Derivative from First Principle

Q.

What is the derivative of $\tan 2x$?

Q.

What is Constant of Integration?

Q.

If ${\cos }^{-1}\left(\frac{3}{5}\right)-{\sin }^{-1}\left(\frac{4}{5}\right)={\cos }^{-1}\left(x\right)$, then $x=?$

1. $0$

2. $1$

3. $-1$

4. $2$

Q.

The domain of the function $f\left(x\right)=\sqrt{\left(x^2-5x+6\right)}+\sqrt{\left(2x+8-x^2\right)}$ is

1. $\left[2,3\right]$

2. $\left[-2,4\right]$

3. $\left[-2,2\right]\cup \left[3,4\right]$

4. $\left[-2,1\right]\cup \left[2,4\right]$

Q.

If $\mathrm{y}={\mathrm{x}}^{\sin \mathrm{x}},$ then$\frac{\mathrm{dy}}{\mathrm{dx}}=$

1. $\frac{{\mathrm{x}}^{\mathrm{sinx}}\left[\mathrm{x}\mathrm{cosx}\mathrm{logx}+\mathrm{sinx}\right]}{\mathrm{x}}$

2. $\frac{{\mathrm{x}}^{\mathrm{sinx}}\left[\mathrm{xcosxlogx}+\cos \mathrm{x}\right]}{\mathrm{x}}$

3. $\frac{\mathrm{y}\left[\mathrm{xsinxlogx}+\mathrm{cosx}\right]}{\mathrm{x}}$

4. None of these

Q.

Which one of the following algebraic expressions is a polynomial in variable $x$?

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

2. $\sqrt{x}+\frac{1}{\sqrt{x}}$

3. $x^2+\frac{3x^{{\displaystyle \frac{3}{2}}}}{\sqrt{x}}$

4. None of these

Q.

A body $A$ starts from rest with an acceleration $a_1$. After$2seconds$, another body $B$ starts from rest with an acceleration$a_2$. If they travel equal distances in the$5thsecond$, after the start of $A$, then the ratio$a_1:a_2$ is equal to

1. $5:9$

2. $5:7$

3. $9:5$

4. $9:7$

Q. Let $f$ be any function defined on $R$ and let it satisfy the condition :
$|f\left(x\right)-f\left(y\right)|\le |(x-y{)}^2|,\forall x,y\in R$
If $f\left(0\right)=1,$ then

1. $f\left(x\right)$ can take any value in $R.$
2. $f\left(x\right)=0,\forall x\in R$
3. $f\left(x\right)>0,\forall x\in R$
4. $f\left(x\right)<0,\forall x\in R$

Q.

The minimum value of $\sin \left(x\right)+\cos \left(x\right)$ is

1. $\sqrt{2}$

2. $-\sqrt{2}$

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

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

5. $1$

Q.

Prove that $\sin 10°\sin 30°\sin 50°\sin 70°=\frac{1}{16}$

Q.

If $z=x+iy$ and $\omega =\frac{1-iz}{z-i}$, then $\left|\omega \right|=1$ implies that, in the complex plane

1. $z$ lies on the imaginary axis

2. $z$ lies on the real axis

3. $z$ lies on the unit circle

4. None of these

Q.

The function $f\left(x\right)={\tan }^{-1}\left(\sin \left(x\right)+\cos \left(x\right)\right)$ is an increasing function in

1. $\left(\frac{\pi }{4},\frac{\mathrm{\pi }}{2}\right)$

2. $\left(-\frac{\pi }{2},\frac{\mathrm{\pi }}{4}\right)$

3. $\left(0,\frac{\mathrm{\pi }}{2}\right)$

4. $\left(-\frac{\pi }{2},\frac{\mathrm{\pi }}{2}\right)$

Q.

$n$th derivative of ${\cos }^2\left(x\right)=$

1. $2^{n-1}\cos \left(n\frac{\pi }{2}+2x\right)$

2. $2^n\cos \left(n\frac{\pi }{2}-2x\right)$

3. $2^{n+1}\cos \left(n\frac{\pi }{2}+2x\right)$

4. $2^{n+2}\cos \left(n\frac{\pi }{2}+2x\right)$

Q.

