L Hospital's Rule

Q.

The value of $\tan \left(40°\right)+\tan \left(20°\right)+\sqrt{3}\tan \left(20°\right)\tan \left(40°\right)$ is

1. $\sqrt{12}$

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

3. $\sqrt{3}$

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

Q.

Find the $270$ degrees value for all the trigonometric angles.

Q.

$\tan (81°)-\tan (63°)-\tan (27°)+\tan (9°)$ equals

1. $6$

2. $0$

3. $2$

4. $4$

Q.

The value of $\tan \left(-945°\right)$ is

1. $-1$

2. $-2$

3. $-3$

4. $-4$

Q.

The value of $\frac{\tan \left(70°\right)-\tan \left(20°\right)}{\tan \left(50°\right)}$ is

1. $2$

2. $1$

3. $0$

4. $3$

Q.

$\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{(cotx-cosx)}{{}_{(\mathrm{\pi }-2x)}3}$

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

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

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

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

Q.

What is $\frac{\tan C}{\sin C}$ is equal to?

Q.

$sec\left(50°\right)+\tan \left(50°\right)$ is equal to

1. $\tan (20°)+\tan (50°)$

2. $2\tan (20°)+\tan (50°)$

3. $\tan (20°)+2\tan (50°)$

4. $2\tan (20°)+2\tan (50°)$

Q.

The value of $\tan \left(20°\right)+2\tan \left(50°\right)-\tan \left(70°\right)$ is

1. $1$

2. $0$

3. $\tan \left(50°\right)$

4. None of these.

Q.

If $\frac{\tan \left(3A\right)}{\tan \left(A\right)}=a$, then $\frac{\sin \left(3A\right)}{\sin \left(A\right)}$ is equal to

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

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

3. $\frac{a}{a+1}$

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

Q. If f(x) has a derivative at x=a, then ${lim}_{x\to a}\frac{xf\left(a\right)-af\left(x\right)}{x-a}$ is equal to

1. $af\left(a\right)+f^{\prime }\left(a\right)$
2. $af\left(a\right)-f^{\prime }\left(a\right)$
3. $f\left(a\right)+f^{\prime }\left(a\right)$
4. $f\left(a\right)-a{f}^{\prime }\left(a\right)$

Q.

If $y={(\sin x)}^{\tan x}$, then $\frac{dy}{dx}$is equal to

1. $\sin x^{\tan x}(1+se{c}^{2}x\log \sin x)$

2. $\tan x{(\sin x)}^{\tan x-1}\cos x$

3. $\sin x^{\cos x}se{c}^{2}x\log \sin x$

4. $\tan x{(\sin x)}^{\tan x-1}$

Q. Let $A\left(t\right)=\left[a_{ij}\right]$ be a matrix of order $3\times 3$ given by $a_{ij}=\{\begin{array}{cc}2\cos t,& \text{if}i=j\\ 1,& \text{if}|i-j|=1\\ 0,& \text{otherwise}\end{array}$ where $\left|A\right(t\left)\right|$ is determinant value of matrix $A\left(t\right).$ Then which of the following is/are CORRECT?

1. $\underset{t\to 0}{lim}\left|A\right(t\left)\right|\left|A\right(4t\left)\right|=16$
2. Maximum value of $\left|A\right(t\left)\right|\left|A\right(3t\left)\right|$ is $16$
3. $\underset{0}{\overset{\pi }{\int }}\left|A\right(t\left)\right|\left|A\right(4t\left)\right|dt=0$
4. $\left|A\left(\frac{\pi }{17}\right)\right|\left|A\left(\frac{4\pi }{17}\right)\right|=1$

Q.

Integrate $\int \frac{1+\tan x}{1-\tan x}dx$

Q.

Find the value of $$tan{\displaystyle \frac{19\pi }{3}}$$.

Q.

If f(x) has a derivative at x=0, then ${lim}_{x\to a}\frac{xf\left(a\right)-af\left(x\right)}{x-a}=$

1. f(a)+af(a)

2. f(a)-af(a)

3. f(a)+af(a)

4. f(a)-af'(a)

Q.

${lim}_{x\to 0^{+}}xlo{g}_{e}x$ will be equal to ___

Q. If f(x) has a derivative at x=a, then ${lim}_{x\to a}\frac{xf\left(a\right)-af\left(x\right)}{x-a}$ is equal to

1. $f\left(a\right)-a{f}^{\prime }\left(a\right)$
2. $af\left(a\right)-f^{\prime }\left(a\right)$
3. $f\left(a\right)+f^{\prime }\left(a\right)$
4. $af\left(a\right)+f^{\prime }\left(a\right)$

Q. $\underset{x\to 1}{lim}[\frac{1}{\ln x}-\frac{1}{(x-1)}]$

1. Does not exist
2. $1$
3. $\frac{1}{2}$
4. $0$

Q. Evaluate : $\underset{x\to 1}{lim}\frac{x^3-3x+1}{x-1}$

Q. Evaluate $\underset{x\to 1}{lim}\frac{x^2-3x+2}{x^3-4x+3}$

Q. $\underset{x\to 0}{lim}\frac{(x+1{)}^5-1}{x}$

Q.

$\sin 120°\cos 150°-\cos 240°\sin 330°$ is equal to

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

2. $-1$

3. $1$

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

Q. $L=\underset{x\to \pi /2}{lim}[x\tan x-\left(\pi /2\right)\sec x]$
Then value of $L+2$ is

Q. ${\lim }_{x\to 0}\frac{x^2-\tan 2x}{\tan x}$

Q.

Write the value of ${lim}_{x\to \frac{\pi }{2}}\frac{2x-\pi }{cosx}.$

Q.

${lim}_{x\to \frac{\pi }{2}}(1-sinx)tanx$ will be equal to _____

1. 1

2. -1

3. 0

4. None of the above

Q.

${lim}_{x\to 0}xlo{g}_{e}x$ will be equal to _____

___

Q. $\underset{x\to \frac{\pi }{2}}{lim}[x\tan x-\frac{\pi }{2}\sec x]$ is equal to

1. $-1$
2. None of these
3. $0$
4. $1$

Q.

Evaluate :$\int \frac{\cos x}{{\sin }^{2}x(\sin x+\cos x)}dx$

1. $\log \left|\frac{(1+\tan x)}{\tan x}\right|-\cot \left(x\right)+C$

2. $\log \left|\frac{t\mathrm{an}\left(x\right)}{1+\tan \left(x\right)}\right|+C$

3. $\log \left|\frac{t\mathrm{an}\left(x\right)}{1+\tan \left(x\right)}\right|-\tan \left(x\right)+C$

4. $\log \left|\frac{\tan \left(x\right)}{(1+\tan x)}\right|+\cot \left(x\right)+C$