Limit

Q.

If $u={\sin }^{-1}\left(\frac{(x^2+y^2)}{(x+y)}\right)$, then $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}$ is equal to

1. $\sin u$

2. $\tan u$

3. $\cos u$

4. $cotu$

Q.

E.m.f of the following cell at $298\mathrm{K}$ in $\mathrm{V}$ is $\mathrm{x}\times {10}^{-2}.$

$\mathrm{Zn}|{\mathrm{Zn}}^{2+}(0.1\mathrm{M}\left)\right|\left|{\mathrm{Ag}}^{+}\right(0.01\mathrm{M}\left)\right|\mathrm{Ag}$

The value of x is _________. (Rounded off to the nearest integer)

$$\begin{gathered} \mathrm{Given}: \\ {\mathrm{E}}_{{\mathrm{Zn}}^{2+}/\mathrm{Zn}}^{°}=-0.76; \\ {\mathrm{E}}_{{\mathrm{Ag}}^{+}/\mathrm{Ag}}^{°}=+0.80\mathrm{V}; \\ 2.303\frac{\mathrm{RT}}{\mathrm{F}}=0.059 \end{gathered}$$

Q.

${lim}_{n\to \infty }\left[\frac{1-2+3-4+5-6+\cdots (2n-1)-2n}{\sqrt{n^2+1}+\sqrt{n^2-1}}\right]$ is equal to

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

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

3. 1

4. -1

Q.

The value of $\underset{n\to \infty }{\lim }\left[\frac{x^n}{x^n+1}\right]$, where $x<-1$ is

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

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

3. $1$

4. None of these

Q.

${}^n{C}_{r-1}=36,{}^n{C}_r=84\&{}^n{C}_{r+1}=126$, then $r$ is equal to:

1. $1$

2. $2$

3. $3$

4. None of these

Q.

Write the value of ${lim}_{x\to 0^{-}}\left[x\right].$

Q.

$\underset{x\to 0}{\lim }{(1-ax)}^{\frac{1}{x}}=$

1. $e^{-a}$

2. $e$

3. $e^a$

4. $1$

Q.

If $(x+y)\sin u=x^2{y}^2$, then $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}$ is equal to:

1. $\sin u$

2. $\mathrm{cosec}u$

3. $\tan u$

4. $3\tan u$

Q.

Find the range of ${\sin }^{-1}\left(x\right)-{\cos }^{-1}\left(x\right)$

Q. Evaluate the given limit :
$\underset{x\to 0}{lim}(\text{cosec}x-\cot x)$

Q.

The number of solutions of the equation $\sin \left(2x\right)+\cos \left(4x\right)=2$ is

1. $0$

2. $1$

3. $2$

4. $\infty$

Q.

If $0<a<b$, then lim_n→∞ $\underset{n\to \infty }{\lim }\left[\frac{(a^n+b^n)}{a^n-b^n}\right]=$

1. $0$

2. $-1$

3. $1$

4. $Doesnotexist$

Q.

The value of $\underset{x\to 2}{\lim }\frac{5}{\sqrt{2}-\sqrt{X}}$ is

1. $10\sqrt{2}$

2. $+\infty$

3. $-\infty$

4. does not exist.

Q.

${lim}_{x\to 2}\frac{x^3-8}{x^2-4}$

Q.

${lim}_{x\to 1}\frac{x^3-3x+1}{x-1}$

Q.

If the integral of $(\sin 2\mathrm{x}-\cos 2\mathrm{x})=\frac{1}{\sqrt{2}}\sin (2\mathrm{x}-\mathrm{a})+\mathrm{b}$ then find $\mathrm{a}$ and $\mathrm{b}$.

Q.

${lim}_{x\to a}\frac{\sqrt{x}+\sqrt{a}}{x+a}$

Q.

Prove that ${lim}_{x\to a^{+}}\left[x\right]$ =[a] for all $a\in R$. Also, prove that ${lim}_{x\to 1^{-}}\left[x\right]=0$

Q. Evaluate the limit : $\underset{x\to 0}{lim}\frac{x^{2/3}-9}{x-27}$

Q. Which of the functions defined below is/are one-one function(s)?

1. $f\left(x\right)=x+1$ for $x>-1$
2. $g\left(x\right)=x+\frac{1}{x}$ for $x>0$
3. $h\left(x\right)=x^2+4x-5$ for $x>0$
4. $f\left(x\right)=e^{-x}$ for $x\ge 0$

Q.

If $a,b,c$ are in H.P., then for all $n\in N$ the true statement is

1. $a^n+c^n<2b^n$

2. $a^n+c^n>2b^n$

3. $a^n+c^n=2b^n$

4. $0$

Q.

$\underset{x\to 0}{\lim }{(\cos ecx)}^{\frac{1}{\log \left(x\right)}}=$

1. $0$

2. $1$

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

4. None of these

Q.

If $2^x.3^{x+4}=7^x$, then $x$ is equal to

1. $\frac{4{\log }_{e}3}{{\log }_{e}7-{\log }_{e}6}$

2. $\frac{4{\log }_{e}3}{{\log }_{e}6-{\log }_{e}7}$

3. $\frac{2{\log }_{e}3}{{\log }_{e}7-{\log }_{e}6}$

4. $\frac{2{\log }_{e}3}{{\log }_{e}6-{\log }_{e}7}$

Q. Let $f\left(x\right)=\left\{x\right\}.$ Then which of the following(s) is/are correct
(where $\{.\}$ represents the fractional part function)

1. $f\left(1^{-}\right)=1$
2. $f\left(1\right)=0$
3. $x=1$ is the point of minimum.
4. $x=1$ is the point of maximum

Q.

If ${\mathrm{S}}_{\mathrm{n}}=1^3+2^3+\dots ..+{\mathrm{n}}^3$ and ${\mathrm{T}}_{\mathrm{n}}=1+2+\dots ..+\mathrm{n}$, then

1. ${\mathrm{S}}_{\mathrm{n}}={\mathrm{T}}_{\mathrm{n}}$

2. ${\mathrm{S}}_{\mathrm{n}}={\mathrm{T}}_{\mathrm{n}}^4$

3. ${\mathrm{S}}_{\mathrm{n}}={\mathrm{T}}_{\mathrm{n}}^2$

4. $S_n=T_n^3$

Q. $\underset{x\to -1}{lim}\frac{(1+x^{})(1-x^2)(1+x^3)(1-x^4)\cdots (1-x^{4n})}{\left[\right(1+x^{}\left)\right(1-x^2\left)\right(1+x^3\left)\right(1-x^4)\cdots (1-x^{2n}){]}^2}$ is equal to

1. ${}^{4n}{C}_{2n}$
2. ${}^{2n}{C}_n$
3. $2\cdot {}^{4n}{C}_{2n}$
4. $2\cdot {}^{2n}{C}_n$

Q.

${lim}_{x\to 0}9$

Q.

${lim}_{x\to 0}\frac{log(2+x)+log0.5}{x}$

Q.

In Triangle ABC. Line joining the Circum-centre and Ortho-centre is Parallel to side AC, then value of tan A tan C is equal to

1. 3

2. 

3. 

4. None of these

Q.

Evaluate ${lim}_{x\to 2}f\left(x\right)$ (if it exists), where f(x)

$=\{\begin{array}{cc}x-\left[x\right],& x<2\\ 4,& x=2\\ 3x-5,& x>2\end{array}$