Composition of Inverse Trigonometric Functions and Trigonometric Functions

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

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

Q. For any positive integer $n$, define $f_n:(0,\infty )\to R$ as
$f_n\left(x\right)=\sum _{j=1}^n{\tan }^{-1}\left(\frac{1}{1+(x+j)(x+j-1)}\right)$ for all $x\in (0,\infty )$
$($Here, the inverse trigonometric function ${\tan }^{-1}x$ assumes values in $(\frac{-\pi }{2},\frac{\pi }{2})$ $)$
Then, which of the following statement(s) is (are) TRUE?

1. $\sum _{j=1}^5{\tan }^2\left(f_j\right(0\left)\right)=55$
2. $\sum _{j=1}^{10}(1+f_j^{\prime }(0\left)\right){\sec }^2\left(f_j\right(0\left)\right)=10$
3. For any fixed positive integer $n,\underset{x\to \infty }{lim}\tan \left(f_n\right(x\left)\right)=\frac{1}{n}$
4. For any fixed positive integer $n,\underset{x\to \infty }{lim}{\sec }^2\left(f_n\right(x\left)\right)=1$

Q.

$\cos A+\cos (240°+A)+\cos (240°-A)=?$

1. $\cos A$

2. $0$

3. $\sqrt{3}\sin A$

4. $\sqrt{3}\cos A$

Q.

The value of integral ${\int }_{-1}^1\left[\frac{\left|x+2\right|}{(x+2)}\right]dx=$

1. $1$

2. $2$

3. $0$

4. $-1$

Q.

Examine that sin |x| is a continuous function.

Q. The value of $co{s}^{-1}\left(cos10\right)$=

1. $10$
2. $4\pi -10$
3. $2\pi +10$
4. $2\pi -10$

Q.

Find the maximum and minimum values, if any, of the following functions given by

$f\left(x\right)=|x+2|-1$

$g\left(x\right)=-|x+1|+3$

$h\left(x\right)=sin\left(2x\right)+5$

$f\left(x\right)=|sin4x+3|$

$h\left(x\right)=x+1,xϵ(-1,1)$

Q. Let $m$ and $M$ be respectively the minimum and maximum values of
$\left|\begin{array}{ccc}{\cos }^{2}x& 1+{\sin }^{2}x& \sin 2x\\ 1+{\cos }^{2}x& {\sin }^{2}x& \sin 2x\\ {\cos }^{2}x& {\sin }^{2}x& 1+\sin 2x\end{array}\right|$
Then the ordered pair $(m,M)$ is equal to:

1. $(–3,–1)$
2. $(–4,–1)$
3. $(1,3)$
4. $(–3,3)$

Q. Number of false relations of the following
$\left(i\right)tan|ta{n}^{-1}x|=|x|\left(ii\right)cot|co{t}^{-1}x|=|x|\left(iii\right)ta{n}^{-1}|tanx|=|x|\left(iv\right)sin|si{n}^{-1}x|=|x|$ is:

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

Q.

Find the absolue maximum value and he absolute minimum value of the following functions in the given intervals:

$f\left(x\right)=x^3,xϵ[-2,2]$

$f\left(x\right)=sinx+cosx,xϵ[0,\pi ]$

$f\left(x\right)=4x-\frac{1}{2}{x}^2,xϵ[-2,\frac{9}{2}]$

$f\left(x\right)=(x-1{)}^2+3,xϵ[-3,1]$

Q.

Show that the function given by $f\left(x\right)=sinx$ is
neither increasing nor decreasing in $(0,\pi )$

Q. The solution set of tan theta = 3 cot theta is

Q.

If ${\theta }_1$ and ${\theta }_2$ are two value lying on [0, 2$\pi$] for which tan$\theta$ = $\lambda$ then tan $\frac{{\theta }_1}{2}$.tan $\frac{{\theta }_2}{2}$ is

1. -1

2. 2

3. 1

4. 0

Q. If $p$ and $q$ are the roots of $6x^2+10x+1=0$ then the value of $|[{\tan }^{-1}p+{\tan }^{-1}q]|$ is
(where [ ] denotes greater integer function)

Q. If y = f(x) = tan x cot 3x, then

1. 
2. 
3. 
4. 

