Definition of Limit

Q. ω is an imaginary cube root of unity . If $(1+w^2{)}^m=(1+w^4{)}^m$ , then least positive integral value of m is

1. 5
2. 4
3. 3
4. 6

Q. A pump ejects 12000 kg of water at speed of 4m/s in 40 seconds. Find the average rate at which the pump is working ?

Q.

How to integrate $x{e}^x$?

Q.

Find the following integrals.
$\int (4e^{3x}+1)dx.$

Q.

The eccentricity of the ellipse $9x^2+16y^2$ = 576 is

1. 

2. 

3. 

4. 

Q. The value of $\frac{1}{\sin {20}^{\circ }}-\frac{1}{\sqrt{3}\cos {20}^{\circ }}$ is

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

Q.

Find the quotient $-20÷4$

Q. $\int \sqrt{x^2+4x+6}dx$ is equal to
(where $C$ is integration constant)

1. $\frac{x+2}{2}\sqrt{x^2+4x+6}+\ln |x+2+\sqrt{x^2+4x+6}|+C$
2. $\ln |x+2+\sqrt{x^2+4x+6}|+C$
3. $\frac{1}{2}{\tan }^{-1}\left(\frac{x+2}{\sqrt{2}}\right)+C$
4. $\frac{x}{2}\sqrt{x^2+4x+6}+\ln |x+\sqrt{x^2+4x+6}|+C$

Q.

If $|a|=9$ and $a$ is an integer, then $a$ is

1. $9$

2. $-9$

3. $9and-9$

4. none of these

Q. $\int \frac{x^3}{(x+1{)}^2}dx$ is equal to
(where $C$ is constant of integration)

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

Q. Evaluate $\int x\ln xdx$
(where $C$ is constant of integration)

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

Q. Evaluate $\int (1+\ln x)dx$
(where $C$ is constant of integration)

1. $\ln x+C$
2. $x\ln x+C$
3. $x+C$
4. $x+\ln x+C$

Q.

Find the quotient.

$21÷(-3)$

Q.

Find the following integrals.
$\int (2x^2+e^x)dx.$

Q.

Add the following integers: $-2320+1320+(-100)+80$

Q. If the square ABCD where A (0, 0), B(2, 0), C(2, 2) and D(0, 2) undergoes the following transformations successively
(i) $f_1$(x, y) $\to$ (y, x).
(ii) $f_2$(x, y) $\to$ (x + 3y, y).
(iii) $f_3$(x, y) $\to$$(\frac{x-y}{2},\frac{x+y}{2})$
then the final figure is a

1. 2x - y = a
2. Square
3. Parallelogram
4. rhombus

Q. Evaluate the following limits:

Q. Consider a function $f\left(x\right)$ where $f:R^{+}\to R$ such that $f\left(3\right)=1$
and $f\left(x\right)f\left(y\right)+f\left(\frac{3}{x}\right)f\left(\frac{3}{y}\right)=2f\left(xy\right)$ $\forall x,y\in R^{+}$, then

1. $f\left(x\right)$ is a cubic polynomial
2. $f\left(x\right)$ is an even function
3. $\frac{f\left(2\right)}{f\left(1\right)+f\left(2\right)}=\frac{1}{2}$
4. If $g\left(x\right)=x^2,$ then number of Solutions of $f\left(x\right)=g\left(x\right)$ is $1$.

Q. Let $g\left(x\right)=\frac{1}{2}\left[f\right(x)-f(-x\left)\right]$ for $-3\le x\le 3$ and $f\left(x\right)=2x^2-2x+1$ then find ${\int }_{-3}^{3}g\left(x\right)dx$.

Q. For every positive integer n, $2^n<n!$ when

1. 
2. 
3. 
4. 

Q. $\underset{0}{\overset{\pi /2}{\int }}(x-[\sin x\left]\right)dx=$ _____ , where $\left[x\right]$ is the greatest integer function.

1. $\frac{{\pi }^2}{8}-2$
2. $\frac{{\pi }^2}{4}-1$
3. $\frac{{\pi }^2}{8}-1$
4. $\frac{{\pi }^2}{8}$

Q. The value of the definite integral ${\int }_{\frac{1}{b}}^b\frac{xlnx}{(a^2+x^2)(a^2{x}^2+1)}dx$ (where b > 0) equals

Q. If ${\int }_0^{\pi }\frac{x^2\cos x}{(1+\sin x{)}^2}dx=A\pi -{\pi }^2$ then A is

Q. The value of ${\int }_1^3{x}^{2}dx$ is:

1. $\frac{26}{3}$
2. $\frac{28}{3}$
3. $\frac{25}{3}$
4. None of these

Q. Evaluate ${\int }_0^{\frac{\pi }{2}}\frac{\sin x}{1+\cos x+\sin x}dx$

Q. Which of the following pair of functions are identical?

1. $\sqrt{1+\sin x},\sin \frac{x}{2}+\cos \frac{x}{2}$
2. $x,\frac{x^2}{x}$
3. $\sqrt{x^2},{\left(\sqrt{x}\right)}^2$
4. $ln{x}^3+ln{x}^2,5lnx$

Q.

Evaluate the definite integrals.
${\int }_4^5{e}^{x}dx.$

Q. Let $N=2^{101}\times \frac{{\int }_0^1{x}^{50}(1-x{)}^{50}dx}{{\int }_0^1{x}^{50}(1-x^{102}{)}^{50}dx}$ then the number of ways in which N can be resolved into two factors which are relatively prime numbers is

Q. ${\int }_0^{\frac{\pi }{2}}\frac{{\sin }^{\frac{3}{2}}x}{{\sin }^{\frac{3}{2}}x+{\cos }^{\frac{3}{2}}x}dx=\frac{\pi }{4}$

1. True
2. False

Q. The value of integral $\int \frac{dx}{(2x-3{)}^{1/3}(2x+1{)}^{5/3}}$ is
(where $C$ is integration constant)

1. $\frac{3}{16}{\left(\frac{2x+1}{2x-3}\right)}^{2/3}+C$
2. $\frac{3}{16}{\left(\frac{2x-3}{2x+1}\right)}^{2/3}+C$
3. $\frac{3}{2}{\left(\frac{2x-3}{2x+1}\right)}^{2/3}+C$
4. $\frac{3}{2}{\left(\frac{2x+1}{2x-3}\right)}^{2/3}+C$