Factorization Method Form to Remove Indeterminate Form

Q.

Find the next term of the sequence ?

$0,6,24,60,120,210,?$

Q. If Rolle’s theorem holds for the function $f\left(x\right)=x^3–a{x}^2+bx–4,x\in [1,2]$ with $f^{\prime }\left(\frac{4}{3}\right)=0,$ then ordered pair (a, b) is equal to :

1. $(-5,8)$
2. $(5,8)$
3. $(5,-8)$
4. $(-5,-8)$

Q.

Sin10°Sin50°Sin60°Sin70°=√3/16

Prove it

Q.

Evaluate the following limits:

${lim}_{x\to \pi }\frac{1+cosx}{ta{n}^{2}x}$

Q.

Consider the following statements.

p: Suman is brilliant.

q: Suman is rich.

r: Suman is honest.

The negation of the statement “Suman is brilliant and dishonest if and only if Suman is rich” can be expressed as

1. $~[q\leftrightarrow (p\wedge ~r\left)\right]$

2. $~q\leftrightarrow p\wedge r$

3. $~(p\wedge r)\leftrightarrow q$

4. $~p\wedge (q\leftrightarrow r)$

Q. $\underset{x\to \frac{\pi }{4}}{lim}\frac{{\cot }^{3}x-\tan x}{\cos (x+\frac{\pi }{4})}$ is:

1. $4\sqrt{2}$
2. $4$
3. $8\sqrt{2}$
4. $8$

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

Q. $\underset{x\to a}{lim}\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}};(a\ne 0)$ is equal to

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

Q.

The ratio of $4^{3.5}:2^5$ is the same as

Q.

${lim}_{x\to \sqrt{2}}\frac{x^2-2}{x^2+\sqrt{2x}-4}$

Q.

${lim}_{x\to \sqrt{3}}\frac{x^4-9}{x^2+4\sqrt{3x}-15}$

Q.

Sin^{​​​​2}​​​24° - Sin^{​​​​2}​​​6°

Q.

the value of ${lim}_{x\to \infty }\frac{\sqrt{1+x^4}+(1+x^2)}{x^2}$ is

1. -1

2. 1

3. 2

4. None of these

Q. Let $f\left(x\right)=5-|x-2|$ and $g\left(x\right)=|x+1|,x\in R.$ If $f\left(x\right)$ attains maximum value at $\alpha$ and $g\left(x\right)$ attains minimum value at $\beta ,$ then $\underset{x\to -\alpha \beta }{lim}\frac{(x-1)(x^2-5x+6)}{x^2-6x+8}$ is equal to:

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

Q. The domain of the function $f\left(x\right)={\log }_e(x-[x\left]\right)$ is (where $[.]$ represents the greatest integer function)

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

Q.

If $\mathrm{f}:\mathrm{R}\to \mathrm{R}$ is defined as $\mathrm{f}\left(\mathrm{x}\right)={(1-\mathrm{x})}^{\frac{1}{3}}$, then ${\mathrm{f}}^{-1}\left(\mathrm{x}\right)$ is equal to

1. ${\left(1-\mathrm{x}\right)}^{\frac{-1}{3}}$

2. ${\left(1-\mathrm{x}\right)}^3$

3. $1-{\mathrm{x}}^3$

4. $1-{\mathrm{x}}^{\frac{1}{3}}$

Q.

$\underset{x\to \infty }{\lim }{\left(1+\frac{4}{x-1}\right)}^{x+3}=$

1. $e^4$

2. $e^2$

3. $e^3$

4. $e$

Q.

If ${\sin }^{-1}a+{\sin }^{-1}b+{\sin }^{-1}c=\pi$, then the value of $a\sqrt{\left(1-a^2\right)}+b\sqrt{\left(1-b^2\right)}+c\sqrt{\left(1-c^2\right)}$ will be

1. $2\mathrm{abc}$

2. $abc$

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

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

Q.

For $x$ belongs to $R$, Find$\underset{x\mapsto \infty }{lim}{\left[\frac{(x-3)}{(x+2)}\right]}^x=?$

1. $e$

2. $e^{-1}$

3. $e^{-5}$

4. $e^5$

Q. Evaluate the given limit :
$\underset{z\to 1}{lim}\frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}$

Q. The value of $\underset{x\to 0}{lim}\left(\frac{x}{\sqrt[8]{81-\sin x}-\sqrt[8]{81+\sin x}}\right)$ is equal to

1. $4$
2. $-4$
3. $-1$
4. $0$

Q.

${lim}_{x\to 1}\frac{x^4-3x^3+2}{x^3-5x^2+3x+1}$

Q.

The $n^{th}$ term of the sequence $\frac{1}{1+\sqrt{x}},\frac{1}{1-x},\frac{1}{1-\sqrt{x}},.......$ is

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

2. $\frac{1+\sqrt{x}\left(n-1\right)}{1+\sqrt{x}}$

3. $\frac{1+\sqrt{x}\left(n-2\right)}{1-x}$

4. $\frac{3-\sqrt{x}\left(n+2\right)}{3\left(1-x\right)}$

Q.

If a and b are two different positive real numbers, then which of the following relation is true.

1. $2\sqrt{ab}>a+b$

2. $2\sqrt{ab}<a+b$

3. $2\sqrt{ab}=a+b$

4. none of these

Q.

${lim}_{x\to 3}\frac{x^4-81}{x^2-9}$

Q. Evaluate the limit: $\underset{x\to 2}{lim}\frac{x^4-16}{x-2}$

Q.

${lim}_{x\to 1}\frac{x^7-2x^5+1}{x^3-3x^2+2}$

Q.

If $a,b,c$are in AP, then $\frac{1}{(\sqrt{a}+\sqrt{b})},\frac{1}{(\sqrt{a}+\sqrt{c})},\frac{1}{(\sqrt{b}+\sqrt{c})}$ are in

Q. $f\left(x\right)=\prod _{k=1}^{\infty }\left(\frac{1+2\cos \left(\frac{2x}{3^k}\right)}{3}\right)$, then the number of points where $\left[xf\right(x\left)\right]+|xf\left(x\right)|+(x-1)|x^2-3x+2|$ is non-differentiable in $x\in (0,3\pi )$ is equal to (where [.] denotes the greatest integer function)

Q.

Which is equivalent to start root $10$ end root superscript three-fourths $x$ ?