Basic Differentiation Rule

Q.

If $A$ and $B$ are two sets, then $(A\cap B)$ is equal to

1. $A\cap B$

2. $AUB$

3. $A\cap B$

4. $AUB$

Q. Find the derivative of $x^5$

1. $4x^4$
2. $4x^5$
3. $5x^4$
4. $5x^5$

Q. Derivative of $f\left(x\right)=4e^x-4^x+2\ln x$ is

1. $4e^x-4^x\ln 4+\frac{2}{x}$
2. $4e^x+4^x\ln 4+\frac{2}{x}$
3. $4e^x-4^x\ln 4-\frac{2}{x}$
4. $4e^x-4^x\ln 4+2x$

Q.

If the equation $x^2+y^2+2gx+2fy+1=0$ represent a pair of straight lines, then

1. $g^2–f^2=1$

2. $f^2–g^2=1$

3. $g^2+f^2=1$

4. $g^2+f^2=\frac{1}{2}$

Q.

$n$th derivative of ${(x+1)}^n$ is equal to

1. $(n-1)!$

2. $(n+1)!$

3. $n!$

4. $n{\left[\right(n+1\left)\right]}^{n-1}$

Q.

If $x=\frac{2t}{1+t^2},y=\frac{1-t^2}{1+t^2}$, then $\frac{dy}{dx}$ equals:

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

2. $\frac{2t}{t^2-1}$

3. $\frac{2t}{1-t^2}$

4. None of these

Q.

$\frac{d^n}{d{x}^n}(\log x)$is equal to

1. $\frac{(n-1)!}{x^n}$

2. $\frac{\left(n\right)!}{x^n}$

3. $\frac{(n-2)!}{x^n}$

4. ${(-1)}^{n-1}\frac{(n-1)!}{x}$

Q.

The electric current in a charging R-C circuit is given is given by $i=i_o{e}^{\frac{-t}{RC}}$ where io, R and C are constant parameter of the circuit and t is time. Find the rate of change of current at (x)t=0, (y)t=RC, (z)t=10 RC

$\left(i\right)\frac{-i_o}{RC}\left(ii\right)0\left(iii\right)\frac{-i_o}{R{C}_e}\left(iv\right)\frac{-i_o}{R{C}_e^{10}}\left(v\right)\frac{-i_o{e}^{10}}{RC}$

1. x-(i); y-(iii); z-(iv)

2. x - (ii) ; y - (i) ; z - (v)

3. x - (iii) ; y - (ii) ; z - (iv)

4. x - (i) ; y - (iii) ; z - (ii)

Q.

If for some $\mathrm{\alpha }$ and $\mathrm{\beta }$ in $R$, the intersection of the following three planes $\mathrm{x}+4\mathrm{y}-2\mathrm{z}=1,\mathrm{x}+7\mathrm{y}-5\mathrm{z}=\mathrm{\beta }$ and $\mathrm{x}+5\mathrm{y}+\mathrm{\alpha z}=5$ is a line in $R^3$, then $\mathrm{\alpha }+\mathrm{\beta }$ is equal to:

1. $0$

2. $10$

3. $-10$

4. $2$

Q.

For the curve $xy=c^2$ the subnormal at any point varies as

1. $x^2$

2. $x^3$

3. $y^2$

4. $y^3$

Q.

If $a_1=a_2=2,a_n=a_{n-1}-1(n>2),$then $a_5$is

1. $1$

2. $-1$

3. $0$

4. $2$

Q.

If $y=3x^2+2xthen\frac{dy}{dx}$=?

1. 8x

2. $3x^2+2x$

3. 6x + 2

4. none of these

Q.

The three lines $px+qy+r=0,qx+ry+p=0,rx+py+q=0$ are concurrent if

1. $p+q+r=0$

2. $p^2+q^2+r^2=pq+qr+pr$

3. $p^3+q^3+r^3=3pqr$

4. None of these

Q.

How do you find the vertical and horizontal asymptotes $?$

Q.

$$\begin{gathered} \mathrm{\_}+\mathrm{\_}=17 \\ ++ \\ \mathrm{\_}-\mathrm{\_}=13 \\ == \\ 1915 \end{gathered}$$

Q.