Q. $\frac{{\alpha }^3}{2}{\text{cosec}}^2\frac{1}{2}\left({\tan }^{-1}\frac{\alpha }{\beta }\right)+\frac{{\beta }^3}{2}{\sec }^2\left(\frac{1}{2}{\tan }^{-1}\frac{\beta }{\alpha }\right)$ is equal to

1. $(\alpha -\beta )({\alpha }^2+{\beta }^2)$
2. $(\alpha +\beta )({\alpha }^2-{\beta }^2)$
3. $(\alpha +\beta )({\alpha }^2+{\beta }^2)$
4. None of the above

Q. In a $\Delta ABC$, angles $A,B,C$ are in $A.P$. Then the value of $\underset{A\to C}{lim}\frac{\sqrt{3-4\sin A\sin C}}{|A-C|}$ is:

Q. The value of ${\tan }^{-1}(-1)$ is

1. ${45}^{\circ }$
2. ${135}^{\circ }$
3. $-{45}^{\circ }$
4. $-{60}^{\circ }$

Q. The value of ${\cot }^{-1}\left(1\right)+{\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right)$ is

[1 mark]

1. $\frac{\pi }{2}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{8}$
4. $\frac{\pi }{6}$

Q. Evaluate the limit:

$\underset{x\to 0}{lim}\frac{1-cos2x+tan{}^{2}x}{xsinx}$

Q. The set of values of $x$ such that $({\cos }^{-1}x{)}^2-3{\cos }^{-1}x+2>0$, is/are

1. $[\cos 2,\cos 1]$
2. $[0,1]$
3. $[-1,\cos 2)$
4. $(\cos 1,1]$

Q. The value of ${\sec }^{-1}\left(\sec 3\right)+{\cos }^{-1}\left(\cos 12\right)+{\text{cosec}}^{-1}\left(\text{cosec}6\right)+{\cot }^{-1}\left(\cot 10\right)$ is equal to

1. $7-\pi$
2. $\pi +7$
3. $7$
4. $\pi -7$

Q. Prove that $2{\sin }^2\frac{\pi }{6}+{\text{cosec}}^2\frac{7\pi }{6}{\cos }^2\frac{\pi }{3}=\frac{3}{2}$

Q. $\underset{-\pi /2}{\overset{\pi /2}{\int }}{\sin }^{4}x{\cos }^{6}xdx$ is equal to

1. $\frac{3\pi }{64}$
2. $\frac{3\pi }{572}$
3. $\frac{3\pi }{256}$
4. $\frac{3\pi }{128}$

Q. $\frac{1}{2}co{s}^{-1}\left(\frac{1-x}{1+x}\right)=$

1. $co{t}^{-1}\sqrt{x}$
2. $ta{n}^{-1}\sqrt{x}$
3. $ta{n}^{-1}x$
4. $co{t}^{-1}x$

Q. Prove that: $2{\sin }^2\frac{3\pi }{4}+2{\cos }^2\frac{\pi }{4}+2{\sec }^2\frac{\pi }{3}=10$

Q. If $I_1=\underset{0}{\overset{\pi /2}{\int }}\cos \left(\sin x\right)dx;I_2=\underset{0}{\overset{\pi /2}{\int }}\sin \left(\cos x\right)dx$ and $I_3=\underset{0}{\overset{\pi /2}{\int }}\cos xdx,$ then

1. $I_1>I_2>I_3$
2. $I_1>I_3>I_2$
3. $I_2>I_3>I_1$
4. $I_3>I_1>I_2$

Q. The value of the integral ${\int }_0^1\frac{dx}{x^2+2x\cos \alpha +1}$, where $0<\alpha <\frac{\pi }{2}$, is equal to

1. $\alpha \sin \alpha$
2. $\sin \alpha$
3. $\frac{3\alpha -\pi }{2\sin \alpha }$
4. $\frac{\alpha }{2}\sin \alpha$

Q. $L{t}_{x\to 0}\frac{\sin 2x+a\sin x}{x^3}$ exists and finite then $a=$

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

Q. The set of value(s) of $k$ for which $x^2-kx+{\sin }^{-1}\left(\sin 4\right)>0$ for all real $x$ is

1. $\varphi$
2. $(-2,2)$
3. $R$
4. $(-1,3)$