Algebra of Derivatives

Q.

Evaluate the expression ${}^{5}C_2$?

Q.

What is ${}^5{C}_3$?

Q.

$\frac{d}{dx}\left({\tan }^{-1}\left(\sec \left(x\right)+\tan \left(x\right)\right)\right)=$

1. $1$

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

3. $\cos \left(x\right)$

4. $\sec \left(x\right)$

Q.

Evaluate $\cos {48}^0-\sin {42}^0$

Q.

${\cos }^{-1}\left(\frac{4}{5}\right)+{\tan }^{-1}\left(\frac{3}{5}\right)=?$

1. ${\tan }^{-1}\left(\frac{27}{11}\right)$

2. ${\sin }^{-1}\left(\frac{11}{27}\right)$

3. ${\cos }^{-1}\left(\frac{11}{27}\right)$

4. None of these.

Q.

$\int \frac{(\cos x+x\sin x)}{(x^2+x\cos x)}dx=$

1. $\log \left|\frac{(\sin x)}{(1+\cos x)}\right|+c$

2. $\log \left|\frac{(\sin x)}{(x+\cos x)}\right|+c$

3. $\log \left|\frac{(2\sin x)}{(x+\cos x)}\right|+c$

4. $\log \left|\frac{(x\sin x)}{(x+\cos x)}\right|+c$

5. $\log \left|\frac{\left(x\right)}{(x+\cos x)}\right|+c$

Q.

Find the solution of the integral $I=\int \frac{\sin \left(2x\right)}{{\sin }^4\left(x\right)+\mathrm{co}{s}^4\left(x\right)}dx$

Q.

Let $\alpha >0,\beta >0$, be such that ${\alpha }^3+{\beta }^2=4$. If the maximum value of the term independent of $x$ in the binomial expansion of ${\left(\alpha x^{\frac{1}{9}}+\beta x^{-\frac{1}{6}}\right)}^{10}$ is $10k$, then $k$ is equal to:

1. $176$

2. $336$

3. $352$

4. $84$

Q.

If $a^x=b^y=c^{z}and{b}^2=ac$, then the value of $y$ is

Q.

If $y=\left(1+\tan A\right)\left(1-\tan B\right)$, where $A-B=\frac{\mathrm{\pi }}{4}$, then ${\left(y+1\right)}^{y+1}$ is equal to

1. $9$

2. $4$

3. $27$

4. $81$

Q.

The value of ${\tan }^{-1}\left(2\right)+{\tan }^{-1}\left(3\right)$ is

1. $\frac{3\mathrm{\pi }}{4}$

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

3. $\frac{\mathrm{\pi }}{3}$

4. ${\tan }^{-1}\left(6\right)$

5. ${\tan }^{-1}\left(5\right)$

Q.

The solutions of the equation$\left|\begin{array}{ccc}1+{\sin }^{2}x& {\sin }^{2}x& {\sin }^{2}x\\ {\cos }^{2}x& 1+{\cos }^{2}x& {\cos }^{2}x\\ 4\sin 2x& 4\sin 2x& 1+4\sin 2x\end{array}\right|=0,\left(0<x<\mathrm{\pi }\right)$ are:

1. $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.,\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$

2. $\raisebox{1ex}{$7\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$12$}\right.,\raisebox{1ex}{$11\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$12$}\right.$

3. $\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$12$}\right.,\raisebox{1ex}{$7\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$12$}\right.$

4. $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$12$}\right.,\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$

Q.

If $y={\cos }^2\left(\frac{3x}{2}\right)-{\sin }^2\left(\frac{3x}{2}\right)$, then $\frac{d^{2}y}{d{x}^2}$is

1. $9y$

2. $-9y$

3. $3\surd (1–y^2)$

4. $-3\surd (1–y^2)$

Q.

The general solution of $\sin x–3\sin 2x+\sin 3x=\cos x–3\cos 2x+\cos 3x$ is

1. $n\pi +\frac{\pi }{8}$

2. $n\frac{\pi }{2}+\frac{\pi }{8}$

3. $(-1{)}^n\frac{n\pi }{2}+\frac{\pi }{8}$

4. $2n\pi +co{s}^{-1}\frac{3}{2}$

Q.

