Substitution Method to Remove Indeterminate Form

Q.

$\int \sqrt{\left(1+\cos x\right)}dx=$

1. $2\sin \left(\frac{x}{2}\right)+c$

2. $\sqrt{2}\sin \left(\frac{x}{2}\right)+c$

3. $\left(\frac{1}{2}\right)\sin \left(\frac{x}{2}\right)+c$

4. $\left(\frac{\sqrt{2}}{2}\right)\sin \left(\frac{x}{2}\right)+c$

5. $2\sqrt{2}\sin \left(\frac{x}{2}\right)+c$

Q.

If ${\sin }^{-1}x+{\sin }^{-1}y=\frac{\pi }{2}$ then $\frac{dy}{dx}$ is equal to

1. $\frac{x}{y}$

2. $-\frac{x}{y}$

3. $\frac{y}{x}$

4. $-\frac{y}{x}$

Q.

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

1. $\frac{\sin a}{{\sin }^2(a+y)}$

2. $\frac{{\sin }^2(a+y)}{\sin a}$

3. $\sin a{\sin }^2(a+y)$

4. $\frac{{\sin }^2(a-y)}{\sin a}$

Q.

The value of $\sin \left(50°\right)-\sin \left(70°\right)+\sin \left(10°\right)$ is

1. $0$

2. $1$

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

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

Q.

What is the formula for ${\sin }^{2}A+{\sin }^{2}B$?

Q.

Evaluate ${}^2{C}_2$

Q.

If in $∆ABC$, $\sin \frac{A}{2}\sin \frac{C}{2}=\sin \frac{B}{2}$ and $2s$ is the perimeter of the triangle, then $s$ is equal to:

1. $2b$

2. $b$

3. $3b$

4. $4b$

Q.

Evaluate: $\sin \frac{2\mathrm{\pi }}{3}$

Q.

Evaluate ${\left(\frac{1}{2}\right)}^{-5}$

Q.

The expression of the complex number $\frac{1}{1+\cos \theta -i\sin \theta }$ in the form $a+ib$ is

Q.

Evaluate :$\int \frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}}dx$

1. $x+2\sin x+\sin 2x+c$

2. $x+2\cos x+\sin 2x+c$

3. $x-2\sin x+\sin 2x+c$

4. $x+2\sin x-\sin 2x+c$

Q.

$co{t}^{-1}(2\times 1^2)+co{t}^{-1}(2\times 2^2)+co{t}^{-1}(2\times 3^2)+...$ up to $\infty$ is equal to

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

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

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

4. $\frac{\mathrm{\pi }}{5}$

Q. Evaluate the limit:
$\underset{x\to 0}{lim}(\cos x+\sin x{)}^{1/x}$

Q.

If $\frac{(2\sin \alpha )}{(1+\cos \alpha +\sin \alpha )}=x$, then $\frac{(1-\cos \alpha +\sin \alpha )}{(1+\sin \alpha )}$ is equal to

1. $\frac{1}{x}$

2. $x$

3. $1-x$

4. $1+x$

Q.

$\frac{d\left(\sin \left(2x^2\right)\right)}{dx}$

1. $4x\cos \left(2x^2\right)$

2. $2\sin \left(x^2\right)\cos \left(x^2\right)$

3. $4x\sin \left(x^2\right)$

4. $4x\sin \left(x^2\right)\cos \left(x^2\right)$

Q. $\underset{x\to 0}{lim}\frac{x\cot \left(4x\right)}{{\sin }^{2}x{\cot }^2\left(2x\right)}$ is equal to :

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

Q.

Derivative of ${\sec }^{-1}\left(\frac{1}{1-2x^2}\right)$ with respect to ${\sin }^{-1}\left(3x-4x^3\right)$ is

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

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

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

4. $1$

Q.

If $\cos \theta +sec\theta =\frac{5}{2}$, then the general value of $\theta$ is

1. $n\pi \pm \frac{\pi }{3}$

2. $2n\pi \pm \frac{\pi }{6}$

3. $n\pi \pm \frac{\pi }{6}$

4. $2n\pi \pm \frac{\pi }{3}$

Q.

How do you simplify $i^{35}$?

Q.

Statement $1$: Both $\sin x$ and $\cos x$ are decreasing function in $\left(\frac{\mathrm{\pi }}{2},\mathrm{\pi }\right)$.

Statement $2$: If a differentiable function decreases in $\left(a,b\right)$ then its derivative also decreases in $\left(a,b\right)$.

1. Statement $1$ is true, Statement $2$ is true; statement $2$ is a correct explanation for statement $1$.

2. Statement $1$ is true, statement $2$ is true; statement $2$ is not a correct explanation for statement $1$.

3. Statement $1$ is true, statement $2$ is false

4. Statement $1$ is false, statement $2$ is true

Q.

The value of $\sin \left(20°\right)\left[4+\sec \left(20°\right)\right]$ is

1. $0$

2. $1$

3. $\sqrt{2}$

4. $\sqrt{3}$

Q.

${lim}_{x\to 0}\frac{a^x+a^{-x}-2}{x^2}$

Q.

${lim}_{x\to \pi }\frac{\sqrt{5+cosx}-2}{(\pi -x{)}^2}$

Q.

${lim}_{x\to a}\frac{asinx-xsina}{a{x}^2-x{a}^2}$

Q.

If $2y={\sin }^{-1}(x+5y)$, then $\frac{dy}{dx}$ is equal to

1. $\cos 2y–5$

2. $\cos 2y+5$

3. $\frac{1}{(2\cos 2y–5)}$

4. None of these

Q.

If $2{\tan }^{-1}(\cos x)={\tan }^{-1}(2\cos ecx)$, then the value of $x$ is

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

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

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

4. None of these

Q.

Evaluate ${\tan }^{-1}5+{\tan }^{-1}3-{\cot }^{-1}\left(\frac{4}{7}\right)$:

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

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

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

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

Q.

If $\sin y=x\sin (a+y)$ and $\frac{dy}{dx}=\frac{A}{1+x^2-2x\cos a}$, then the value of $A$ is

1. $2$

2. $\cos a$

3. $\sin a$

4. None of these

Q.

The period of the function $f\left(x\right)=\left|\cos \left(\frac{x}{2}\right)\right|+\left|\sin x\right|$ is:

1. $2\mathrm{\pi }$

2. $\mathrm{\pi }$

3. $4\mathrm{\pi }$

4. None of these

Q.

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