The graph of the function $\mathrm{y}=\mathrm{f}\left(\mathrm{x}\right)$ passing through the point $(0,1)$ and satisfying the differential equation$\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}.\cos \mathrm{x}=\cos \mathrm{x}$ is such that

1. It is a constant function

2. It is periodic

3. It is neither an even nor an odd function

4. It is continuous and differentiable for all values of $\mathrm{x}$

Q. What is the meaning of derivative?

Q. If $y=e^{2^x}-\cos \left(\ln x\right)$, then $\frac{dy}{dx}$ is

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

Q. A cylinder of radius $x$ and height $2h$ is to be inscribed in a sphere of radius $R$ centred at $O$ as shown in figure.


i. Evaluate the volume of cylinder in terms of $h$.
ii. The cylinder has maximum volume when $h=?$.

1. i. $V=2\pi h(R^2-h^2)$
   ii. $h=\frac{R}{\sqrt{3}}$
2. i. $V=2\pi h(R^2-h^3)$
   ii. $h=\frac{R}{\sqrt{3}}$
3. i. $V=2\pi h(R^2-2h^2)$
   ii. $h=\frac{R}{\sqrt{5}}$
4. i. $V=3\pi h(R^2-h^2)$
   ii. $h=\frac{2R}{\sqrt{3}}$

Q. If sides of a rectangle with given perimeter are $a$ & $b$, then find the relation between $a$ & $b$ for which area of the given rectangle is maximum.

1. $a+b=0$
2. $a=b$
3. $a.b=1$
4. $a=4b$

Q. The value of the derivative of the function $y=16x^{3}atx=-1$ is

1. -1
2. -16
3. -48
4. 48

Q.

The electric current in a charging R-C circuit is given by $i=i_o{e}^{\frac{-t}{RC}}$ where io, R and C are constant parameter of the circuit and t is time. Find the rate of change of current at (x)t=0, (y)t=RC, (z)t=10 RC

$\left(i\right)\frac{-i_o}{RC}\left(ii\right)0\left(iii\right)\frac{-i_o}{R{C}_e}\left(iv\right)\frac{-i_o}{R{C}_e^{10}}\left(v\right)\frac{-i_o{e}^{10}}{RC}$

1. x-(i); y-(iii); z-(iv)

2. x - (ii) ; y - (i) ; z - (v)

3. x - (iii) ; y - (ii) ; z - (iv)

4. x - (i) ; y - (iii) ; z - (ii)

Q. If $y=e^x\cot \left(x\right)$, then $\frac{dy}{dx}$ will be

1. $e^x\cot \left(x\right)-{\csc }^2\left(x\right)$
2. $e^x{\csc }^2\left(x\right)$
3. $e^x\left(\cot \left(x\right)-{\csc }^2\left(x\right)\right)$
4. $e^x\cot \left(x\right)$

Q.

If $y=\left[\frac{x^2+1}{x+1}\right],then\frac{dy}{dx}$=?

1. 

2. 

3. 

4. none of these

Q.

If $y=3x^2+2xthen\frac{dy}{dx}$=?

1. 8x

2. $3x^2+2x$

3. 6x + 2

4. none of these

Q. If $S=e^x-\sin x$, find $\frac{d^{2}S}{d{x}^2}$

1. $e^x+\cos x$
2. $e^x-\sin x$
3. $e^x+\sin x$
4. $e^x-\cos x$

Q. If surface area of a cube is changing at a rate of $6m^2/s$ , find the rate at which length of the body diagonal changes, at a moment when side length is $1m$.

1. $6\text{m/s}$
2. $3\sqrt{3}\text{m/s}$
3. $\frac{3}{2}\text{m/s}$
4. $\frac{\sqrt{3}}{2}\text{m/s}$

Q. If $y=1/x^4,$ then $\frac{dy}{dx}$ will be

1. $\frac{4}{x^3}$
2. $4x^3$
3. $-\frac{4}{x^5}$
4. $\frac{4}{x^5}$

Q.

Find the value of $\mathrm{y}$ when $\mathrm{x}=-5.$

$\mathrm{y}=11-\mathrm{x}$

Q. Find derivative of $f\left(x\right)=x+\frac{1}{x}+1$

1. $\frac{(x+1)(x-1)}{x}$
2. $\frac{(x+1)(x-1)}{x^2}$
3. $\frac{(1+x)(1-x)}{x^2}$
4. $\text{None of these}$