Integration as Antiderivative

Q.

What is differentiation in simple words?

Q. Let $f:[0,1]\to R$ be such that $f\left(xy\right)=f\left(x\right)\cdot f\left(y\right)$, for all $x,y\in [0,1]$, and $f\left(0\right)\ne 0$. If $y=y\left(x\right)$ satisfies the differential equation, $\frac{dy}{dx}=f\left(x\right)$ with $y\left(0\right)=1,\text{then}y\left(\frac{1}{4}\right)+y\left(\frac{3}{4}\right)$ is equal to :

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

Q.

A differentiable function $\mathrm{f}\left(\mathrm{x}\right)$ is defined for all$x>0$, and satisfies $\mathrm{f}\left({\mathrm{x}}^3\right)=4{\mathrm{x}}^4$, for all$x>0$. The value of$\mathrm{f}’\left(8\right)$ is

1. $\frac{16}{3}$

2. $\frac{32}{3}$

3. $\frac{16\sqrt{2}}{3}$

4. $\frac{32\sqrt{2}}{3}$

Q. If $y=x\tan y,$ then $\frac{dy}{dx}$ is equal to

1. $\frac{xy}{x-x^2-y^2}$
2. $\frac{y}{x-x^2-y^2}$
3. $\frac{\tan y}{1-x{\sec }^{2}y}$
4. $\frac{y}{1-x{\sec }^{2}y}$

Q.

If $\underset{x\to 0}{lim}\frac{(e^{kx}-1)\sin kx}{x^2}=4$, then $k=?$

1. $2$

2. $-2$

3. $\pm 2$

4. $\pm 4$

Q.

If$y=a+b{x}^2$:$a,b$are arbitrary constants, then

1. $\frac{d^{2}y}{d{x}^2}=2xy$

2. $\frac{x{d}^{2}y}{d{x}^2}=\frac{dy}{dx}$

3. $x\frac{d^{2}y}{d{x}^2}–\frac{dy}{dx}+y=0$

4. $\frac{x{d}^{2}y}{d{x}^2}=2xy$

Q.

Find $\frac{dy}{dx}$ if ${\sin }^2\left(x\right)+{\cos }^2\left(y\right)=1$.

Q.

The value of $I={\int }_0^{1}x|x-\frac{1}{2}|dx$ is

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

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

3. $\frac{1}{8}$

4. None of these

Q. Find the multiplicative inverse of $\sqrt{5}+3i$

Q. If $f\left(x\right)=a{x}^5+b{x}^3+cx+d$ is an odd function, then

1. $a=0$
2. $b=0$
3. $c=0$
4. $d=0$

Q. Differentiate the function ${\sin }^{3}x+{\cos }^{6}x$ w.r.t. $x$

Q.

The maximum slope of the curve $y=-x^3+3x^2+2x-27$ is

1. $5$

2. $-5$

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

4. None of these

Q.

What is the meaning of partial integration?

Q. If $z=x+iy$ satisfies $|z|-2=0$ and $|z-i|-|z+5i|=0$, then

1. $x^2-y+3=0$
2. $x+2y-4=0$
3. $x^2+y-4=0$
4. $x+2y+4=0$

Q. If $n$ is a positive integer for the function $f\left(x\right)=5x(x-2{)}^n,x\in [0,2].$ By Rolle's theorem value of $c$ is $\frac{1}{5},$ then $n$ is equal to

1. $3$
2. $6$
3. $9$
4. $10$

Q. If $y=e^{\sqrt{\cot x}},$ then $\frac{dy}{dx}=$

1. $-\frac{e^{\sqrt{\cot x}}\times {\text{cosec}}^{2}x}{2\sqrt{\cot x}}$
2. $\frac{e^{\sqrt{\cot x}}\times \text{cosec}x}{2\sqrt{\cot x}}$
3. $-\frac{e^{\cot x}\times {\text{cosec}}^{2}x}{2\sqrt{\cot x}}$
4. $-\frac{e^{\sqrt{\cot x}}\times {\text{cosec}}^{2}x}{\sqrt{\cot x}}$

Q. If $y=\tan (x^{\circ }+{45}^{\circ }),$ then $\frac{dy}{dx}=$

1. $\frac{\pi }{180}{\sec }^2(x^{\circ }+{45}^{\circ })$
2. ${\sec }^2(x^{\circ }+{45}^{\circ })$
3. $\frac{\pi }{180}{\sec }^2(x+{45}^{\circ })$
4. $\frac{\pi }{4}{\sec }^2(x^{\circ }+{45}^{\circ })$

Q.

