Differentiation of Inverse Trigonometric Functions

Q.

What is integral of $\cos 2x$?

Q.

If $sec\left(\theta \right)=x+\frac{1}{4x}$ then prove that, $sec\left(\theta \right)+\tan \left(\theta \right)=2x$ or $\frac{1}{2x}$.

Q.

The sum of the series $ta{n}^{-1}\frac{1}{2}+ta{n}^{-1}\frac{1}{8}+ta{n}^{-1}\frac{1}{18}+ta{n}^{-1}\frac{1}{32}+\cdots +\infty$ is

1. 
   

2. 
   

3. 
   

4. None of these

Q.

Differentiate the following functions with respect to x

sin (ax + b).

Q.

The solution of the differential equation $x\frac{dy}{dx}+y=x\cos \left(x\right)+\sin \left(x\right)$, given that $y=1$ when $x=\frac{\pi }{2}$ is

1. $y=\sin \left(x\right)-\cos \left(x\right)$

2. $y=\cos \left(x\right)$

3. $y=\sin \left(x\right)$

4. $y=\sin \left(x\right)+\cos \left(x\right)$

Q.

If $y={(xco{t}^{3}x)}^{3/2}$, then $\frac{dy}{dx}=?$

1. $\frac{3}{2}{\left(x{\cot }^{3}x\right)}^{1/2}\left[{\cot }^{3}x-3x{\cot }^{2}x{\mathrm{cosec}}^{2}x\right]$

2. $\frac{3}{2}{\left(x{\cot }^{3}x\right)}^{1/2}\left[{\cot }^{2}x-3x{\cot }^{2}x{\mathrm{cosec}}^{2}x\right]$

3. $\frac{3}{2}{\left(x{\cot }^{3}x\right)}^{3/2}\left[{\cot }^{3}x-3x{\mathrm{cosec}}^{2}x\right]$

4. None of these

Q.

If $y={\sin }^{-1}\left(\frac{5x+12\sqrt{1-x^2}}{13}\right)$, then $\frac{dy}{dx}$is equal to

1. $\frac{1}{\sqrt{1-x^2}}$

2. $\frac{-1}{\sqrt{1-x^2}}$

3. $\frac{3}{\sqrt{1-x^2}}$

4. $\frac{x}{\sqrt{1-x^2}}$

Q.

$\cos ecA-2cot2A\cos A=?$

1. $2\sin A$

2. $secA$

3. $2\cos AcotA$

4. None of these

Q.

$\left[\frac{\sin 2A}{1+\cos 2A}\right]\left[\frac{\cos 2A}{1+\cos A}\right]=$

1. $\tan \left(\frac{A}{2}\right)$

2. $\cot \left(\frac{A}{2}\right)$

3. $\sec \left(\frac{A}{2}\right)$

4. $\csc \left(\frac{A}{2}\right)$

Q.

If $\sin A+\cos A=\sqrt{2}$, then the value of ${\cos }^{2}A$ is

1. $\sqrt{2}$

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

3. $4$

4. $-1$

Q.

$\cos ec15°-sec15°$ is equal to

1. $2\sqrt{2}$

2. $\sqrt{6}$

3. $2\sqrt{6}$

4. $\sqrt{6}+\sqrt{2}$

Q.

If${\sum }_{i-l}^{20}si{n}^{-1}{x}_i=10\pi$, then ${\sum }_{i-l}^{20}{x}_i$ is equal to

1. 11

2. 21

3. 20

4. 10

Q.

If $x^y=y^x$, then $x(x-y\log x)\frac{dy}{dx}$ is equal to:

1. $y(y–x\log y)$

2. $y(y+x\log y)$

3. $x(x+y\log x)$

4. $x(y–x\log y)$

Q.

If $\alpha +\beta +\gamma =\pi$, then ${\sin }^2\alpha +{\sin }^2\beta -{\sin }^2\gamma$ is equal to

1. $2\sin \alpha \sin \beta \cos \gamma$

2. $2\cos \alpha \cos \beta \cos \gamma$

3. $2\sin \alpha \sin \beta \sin \gamma$

4. None of these

Q. The number of points where function $f\left(x\right)=max\{|\tan x|,\cos |x|\}$ is non-differentiable in the interval $(-\pi ,\pi ),$ is

1. $2$
2. $3$
3. $4$
4. $5$

Q. If $y=\frac{\sqrt{1-x}}{\sqrt{1+x}}$, prove that $(1-x^2)\frac{dy}{dx}+y=0$

Q. Let $f\left(\theta \right)=\sin \left({\tan }^{-1}\left(\frac{\sin \theta }{\sqrt{\cos 2\theta }}\right)\right);-\frac{\pi }{4}<\theta <\frac{\pi }{4}$, Then the value of $\frac{d}{\left(d\right(\tan \theta \left)\right)}\left(f\right(\theta \left)\right)$ is

Q. If f(x) = x + tan x and f is inverse of g, then g’(x) is equal to

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

Q. If $y=e^{-x}\cos 2x$ then which of the following differential equations is satisfied?

1. $\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}+5y=0$
2. $\frac{d^{2}y}{d{x}^2}+5\frac{dy}{dx}+2y=0$
3. $\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}-5y=0$
4. $\frac{d^{2}y}{d{x}^2}-5\frac{dy}{dx}-2y=0$

Q. Find the solution of differential equation
$\frac{dy}{dx}=\frac{(\sin y+e^x)}{(\ln y-x\cos y)}$ is

1. $y\left(\right(\ln y)-1)=e^x+x\sin y+c$
2. $\ln y=x\sin y+c$
3. $y\left(\right(\ln y)+1)=e^x-x\sin y+c$
4. $x\left(\ln y\right)=e^x-x\sin y+c$

Q.

Differentiate the given functions w.r.t. x.

$x^{xcosx}+\frac{x^2+1}{x^2-1}.$

Q.

$ta{n}^{-1}\frac{63}{16}=si{n}^{-1}\frac{5}{13}+co{s}^{-1}\frac{3}{5}$

Q. The function $f\left(x\right)=2|x|+|x+2|-||x+2|-2|x||$ has a local minimum or a local maximum at $x=$

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

Q.

Differentiate given problems w.r.t.x.

$(5x{)}^{3cos2x}$

Q.

Evaluate $\sin {105}^0$

Q. If ${\cot }^{-1}x+{\cot }^{-1}y+{\cot }^{-1}z=\frac{\pi }{2}$, then $x+y+z$ is equal to

1. $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$
2. $xyz$
3. $xy+yz+zx$
4. $2(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})$

Q. The derivative of $co{s}^{-1}\left(\frac{x^{-1}-x}{x^{-1}+x}\right)atx=-1$ is

1. - 2
2. - 1
3. 1
4. 0

Q.

The value of ${\int }_{-2}^2\left[1+2\sin \left(x\right)\right]e^{\left|x\right|}dx$ is

1. $0$

2. $e^2-1$

3. $2\left(e^2-1\right)$

4. $1$

Q.

The sum of the series sin$\theta$.sec3$\theta$ + sin3$\theta$.$sec{3}^2$$\theta$ + $sin{3}^2$$\theta$.sec$3^3$$\theta$ + ...........n terms is

1. 

2. 

3. 

4. 

Q.

${\tan }^2\theta -se{c}^2\theta$ is equal to

1. $1$

2. $-1$

3. $0$

4. $noneofthese$