Find the antiderivative of ${\tan }^2\left(x\right)dx$.

Q.

The value of $\cos \left({\tan }^{-1}\left(\tan \left(2\right)\right)\right)$ is

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

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

3. $\cos \left(2\right)$

4. $-\cos \left(2\right)$

Q.

Prove that:$\frac{(sec8A-1)}{(sec4A-1)}=\frac{\tan 8A}{\tan 2A}.$

Q.

If $y=x\left[\left(\cos \left(\frac{x}{2}\right)+\sin \left(\frac{x}{2}\right)\right)\left(\cos \left(\frac{x}{2}\right))-\sin \left(\frac{x}{2}\right)\right)+\sin x\right]+\frac{1}{2\sqrt{x}}$, then $\frac{dy}{dx}$

1. $(1+x)\cos x+(1-x)\sin x-\frac{1}{4x\sqrt{x}}$

2. $(1-x)\cos x+(1-x)\sin x+\frac{1}{4x\sqrt{x}}$

3. $(1+x)\cos x+(1+x)\sin x-\frac{1}{4x\sqrt{x}}$

4. none of these

Q.

The solution of the differential equation $\frac{dy}{dx}={(4x+y+1)}^2$ is

1. $(4x+y+1)=\tan (2x+C)$

2. ${(4x+y+1)}^2=2\tan (2x+C)$

3. ${(4x+y+1)}^3=3\tan (2x+C)$

4. $(4x+y+1)=2\tan (2x+C)$

Q.

If $\frac{1}{(p+q)},\frac{1}{(r+p)},\frac{1}{(q+r)}$ are in $A.P$, then

1. $p,q,r$ are in $AP$

2. $p^2,q^2,r^2$ are in $AP$

3. $\frac{1}{p},\frac{1}{q},\frac{1}{r}$ are in $AP$

4. None of these

Q. Let $f:R\to R$ be a function defined by $f\left(x\right)=\frac{e^{|x|}-e^{-x}}{e^x+e^{-x}}.$ Then $f$ is

1. a one one but not an onto function
2. an onto but not a one-one function
3. a bijective function
4. neither an one-one nor an onto function

Q.

The value of $\sin \left[2{\tan }^{-1}\left(\frac{1}{3}\right)\right]+\cos \left[{\tan }^{-1}\left(2\sqrt{2}\right)\right]$ is

1. $\frac{16}{15}$

2. $\frac{14}{15}$

3. $\frac{12}{15}$

4. $\frac{11}{15}$

Q.

$\frac{d}{dx}\left({\tan }^{-1}\left(\frac{x}{\sqrt{a^2-x^2}}\right)\right)=$

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

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

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

4. none of these

Q.

Check if $(0,2)$ is the solution of the equation $x-2y=4$or not.

Q.

Differentiate the following functions with respect to x :

$3^x+x^3+3^3$

Q.

Prove that $\frac{\sin A}{1+\cos A}=\frac{1-\cos A}{\sin A}$

Q.

The order and degree of differential equation $[1+{\left(\frac{dy}{dx}\right)}^2]=\frac{d^{2}y}{d{x}^2}$ are
a) $2,\frac{3}{2}$
b) 2, 3
c) 2, 1
d) 3, 4

Q.

What is the factorization of a quadratic polynomial?

Q.

Let $f\left(x\right)=x^2+\left(\frac{1}{x^2}\right)$ and $g\left(x\right)=x-\frac{1}{x},x\in R-\left\{-1,0,1\right\}$. If $h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$, then the local minimum value of $h\left(x\right)$ is:

1. $-2\sqrt{2}$

2. $2\sqrt{2}$

3. $3$

4. $-3$

Q. The degree and the order of the differential equation $\frac{d^{2}y}{d{x}^2}=\sqrt[3]{31+{\left(\frac{dy}{dx}\right)}^2}$ respectively are

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