

Q. A function $f\left(x\right)$ satisfies $f\left(x\right)=\sin x+{\int }_0^x{f}^{\prime }\left(t\right)(2\sin t-{\sin }^{2}t)dt$, then $f\left(x\right)$ is

1. $\frac{\sin x}{1-\sin x}$
2. $\frac{\tan x}{1-\sin x}$
3. $\frac{1-\cos x}{\cos x}$
4. $\frac{x}{1-\sin x}$

Q. If ${\tan }^{-1}\left(x\right)+{\tan }^{-1}\left(\sqrt{3}\right)=\frac{\pi }{2}$ then x is equal to

यदि ${\tan }^{-1}\left(x\right)+{\tan }^{-1}\left(\sqrt{3}\right)=\frac{\pi }{2}$, तब x का मान बराबर है

1. $-\frac{1}{\sqrt{3}}$
2. $-\sqrt{3}$
3. $\frac{1}{\sqrt{3}}$
4. 1

Q. Which of the following is incorrect for the function f(x) = sin^–1x?

निम्नलिखित में से कौनसा विकल्प फलन f(x) = sin^–1x के लिए गलत है?

1. Domain = $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$
   
   प्रांत = $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$
2. Range = [–1, 1]
   
   परिसर = [–1, 1]
3. Domain = [–1, 1]
   
   प्रांत = [–1, 1]
4. Range = $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$
   
   परिसर = $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$

Q. ${\sin }^{-1}\left(\sin \frac{2\pi }{3}\right)+{\cos }^{-1}\left(\cos \frac{7\pi }{6}\right)+{\tan }^{-1}\left(\tan \frac{3\pi }{4}\right)$ is equal to:

1. $\frac{11\pi }{12}$
2. $\frac{31\pi }{12}$
3. $-\frac{3\pi }{4}$
4. $\frac{17\pi }{12}$

Q. Let $\cos (\alpha +\beta )=\frac{4}{5}$ and $\sin (\alpha -\beta )=\frac{5}{13}$ , where $0\le \alpha ,\beta \le \frac{\pi }{4}$ , then $\tan 2\alpha$ is equal to

1. $\frac{25}{16}$
2. $\frac{56}{33}$
3. $\frac{19}{12}$
4. $\frac{20}{7}$

Q. The range of f(x) = 1 + cos^–1x + (cos^–1x)² + … ∞; x ∈ (cos1, 1] is

f(x) = 1 + cos^–1x + (cos^–1x)² + … ∞; x ∈ (cos1, 1] का परिसर है

1. [1, ∞)
2. (1, ∞)
3. [1, ∞) – {2}
4. (1, ∞) – {2}

Q. If ${\cot }^{-1}x+{\tan }^{-1}3=\frac{\pi }{2}$, then $x=$

1. $\frac{1}{3}$
2. $\frac{1}{4}$
3. $3$
4. $4$

Q. If $\sin \left(\sum _{r=1}^{40}{\cot }^{-1}\right(1+r+r^2\left)\right)=\frac{a}{b},$ where $a$ and $b$ are co-prime, then $a+b$

1. is a prime number
2. has odd number of factors
3. is divisible by $5$
4. None of the above