Graphs of Basic Inverse Trigonometric Functions

Q. Area in the first quadrant between the ellipses $x^2+2y^2=a^2$ and $2x^2+y^2=a^2$ is

1. $\frac{a^2}{\sqrt{2}}{\tan }^{-1}\frac{1}{\sqrt{2}}$
2. $\frac{3a^2}{4}{\tan }^{-1}\frac{1}{2}$
3. $\frac{5a^2}{2}{\sin }^{-1}\frac{1}{2}$
4. $\frac{9\pi a^2}{2}$

Q. $3co{s}^{-1}x-\pi x-\frac{\pi }{2}=0$ has :

1. One solution
2. Infinite solutions
3. No solution
4. None of these

Q.

The ratio in which the area enclosed by the curve $y=\cos x(0\le 0\le \frac{\pi }{2})$ in the first quadrant is divided by the curve $y=\sin x$, is:

1. $(\sqrt{2}-1):1$

2. $(\sqrt{2}+1):1$

3. $\sqrt{2}:1$

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

Q. Evaluate $\int (x\cdot \cos x\cdot \cos 2x)dx$
(where $C$ is constant of integration)

1. $x(\sin x-\frac{2}{3}{\sin }^{3}x)+\frac{\cos x}{2}-\frac{\cos 3x}{18}+C$
2. $x(\sin x+\frac{2}{3}{\sin }^{3}x)+\frac{\cos x}{2}-\frac{\cos 3x}{18}+C$
3. $x(\sin x-\frac{2}{3}{\sin }^{3}x)+\frac{\cos x}{2}+\frac{\cos 3x}{18}+C$
4. $x(\sin x+\frac{2}{3}{\sin }^{3}x)-\frac{\cos x}{2}+\frac{\cos 3x}{18}+C$

Q. Which among the following represents the graph of $y=se{c}^{-1}\left(x\right)$

1. 
2. 
3. 
4. 

Q. The number of real solution(s) of the equation $\sqrt{1+\cos 2x}=\sqrt{2}{\sin }^{-1}\left(\sin x\right),-\pi \le x\le \pi$ is

1. $0$
2. $1$
3. $2$
4. Infinite

Q. The complete set of values of $x$ satisfying $\sin x>\frac{1}{2}$ is

1. $2n\pi <x<2n\pi +\frac{\pi }{6},n\in Z$
2. $2n\pi +\frac{\pi }{6}<x<2n\pi +\frac{5\pi }{6},n\in Z$
3. $2n\pi +\frac{\pi }{6}<x<2n\pi +\frac{\pi }{2},n\in Z$
4. None of these

Q. Area of the region bounded by the curve $y=\cos x,x=0,$ and $x=\pi$ is

1. $2\text{sq. units}$
2. $4\text{sq. units}$
3. $3\text{sq. units}$
4. $1\text{sq. units}$

Q. The graph of the inverse trigonometric function can be obtained from the graph of their corresponding trigonometric function by interchanging $x$ and $y$ axes. State True/False.

Q. The inequality $si{n}^{-1}x>co{s}^{-1}x$vholds for

1. no value of x
2. x $\in$ $(0,\frac{1}{\sqrt{2}})$
3. all values of x
4. x $\in$ $(\frac{1}{\sqrt{2}},1)$

Q. Given the graph of $y=4\cos (x+\frac{\pi }{4})\forall x\in R.$ Select the correct statements for this function.

1. Range: $(-5,5)$
2. $y=0\text{at}x=\frac{-3\pi }{4},\frac{\pi }{4},\frac{5\pi }{4}\forall x\in [-\pi ,2\pi ]$
3. Period: $2\pi$
4. Period: $\pi$

Q.

How do you simplify $\cos (x+\mathrm{\pi })$?

Q. If $x={\sin }^{-1}\left(\sin 10\right)$ and $y={\cos }^{-1}\left(\cos 10\right)$, then $y-x$ is equal to:

1. $\pi$
2. $7\pi$
3. $0$
4. $10$

Q. Which among the following represents the graph of $y=se{c}^{-1}\left(x\right)$

1. 
2. 
3. 
4. 

Q.

Find the area bounded by the curve y = sin x between x = 0 and $x=2\pi$.

Q. If ${\sin }^{-1}(1-x)-2{\sin }^{-1}x=\frac{\pi }{2}$, then $x$ equals

1. $0$
2. $\frac{1}{2}$
3. (a) and (b) above.
4. $\frac{1}{4}$

Q. Sort the following values of $\sin x$ in ascending order.

1. $\sin 60°$
2. $\sin {0.25}^c$
3. $\sin {360}^{\circ }$
4. $\sin 4^c$
5. $\sin 150°$

Q. Which of the following is an odd function in their domain ?

1. $si{n}^{-1}\left(x\right)$
2. $co{t}^{-1}\left(x\right)$
3. $se{c}^{-1}\left(x\right)$
4. $co{s}^{-1}\left(x\right)$

Q. If $f\left(x\right)=|x|,g\left(x\right)=\left\{\begin{array}{ccc}4-(x+4{)}^2& ;& xϵ(-5,-1)\\ x^3-x^2& ;& xϵ[-1,1)\end{array}\right\}$

Number of values of x satisfying the equation f(g(f(g(x))))=0, is equal to

1. 3
2. 5
3. 2
4. 8

Q. a(1) cos1sim1 -2

Q. The number of real solution(s) of the equation $\sqrt{1+\cos 2x}=\sqrt{2}{\sin }^{-1}\left(\sin x\right)$, where $x\in [-\pi ,\pi ]$ is

1. $0$
2. $1$
3. $2$
4. infinite

Q. Find the possible value of cos x , if cot x +cosec x= 5.

Q. The complete solution set of the inequality ${\left[{\cot }^{-1}x\right]}^2-6\left[{\cot }^{-1}x\right]+9\le 0$, where $[\cdot ]$ dentoes greatest integer function is

1. $[-\infty ,\cot 3]$
2. $[\cot 3,\cot 2]$
3. $[\cot 3,\infty )$
4. $noneofthese$

Q. Let $f:[0,4\pi ]\to [0,\pi ]$ be defined by $f\left(x\right)=co{s}^{-1}\left(cosx\right)$. The number of points $x\in [0,4\pi ]$ satisfying the equation $f\left(x\right)=\frac{10-x}{10}$ is ................

Q. The least value of $a$ for which the equation $\frac{4}{\sin x}+\frac{1}{1-\sin x}=a$ has atleast one solution in the interval $(0,\frac{\pi }{2})$ is

1. $5$
2. $7$
3. $8$
4. $9$

Q. $3co{s}^{-1}x-\pi x-\frac{\pi }{2}=0$ has :

1. One solution
2. Infinite solutions
3. No solution
4. None of these

Q. $3co{s}^{-1}x-\pi x-\frac{\pi }{2}=0$ has :

1. Infinite solutions
2. No solution
3. None of these
4. One solution

Q. Solve for x: If $\left[{\sin }^{-1}{\cos }^{-1}{\sin }^{-1}{\tan }^{-1}x\right]=1$ where[.] denotes the greatest integer function.

Q. Which among the following represents the graph of $y=se{c}^{-1}\left(x\right)$

1. 
2. 
3. 
4. 

Q. The graph of inverse trigonometric function can be obtained from the graph of their corresponding trigonometric function by interchanging $x$ and $y$ axes.

1. True
2. False