Functions

Q.

Solve and find the value of $\sin {36}^{\circ }$.

Q.

What is the derivative of ${\tan }^{-1}\left\{\frac{\left(1+x\right)}{\left(1-x\right)}\right\}$ with respect to $x$ ?

1. $1+x^2$

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

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

4. None of these

Q. The solution set of $\frac{2x}{x^2-9}\le \frac{1}{x+2}$ is

1. $(-\infty ,-2)\cup (3,\infty )$
2. $(-\infty ,-3)\cup (-2,3)$
3. $(-3,0)\cup (3,\infty )$
4. $(-3,-2)\cup (3,\infty )$

Q. let the function f:r>r be f x =x^3/3 -x^2+3x+asinx the set of values of x for which f x is invertible is /are (0, 1) [-1, 1] (-infinity, 0) (0, infinity)

Q.

Let $e$ denote the base of the natural logarithm. The value of the real number $a$ for which the right-hand limit $\underset{x\to 0^{+}}{\lim }\frac{{\left(1-x\right)}^{{\displaystyle \frac{1}{x}}}-e^{-1}}{x^a}$is equal to a nonzero real number, is _____

Q.

If $\frac{d}{dx}(f(x\left)\right)=\frac{1}{(1+x^2)}$, then $\frac{d}{dx}(f(x^3\left)\right)=$

1. $\frac{3x}{(1+x^3)}$

2. $\frac{3x^2}{(1+x^6)}$

3. $-\frac{6x^5}{{(1+x^6)}^2}$

4. $-\frac{6x^5}{(1+x^6)}$

Q.

If $\mathrm{y}=\mathrm{xlog}\left(\frac{\mathrm{x}}{\mathrm{a}+\mathrm{bx}}\right)$ then ${\mathrm{x}}^3\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}=?$

1. $\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}-\mathrm{y}$

2. ${\left(\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}-\mathrm{y}\right)}^2$

3. $\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}}-\mathrm{x}$

4. ${\left(\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}}-\mathrm{x}\right)}^2$

Q. What is the ${20}^{th}$ term of the sequence defined by $a_n=(n-1)(2-n)(3+n)?$

Q. If ${\sum }_{n=1}^{\infty }{\tan }^{-1}\left(\frac{18}{n^2}\right)=\frac{k\pi }{\lambda },$ where $k$ and $\lambda$ are natural numbers which are coprime, then value of $|k-\lambda |$ is

Q.

The number ${\log }_2\left(7\right)$ is

1. Integer

2. Rational number

3. Irrational number

4. Prime number

Q.

Integrate the following functions.
$\int \frac{1}{(1-tanx)}dx.$

Q.

Integrate the following functions.
$\int \frac{x^2}{1-x^6}dx.$

Q. Function$f\left(x\right)=x-\left[x\right]$ where [ ] shows a greatest integer. This function is

1. A periodic function whose period is1/2
2. A periodic function whose period is 1
3. Not a periodic function
4. A periodic function

Q. Solution of the differential equation $\frac{dy}{dx}=\frac{y}{2y\ln y+y-x},y>0$ is:
(where $C$ is integration constant)

1. $x=y^2\ln y+\frac{C}{y^2}$
2. $x=y\ln y+\frac{C}{y}$
3. $x=y\ln y+C$
4. $x=y^2\ln y+\frac{C}{y}$

Q. If $I_n=\int ta{n}^{n}xdx$ then which of the following relation is correct -

1. 
2. 
3. 
4. 

Q. Let a relation $f$ defined on $(0,\infty )$ as $f\left(x\right)=|1-\frac{1}{x}|$. Then which among the following is true

1. $f(-1)=2$
2. $f$ is many-one function
3. $f$ is one-one function
4. Relation $f$ is not a function

Q. $\int e^{2x}\sin 3xdx$ is equal to
( where $C$ is the constant of integration)

1. $\frac{e^{2x}}{13}(2\sin 3x-3\cos 3x)+C$
2. $\frac{e^{2x}}{9}(2\sin 3x-3\cos 3x)+C$
3. $\frac{e^{2x}}{13}(3\sin 3x-2\cos 3x)+C$
4. $\frac{e^{2x}}{13}(2\sin 3x+3\cos 3x)+C$

Q.

Evaluate $\tan (\mathrm{x}+\frac{\mathrm{\pi }}{4})$

Q.

Evaluate the definite integrals.
${\int }_0^{1}x{e}^{x^2}dx.$

Q. Let $f\left(x\right)$ be a quadratic polynomial such that $f(-1)+f\left(2\right)=0$. If one of the roots of $f\left(x\right)=0$ is $3,$ then its other root lies in:

1. $(0,1)$
2. $(1,3)$
3. $(-1,0)$
4. $(-3,-1)$

Q. Find the indicated term in the sequence, whose $n^{th}$ term is: $a_n=\frac{n(n-2)}{n+3};a_{20}$

Q.

${\int }_2^k\left(2x+1\right)dx=6$, then $k=$

1. $4$

2. $-2$

3. $-3$

4. $3$

Q. The sum of inradius and circumradius of incircle and circumcircle of a regular polygon of $n$ sides and each of length $a$ is

1. $\frac{a}{4}\cot \frac{\pi }{2n}$
2. $a\cot \frac{\pi }{n}$
3. $\frac{a}{2}\cot \frac{\pi }{2n}$
4. $a\cot \frac{\pi }{2n}$

Q. Integrate the rational function: $\frac{x}{(x-1)(x-2)(x-3)}$

Q. Solution of the differential equation $(x^3-3x{y}^2)dx=(y^3-3x^{2}y)dy$ is:
(where $C$ is integration constant)

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

Q.

Integrate the function.
$\int xco{s}^{-1}xdx.$

Q.

If $(1+\sin A)(1+\sin B)(1+\sin C)=(1–\sin A)(1–\sin B)(1–\sin C)$ then each side is equal to

1. $\pm \sin A\sin B\sin C$

2. $\pm \cos A\cos B\cos C$

3. $\pm \sin A\cos B\cos C$

4. $\pm \cos A\sin B\sin C$

Q.

Integrate $\int e^{2x}dx$.

Q. Domain of the function $f\left(x\right)=\sqrt{\left[x\right]-x}+\frac{1}{\sqrt{25-x^2}}$ contains the following subset(s), where $[.]$ represents greatest integer function

1. $\{0,2,4\}$
2. $\{-2,3,5\}$
3. $\{-4,-2,1\}$
4. $\{-3,0,3\}$

Q.

Integrate the following functions w.r.t. x.

$\int \frac{1}{x^{1/2}+x^{1/3}}dx.$