Rationalization Method to Remove Indeterminate Form

Q.

The value of $\sin \left[2{\cos }^{-1}\left(\frac{\sqrt{5}}{3}\right)\right]$ is

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

2. $\frac{2\sqrt{5}}{3}$

3. $\frac{4\sqrt{5}}{9}$

4. $\frac{2\sqrt{5}}{9}$

Q. $\underset{x\to 1^{-}}{lim}\frac{\sqrt{\pi }-\sqrt{2{\sin }^{-1}x}}{\sqrt{1-x}}$ is equal to :

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

Q.

How to find permutation or combination of ${}^7{C}_4$

Q.

$\sin \left[3{\sin }^{-1}\left(\frac{1}{5}\right)\right]=$

1. $\frac{71}{125}$

2. $\frac{74}{125}$

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

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

Q.

Evaluate: $limi{t}_{x\to \infty }\left(\frac{x^3}{3x^2-4}-\frac{x^2}{3x+2}\right)$

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

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

3. $0$

4. $\frac{2}{9}$

Q. If $\underset{x\to \infty }{lim}(\sqrt{x^2-x+1}-ax-b)=0,$ then for $k\ge 2,$ $\underset{n\to \infty }{lim}{\sec }^{2n}(k!\pi b)$ is equal to

1. $-a$
2. $a$
3. $-b$
4. $b$

Q.

The value of $\underset{x\to \infty }{\lim }\sqrt{a^2{x}^2+ax+1}-\sqrt{a^2{x}^2+1}$is

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

2. $1$

3. $2$

4. None of these.

Q.

${lim}_{x\to 1}\frac{\sqrt{3+x}-\sqrt{5-x}}{x^2-1}$

Q.

$\left(1+\cos \frac{\pi }{8}\right)\left(1+\cos \frac{3\pi }{8}\right)\left(1+\cos \frac{5\pi }{8}\right)\left(1+\cos \frac{7\pi }{8}\right)$

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

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

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

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

Q.

${\sin }^{-1}\left(\frac{1}{\sqrt{5}}\right)+{\cot }^{-1}\left(3\right)$ is equal to

1. $\frac{\pi }{6}$

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

3. $\frac{\pi }{3}$

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

Q.

${lim}_{x\to 0}\frac{\sqrt{1+x}-\sqrt{1-x}}{2x}$

Q.

If $\underset{x\to 0}{lim}\frac{ax-\left(e^{4x}-1\right)}{ax\left(e^{4x}-1\right)}$exists and is equal to $b$, then the value of $a-2b$ is:

Q.

Let f(x) is a continuous function which takes positive values for x (x>0), and satisfy ${\int }_0^{x}f\left(t\right)dt=x\sqrt{f\left(x\right)}$ with $f\left(1\right)=\frac{1}{2}$. Then the value of $f(\sqrt{2}+1)$ equals

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

Q.

${lim}_{x\to 0}\frac{\sqrt{1+x^2}-\sqrt{1+x}}{\sqrt{1+x^3}-\sqrt{1+x}}$

Q.

$\underset{x\to -\infty }{\lim }\left[\frac{\left(2x-1\right)}{\left(\sqrt{x^2+2x+1}\right)}\right]=$

1. $2$

2. $-2$

3. $1$

4. $-1$

5. $0$

Q.

If $z_1=\sqrt{2}\left(\cos \frac{\mathrm{\pi }}{4}+i\sin \frac{\mathrm{\pi }}{4}\right)$, $z_2=\sqrt{3}\left(\cos \frac{\mathrm{\pi }}{3}+i\sin \frac{\mathrm{\pi }}{3}\right)$, then $\left|z_1{z}_2\right|$ is equal to

1. $6$

2. $\sqrt{2}$

3. $\sqrt{6}$

4. $\sqrt{3}$

5. $\sqrt{2}+\sqrt{3}$

Q.

