First Principle of Differentiation

Q.

$\sin 2A=2\sin A$ is true when $A=$

1. $30°$

2. $0°$

3. $45°$

4. $90°$

Q.

Is $\sin 2x$ the same as $2\sin x$?

Q.

Solve :

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

Q.

What is the cosine inverse of $0$?

Q.

$\int \text{cosec}(x-a)\text{cosec}xdx=$

1. $\left(\frac{1}{\sin a}\right)\log \left|\frac{\sin (x-a)}{\sin x}\right|+C$

2. $\left(\frac{-1}{\sin a}\right)\log \sin (x-a)\sin x+C$

3. $\left(\frac{1}{\sin a}\right)\log \left(\sin (x-a)+\text{cosec}x\right)+C$

4. $\left(\frac{1}{\sin a}\right)\log \left(\sin (x-a)\sin x\right)+C$

Q.

Integrate the function.
$\int e^x\left(\frac{1+sinx}{1+cosx}\right)dx.$

Q. Total number of points of non-differentiability of $f\left(x\right)=[3+4sinx]$ in $[\pi ,2\pi ]$ where [.] denote the g.i.f are

1. 5
2. 6
3. 9
4. 8

Q.

If $sec\left(A\right)+\tan \left(A\right)=p$ then find the value of $\cos ec\left(A\right)$.

Q.

Express $\cos \left(4\theta \right)$ in terms of $\cos \left(\theta \right)$.

Q.

$\frac{d\left[{\sin }^{-1}\right(3x-4x^3\left)\right]}{dx}=$

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

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

3. $1$

4. None of These

Q. If $f\left(x\right)=|x^2-5x+6|$, then $f^{\prime }\left(x\right)$ equals

1. $2x-5$ for $2<x<3$
2. $5-2x$ for $2<x<3$
3. $2x-5$ for $x<2$
4. $5-2x$ for $x<3$

Q.

If $\sin A=n\sin B$, then $\left[\frac{(n-1)}{(n+1)}\right]\tan \frac{(A+B)}{2}=$

1. $\sin \left(\frac{A-B}{2}\right)$

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

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

4. None of these

Q.

If ${\cos }^{-1}\left(\frac{x}{a}\right)+{\cos }^{-1}\left(\frac{y}{b}\right)=\alpha$, then $\left(\frac{x^2}{a^2}\right)-\left(\frac{2xy}{ab}\right)\cos \alpha +\left(\frac{y^2}{b^2}\right)=$

1. ${\sin }^2\alpha$

2. ${\cos }^2\alpha$

3. ${\tan }^2\alpha$

4. $co{t}^2\alpha$

Q. The total number of points of non-differentiability of $f\left(x\right)=\text{min}\{|\sin x|,|\cos x|,\frac{1}{4}\}$ in $(0,2\pi )$ is -

1. $9$
2. $10$
3. $11$
4. $8$

Q.

$\cos 2(\theta +\Phi )+4\cos (\theta +\Phi )\sin \theta \sin \Phi +2{\sin }^2\Phi =?$

1. $\cos 2\theta$

2. $\cos 3\theta$

3. $\sin 2\theta$

4. $\sin 3\theta$

Q.

The derivative of $f\left(x\right)=\left|x^3\right|$ at$x=0$ is

1. $0$

2. 1$0$

3. $-1$

4. not defined

Q.

$\frac{d^{20}(2\cos x\cos 3x)}{d{x}^{20}}=$

1. $2^{20}(\cos 2x–2^{20}\cos 4x)$

2. $2^{20}(\cos 2x+2^{20}\cos 4x)$

3. $2^{20}(\sin 2x+2^{20}\sin 4x)$

4. $2^{20}(\sin 2x–2^{20}\sin 4x)$

Q.