The function $\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}\frac{\mathrm{\pi }}{4}+{\tan }^{-1}\mathrm{x},\left|\mathrm{x}\right|\le 1\\ \frac{1}{2}\left(\left|\mathrm{x}\right|-1\right),\left|\mathrm{x}\right|>1\end{array}\right.$is

1. both continuous and differentiable on $\mathrm{R}-\left\{-1\right\}$

2. continuous on$\mathrm{R}-\left\{-1\right\}$ and differentiable on $\mathrm{R}-\left\{-1,1\right\}$

3. continuous on $\mathrm{R}-\left\{1\right\}$ and differentiable on $\mathrm{R}-\left\{-1,1\right\}$

4. both continuous and differentiable on $\mathrm{R}-\left\{1\right\}$

Q.

Let $f:R\to R$ be defined as $f\left(x\right)=e^{-x}\sin x$. If $F:\left[0,1\right]R\to$is a differentiable function such that $F\left(x\right)={\int }_0^{x}f\left(t\right)dt,$ then the value of ${\int }_0^1\left[F\left(x\right)+f\left(x\right)\right]e^{x}dx$ lies in the interval.

1. $\left[\frac{330}{360},\frac{331}{360}\right]$

2. $\left[\frac{327}{360},\frac{329}{360}\right]$

3. $\left[\frac{331}{360},\frac{334}{360}\right]$

4. $\left[\frac{335}{360},\frac{336}{360}\right]$

Q. Evaluate the limit:
$\underset{x\to -\infty }{lim}(\sqrt{x^2-8x}+x)$

Q.

$\underset{x\to 0}{\lim }\frac{[1-{\cos }^{3}x]}{x\sin x\cos x}=$

1. $\frac{2}{5}$

2. $\frac{3}{5}$

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

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

Q.

The complex numbers $z=x+iy$ which satisfy the equation $\left|\frac{\left(z-5i\right)}{\left(z+5i\right)}\right|=1$ lie on which of the following.

1. The x - axis

2. The straight line $y=5$

3. a circle passing through the origin

4. None of these

Q.

${\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}\left[\frac{2x\left(1+\sin x\right)}{\left(1+{\cos }^{2}x\right)}\right]dx=$

1. $\frac{{\mathrm{\pi }}^2}{4}$

2. ${\mathrm{\pi }}^2$

3. zero

4. $\frac{\mathrm{\pi }}{2}$

Q.

The derivative of $\tan x-x$ with respect to $x$is

1. $1-{\tan }^{2}x$

2. $\tan x$

3. $-{\tan }^{2}x$

4. ${\tan }^{2}x$

Q.

If $y=a\cos (\log x)+b\sin (\log x)$where $a,b$are parameters then $x^{2}y+xy$ equals to ?

1. $y$

2. $-y$

3. $-2y$

4. $2y$

Q.

Differentiate $secx$ by the first principle.

Q.

The derivative of ${\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ with respect to ${\cot }^{-1}\left(\frac{1-3x^2}{3x-x^3}\right)$

1. $1$

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

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

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

Q.

$\frac{d}{dx}\left(x^3{\tan }^2\left(\frac{x}{2}\right)\right)=$

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

2. $x^3{\tan }^2\left(\frac{x}{2}\right){\sec }^2\left(\frac{x}{2}\right)+3x^2{\tan }^2\left(\frac{x}{2}\right)$

3. $x^3\tan \left(\frac{x}{2}\right){\sec }^2\left(\frac{x}{2}\right)+3x^2{\tan }^2\left(\frac{x}{2}\right)$

4. none of these

Q. Let $f:R\to R$ be twice continuously differentiable (or $f^{\prime \prime }$ exists and is continuous) such that $f\left(0\right)=f\left(1\right)=f^{\prime }\left(0\right)=0$. Then

1. $f^{\prime \prime }\left(c\right)=0$ for some $c\in R$
2. there is no point for which $f^{\prime \prime }\left(x\right)=0$
3. at all points $f^{\prime \prime }\left(x\right)>0$
4. at all points $f^{\prime \prime }\left(x\right)<0$

Q. $\text{If}{x}^y-y^x=a^b,$ find $\frac{dy}{dx}.$

Q.

Simplify and express each of the following in exponential form: (a⁵/a³) × a⁸

Q.

The function $f\left(x\right)=\tan \left(x\right)-x$ is:

1. Always increasing

2. Always decreasing

3. Never decreasing

4. Sometimes decreasing and sometime decreasing

Q.

For every value of $x$ function $f\left(x\right)=e^x$ is

Q.

If $f\left(a\right)=2,f\left(a\right)=1,g\left(a\right)=3,g\left(a\right)=-1$then $\underset{x\to a}{\lim }\frac{[f(a)g(x)-f(x)g(a\left)\right]}{(x-a)}=$

1. $6$

2. $1$

3. $-1$

4. $-5$