Algebra of Limits

Q.

How do you simplify $\frac{(1+\tan x)}{(1-\tan x)}$ ?

Q.

How to find the derivative of ${\mathrm{cosec}}^2\mathrm{x}$?

Q.

The function $f:\mathrm{ℝ}\to \left[-\frac{1}{2},\frac{1}{2}\right]$ is defined as $f\left(x\right)=\frac{x}{1+x^2}$ is

1. Injective but not Surjective.

2. Surjective but not Injective.

3. Neither Injective nor Surjective.

4. Invertible.

Q.

What is the difference between / and % operator?

Q.

The value of the integral${\int }_0^{\mathrm{\pi }}|\sin 2x|dx$ is

Q.

If $z=x-iy$ and $z^{\frac{1}{3}}=p+iq$, then $\frac{{\displaystyle \frac{x}{p}}+{\displaystyle \frac{y}{q}}}{p^2+q^2}$ is equal to

1. $1$

2. $-1$

3. $2$

4. $-2$

Q. Find the derivative of $\sin x$ at $x=0.$

Q.

The value of $\underset{x\to 0}{\lim }\frac{\left[e^{ax}-e^{bx}\right]}{x}$ is

1. $a+b$

2. $a-b$

3. $e^{ab}$

4. $1$

Q.

What is $\log \left(200\right)$?

Q. Evaluate the limit:
$\underset{x\to \infty }{lim}\sqrt{x^2+7x}-x$

Q.

If the fourth term in the expansion of ${\left(x+x^{{\log }_{2}x}\right)}^7$ is $4480$, then the value of $x$ where $x\in N$ is equal to

1. $4$

2. $3$

3. $2$

4. $1$

Q.

Three vectors $7i-11j+k$, $5i+3j-2k$ and $12i-8j-k$ forms?

1. An equilateral triangle

2. An isosceles triangle

3. A right-angled triangle

4. None of these

Q.

Given a square matrix $A$ of order $3\times 3$ such that $\mid A\mid =12$, find the value of $|AadjA|$

Q. Evaluate the limit:
$\underset{x\to \infty }{lim}\frac{x}{\sqrt{4x^2+1}-1}$

Q.

$\frac{d\left({\tan }^{-1}\left({\displaystyle \frac{ax-b}{bx+a}}\right)\right)}{dx}=$

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

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

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

4. none of these

Q.

${\mathrm{CdSO}}_4$ has a solubility in water$8.0\times {10}^{-4}\mathrm{mol}{\mathrm{L}}^{-1}$. What is its solubility $0.01\mathrm{M}$ ${\mathrm{H}}_2{\mathrm{SO}}_4$?(Round off it to the nearest integer)

Q.

The function $f\left(x\right)=\sin x+\cos x$ is increasing in the interval

1. $\left[\frac{3\mathrm{\pi }}{4},\frac{7\mathrm{\pi }}{4}\right]$

2. $\left[0,\frac{3\mathrm{\pi }}{4}\right]$

3. $\left[\frac{\mathrm{\pi }}{4},\frac{3\mathrm{\pi }}{4}\right]$

4. None of these

Q.

$limi{t}_{x\to \infty }\left(\sqrt{x^2+2x-1}\right)-x=$

1. $\infty$

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

3. $4$

4. $1$

Q. Let $f\left(x\right)$ be a polynomial of degree $4$ having extreme values at $x=1$ and $x=2$. If $\underset{x\to 0}{lim}(\frac{f\left(x\right)}{x^2}+1)=3$ then $f(-1)$ is equal to :

1. $\frac{3}{2}$
2. $\frac{5}{2}$
3. $\frac{1}{2}$
4. $\frac{9}{2}$

Q. The fundamental period of the function $f\left(x\right)=\left[x\right]+\left[2x\right]+\left[3x\right]+\cdots +\left[nx\right]-\frac{n(n+1)}{2}x,n\in N,$ is $($ where $[.]$ is the greatest integer function $)$

Q.

Let $y=y\left(x\right)$ be the solution of the differential equation $\cos x(3\sin x+\cos x+3)dy=(1+y\sin x(3\sin x+\cos x+3\left)\right)dx,0\le x\le \frac{\mathrm{\pi }}{2},y\left(0\right)=0.$ Then, $y\left(\frac{\mathrm{\pi }}{3}\right)$ is equal to:

1. $2{\log }_e\left(\frac{2\sqrt{3}+10}{11}\right)$

2. $2{\log }_e\left(\frac{\sqrt{3}+7}{2}\right)$

3. $2{\log }_e\left(\frac{3\sqrt{3}-8}{4}\right)$

4. $2{\log }_e\left(\frac{2\sqrt{3}+9}{6}\right)$

Q. If {x} and [x] are the fractional part function and greatest integer functions of x respectively, then $\underset{x\to \left[a\right]}{lim}\frac{e^{\left\{x\right\}}-\left\{x\right\}-1}{\{x{\}}^2}$

1. 
2. 
3. 
4. Does not exist

Q.

The number of ways in which one can select three distinct integers between $1$ and $30$, both inclusive, whose sum is even, is

1. $455$

2. $1575$

3. $1120$

4. $2030$

5. $1930$

Q.

Find the value of $x$ in exponents and powers when the $2^{x+1}\times 2^5=2^7$.

Q.

The real number which most exceeds its cube is

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

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

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

4. None of these

Q.

If ${}^{10}P_r=5040$ . Then, find out $r$.

Q.

If $\mathrm{y}={\left({\mathrm{x}}^{\log \mathrm{x}}\right)}^{\log \log \mathrm{x}},$then $\frac{\mathrm{dy}}{\mathrm{dx}}=?$

1. $\left\{{\left(\mathrm{xlogx}\right)}^{\log \mathrm{logx}}\left\{\left(\frac{1}{\mathrm{xlogx}}\right)\left(\mathrm{logx}+\log \mathrm{logx}\right)\right\}+\left(\log \mathrm{logx}\right)\left[\left(\frac{1}{\mathrm{x}}\right)+\left(\frac{1}{\mathrm{xlogx}}\right)\right]\right\}$

2. $\left\{{\left(\mathrm{xlogx}\right)}^{\log \mathrm{logx}}\left[\log (\log x)\right]\left[\left(\frac{2}{\mathrm{logx}}\right)+\frac{1}{x}\right]\right\}$

3. ${\left(\mathrm{xlogx}\right)}^{\mathrm{xlogx}}\left[\frac{\left(\log \mathrm{logx}\right)}{\mathrm{x}}\right]\left[\left(\frac{1}{\mathrm{logx}}\right)+1\right]$

4. $\left[\frac{\mathrm{ylogy}}{\mathrm{xlogx}}\right]\left[2\log \left(\mathrm{logx}\right)+1\right]$

Q. Evaluate the limit:
$\underset{x\to \infty }{lim}(a^{1/x}-1)x$

Q. For $x,y,z>0$ and $x>m,y>n,z>r,$ if $\left|\begin{array}{ccc}x& n& r\\ m& y& r\\ m& n& z\end{array}\right|=0,$ then the greatest value of $\frac{27xyz}{(x-m)(y-n)(z-r)}$ is

Q.

If $\mathrm{f}:\mathrm{R}\to \mathrm{R}$ is defined by $\mathrm{f}\left(\mathrm{x}\right)=3\mathrm{x}+\left|\mathrm{x}\right|$, then $\mathrm{f}\left(2\mathrm{x}\right)-\mathrm{f}(-\mathrm{x})-6\mathrm{x}=$

1. f(x)

2. 2f(x)

3. -f(x)

4. f(-x)