${lim}_{x\to 0}\frac{\sqrt{1+3x}-\sqrt{1-3x}}{x}$

Q.

$\sin \left[\left(\frac{\pi }{2}\right)-{\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]=$

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

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

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

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

Q.

$d/dx\log \left(\sqrt{(x-}a\right)+\sqrt{(x-b}\left)\right)=$

1. $1/2\sqrt{(x-a})+\sqrt{(x-b})$

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

3. $1/\sqrt{(x-a)}+\sqrt{(x-b)}$

4. $noneofthese$

Q.

If$y=\cos (\sin x^2)$, then at $x=\sqrt{\frac{\mathrm{\pi }}{2}},\frac{dy}{dx}$ is equals to ?

1. $-2$

2. $2$

3. $0$

4. none of these

Q.

${lim}_{x\to \infty }\frac{\sqrt{x^2+a^2}-\sqrt{x^2+b^2}}{\sqrt{x^2+c^2}-\sqrt{x^2+d^2}}$

Q.

If $\alpha$ is positive root of the equation, $\begin{array}{rcl}p\left(x\right)& =& x^2-x-2=0\end{array}$, $\underset{x\to {\alpha }^{+}}{\lim }\frac{\sqrt{1-\cos p\left(x\right)}}{x+\alpha -4}$ is equal to:

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

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

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

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

Q.

If $f\left(x\right)={\tan }^{-1}\left(secx+\tan x\right)$, $\left(\frac{-\mathrm{\pi }}{2}\right)<x<\left(\frac{\mathrm{\pi }}{2}\right)$, and $f\left(0\right)=0$, then $f\left(1\right)$ is equal to

1. $\frac{\left(\mathrm{\pi }+1\right)}{4}$

2. $\frac{\left(\mathrm{\pi }+2\right)}{4}$

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

4. $\frac{\left(\mathrm{\pi }-1\right)}{4}$

Q.

${lim}_{x\to \sqrt{2}}\frac{\sqrt{3+2x}-(\sqrt{2}+1)}{x^2-2}$

Q.

$\int \sqrt{\left(1+\sin \left(\frac{x}{4}\right)\right)}dx=$

1. $8\left[\sin \left(\frac{x}{8}\right)+\cos \left(\frac{x}{8}\right)\right]+c$

2. $8\left[\sin \left(\frac{x}{8}\right)-\cos \left(\frac{x}{8}\right)\right]+c$

3. $8\left[\sin \left(\frac{x}{8}\right)+\sin \left(\frac{x}{8}\right)\right]+c$

4. $\left(\frac{1}{8}\right)\left[\sin \left(\frac{x}{8}\right)-\cos \left(\frac{x}{8}\right)\right]+c$

Q.

The value of $\underset{x\to 0}{\lim }\frac{1}{x^3}{\int }_0^x\frac{t\ln \left(1+t\right)}{t^4+4}dt$ is,

1. $0$

2. $\frac{1}{12}$

3. $\frac{1}{24}$

4. $\frac{1}{64}$

Q.

${lim}_{x\to -\infty }(\sqrt{4x^2-7x}+2x)$

Q.

The value of ${lim}_{x\to 0}\frac{\sqrt{a^2-ax+x^2}-\sqrt{a^2+ax+x^2}}{\sqrt{a+x}-\sqrt{a-x}}$ is

1. $-\sqrt{a}$
   

2. $\sqrt{a}$

3. -a

4. a

Q.

If $cotx+\cos ecx=\surd 3$ , then the principal value of $\left(x-\frac{\pi }{6}\right)$

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

2. $\frac{\mathrm{\pi }}{4}$

3. $\frac{\mathrm{\pi }}{2}$

4. $\frac{\mathrm{\pi }}{6}$

Q.

If ${\sin }^{-1}\frac{3}{x}+{\sin }^{-1}\frac{4}{x}=\frac{\pi }{2}$, then $x=$

1. $3$

2. $5$

3. $7$

4. $11$