$\int e^{kx}$ [f(kx) + f'(kx)]dx will be equal to $\frac{1}{k}{e}^{kx}$ f(kx) + C only when k = 1.

1. True

2. False

Q. If $f\left(x\right)=\{\begin{array}{cc}x,& x<0\\ 1,& x=0\\ x^2,& x>0\end{array},$ then $\underset{x\to 0}{lim}f\left(x\right)$ is equal to:

1. $0$
2. $1$
3. $2$
4. does not exist

Q. Cos 9A=sinA and 9A<90^°, then value of tanA?

Q. If $y=e^{\sin \sqrt{x}},$ then $\frac{dy}{dx}=$

1. $\frac{e^{\cos \sqrt{x}}\sin \sqrt{x}}{2\sqrt{x}}$
2. $\frac{e^{\sin \sqrt{x}}}{2\sqrt{x}}$
3. $\frac{e^{\sin \sqrt{x}}\cos \sqrt{x}}{2\sqrt{x}}$
4. $\frac{e^{\sin \sqrt{x}}\cos \sqrt{x}}{\sqrt{x}}$

Q. The value of constants $m$ and $c$ for which $y=mx+c$ is a solution of the differential equation $D^{2}y+3Dy+4y=4x$
$(D^{2}y=\frac{d^{2}y}{d{x}^2},Dy=\frac{dy}{dx})$

1. $m=-1$
2. $c=-\frac{3}{4}$
3. $c=\frac{3}{4}$
4. $m=1$

Q. The solution of differential equation $(2x-y+1)dx+(2y-x+1)dy=0$ is
(where $c$ is integration constant)

1. ${(y-1)}^2-(y+1)(x+1)+{(x-1)}^2=c^2$
2. ${(y+1)}^2-(y+1)(x+1)+{(x+1)}^2=c^2$
3. ${(y+1)}^2-(y-1)(x-1)+2{(x+1)}^2=c^2$
4. $3{(y+1)}^2-(y+1)(x+1)+2{(x+1)}^2=c^2$

Q. If $f\left(x\right)$ is invertible and twice differentiable function satisfying $f^{\prime }\left(x\right)=\stackrel{f\left(x\right)}{\underset{0}{\int }}{f}^{-1}\left(t\right)dt,\forall x\in R$ and $f^{\prime }\left(0\right)=1$, then $f^{\prime }\left(1\right)$ can be

1. $e$
2. $e^2$
3. $\frac{1}{e}$
4. $\sqrt{e}$

Q. $\text{For}x\in [-2\pi ,2\pi ],f\left(x\right)=\sin x;$
$g\left(x\right)=\cos x.$ Then, select the correct statements.

1. $f(-x)\&-g\left(x\right)$ will have $4$ points of intersection
2. $f\left(x\right)\&g\left(x\right)$ will have $4$ points of intersection
3. $f(-x)\&g\left(x\right)$ will have $4$ points of intersection
4. $f\left(x\right)\&-g\left(x\right)$ will have $4$ points of intersection

Q. Find $\frac{dy}{dx}$, if $y={\sin }^{-1}x+{\sin }^{-1}\sqrt{1-x^2},0<x<1$

Q. $\int \frac{e^{2x}-1}{e^{2x}+1}dx$ is equal to
(where $C$ is constant of integration)

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

Q.

What is integration in simple words?

Q. Differentiate the function $\cos (a\cos x+b\sin x)$ w.r.t. $x$ for some constants $a$ and $b.$

Q. The value of $\int {\cot }^{4}xdx$ is

1. $\frac{{\cot }^{3}x}{3}+\cot x+x+C$
2. $\frac{1}{3}\cot x+C$
3. $-\frac{{\cot }^{3}x}{3}+{\cot }^2+x+C$
4. $-\frac{{\cot }^{3}x}{3}+\cot x+x+C$