If $f\left(x\right)={\left(\frac{a+x}{b+x}\right)}^{a+b+2x}$ then, $f\left(0\right)=$

1. $2\log \frac{a}{b}+\frac{b^2-a^2}{ab}$

2. ${\left(\frac{a}{b}\right)}^{a+b}\left[2\log \frac{a}{b}+\frac{b^2-a^2}{ab}\right]$

3. ${\left(\frac{a}{b}\right)}^{a+b}\left[\frac{b^2-a^2}{ab}\right]$

4. None of these

Q. If $f(x+y)=f\left(x\right)+kxy-2y^2\forall x,y\in R$ and $f\left(1\right)=2,f\left(2\right)=6,$ then which of the following is/are true :

1. $k=6$
2. $f^{\prime }\left(3\right)=18$
3. $f^{\prime \prime }\left(100\right)=600$
4. $f^{\prime \prime \prime }\left(100\right)=0$

Q. The area of the smaller portion enclosed by the curves $x^2+y^2=9$ and $y^2=8x$ is

1. $\frac{\sqrt{2}}{3}+\frac{9\pi }{4}-\frac{9}{2}si{n}^{-1}\left(\frac{1}{3}\right)$
2. $2(\frac{\sqrt{2}}{3}+\frac{9\pi }{4}-\frac{9}{2}si{n}^{-1}\left(\frac{1}{3}\right))$
3. $2[\frac{\sqrt{2}}{3}+\frac{9\pi }{4}+\frac{9}{2}si{n}^{-1}\left(\frac{1}{3}\right)]$
4. $[\frac{\sqrt{2}}{3}+\frac{9\pi }{4}+\frac{9}{2}si{n}^{-1}\left(\frac{1}{3}\right)]$
   

Q. Let $f\left(x\right)=\underset{h\to 0}{lim}\frac{\left(\sin \right(x+h){)}^{\ln (x+h)}-(\sin x{)}^{\ln x}}{h}$ then $f\left(\frac{\pi }{2}\right)$ is

1. Equal to $0$
2. Equal to $1$
3. $\ln \frac{\pi }{2}$
4. Non - existent

Q. Let $f\left(x\right)=x^2+ax+3,g\left(x\right)=x+b$ and $F\left(x\right)=\underset{n\to \infty }{lim}\frac{f\left(x\right)+x^{2n}g\left(x\right)}{1+x^{2n}}$. If $F\left(x\right)$ is continuous $\forall x\in R,$ then

1. $a=5,b=4$
2. $a=1,b=3$
3. $a=4,b=1$
4. $a=1,b=4$

Q.

In a $∆ABC$, if ${\sin }^2\left(\frac{A}{2}\right),{\sin }^2\left(\frac{B}{2}\right)$ and ${\sin }^2\left(\frac{C}{2}\right)$ are in $\mathrm{HP}$. Then, $a,b$ and $c$ will be in

1. $\mathrm{AP}$

2. $\mathrm{GP}$

3. $\mathrm{HP}$

4. None of these

Q. $\mathrm{The}\mathrm{function}f\left(x\right)=\sec x\mathrm{is}\mathrm{continuous}\mathrm{for}\mathrm{all}x\in R.$

1. True
2. False

Q.

Find the derivative of $x^2-2$ at $x=10$ from the first principle.

Q.

Differentiate the following questions w.r.t. x.

$sin\left(ta{n}^{-1}{e}^{-x}\right).$

Q.

If $f\left(x+y\right)=2f\left(x\right)f\left(y\right),f\left(5\right)=1024\left(\log 2\right)$ and $f\left(2\right)=8$ then the value of $f\left(3\right)$ is

1. $64\left(\log 2\right)$

2. $128\left(\log 2\right)$

3. $256$

4. $256\left(\log 2\right)$

Q.

$\tan {75}^{°}-\cot {75}^{°}=$

1. $2\sqrt{3}$

2. $2+\sqrt{3}$

3. $2-\sqrt{3}$

4. None of these

Q. Which of the following functions is/are continuous everywhere?

1. $\sin x$
2. $\tan x$
3. $|x|$
4. $a^x,\text{where}a>0$
5. Polynomial function

Q.

Find the value of $\tan {15}^{\circ }$.