Parametric Differentiation

Q.

If $x=a\left(t-\sin t\right)$ and $y=a\left(1-\cos t\right)$, then $\frac{dy}{dx}=$

1. $\tan \left(\frac{t}{2}\right)$

2. $-\tan \left(\frac{t}{2}\right)$

3. $\cot \left(\frac{t}{2}\right)$

4. $-\cot \left(\frac{t}{2}\right)$

Q.

Prove that${\sin }^{6}A+{\cos }^{6}A=1-3{\sin }^{2}A{\cos }^{2}A$

Q.

How do you find the value of $\tan \left(-135°\right)$?

Q.

The value of $\sin \left[{\tan }^{-1}\left(\frac{1-x^2}{2x}\right)+{\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right]$ is

1. $1$

2. $0$

3. $-1$

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

Q.

Evaluate $\frac{1+\sin x}{1-\sin x}$

Q.

$\cos \left[{\cos }^{-1}\left(-\frac{1}{7}\right)+{\sin }^{-1}\left(-\frac{1}{7}\right)\right]=?$

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

2. $0$

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

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

Q.

If $x=\frac{2at}{1+t^3}$ and $y=\frac{2a{t}^2}{{\left(1+t^3\right)}^2}$, then $\frac{dy}{dx}$ is equal to:

1. $ax$

2. $a^2{x}^2$

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

4. $\frac{x}{2a}$

Q.

${10}^{-x\tan x}\left[\frac{d}{dx}\left({10}^{x\tan x}\right)\right]$ is equal to

1. $\tan x–xse{c}^{2}x$

2. $\log 10[\tan x+xse{c}^{2}x]$

3. $x\tan x\log 10$

4. None of these

Q.

Express $\sin \frac{x}{2}$ in terms of $\cos x$ using the double angle identity.

Q.

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

1. $1$

2. $2$

3. $3$

4. $4$

Q.

Find the value of $\mathrm{cosec}(-1410°).$

Q.

If $y=se{c}^{-1}\left(\frac{\left(x+1\right)}{\left(x-1\right)}\right)+{\sin }^{-1}\left(\frac{\left(x-1\right)}{\left(x+1\right)}\right)$, then $\frac{dy}{dx}=$

1. $1$

2. $0$

3. $\frac{x-1}{x+1}$

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

Q.

We have to prove that

Prove that $(1+cotA-\cos ecA)(1+\tan A+secA)=2$

Q.

If $p={\sin }^{2}x+{\cos }^{4}x$, then

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

2. $\frac{3}{16}\le p\le 1/4$

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

4. $1\le p\le 2$

Q.

Given that $\sin \left(\mathrm{\theta }\right)+2\cos \left(\mathrm{\theta }\right)=1$, then prove that $2\sin \left(\mathrm{\theta }\right)-\cos \left(\mathrm{\theta }\right)=2$.

Q.

If ${\int }_0^p\frac{dx}{\left(1+4x^2\right)}=\frac{\mathrm{\pi }}{8}$, then the value of $p$ is

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

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

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

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

Q.

$\frac{1}{(3^2-1)}+\frac{1}{(5^2-1)}+\frac{1}{(7^2-1)}+\dots \dots .+\frac{1}{({201}^2-1)}$ is equal to

1. $\frac{101}{404}$

2. $\frac{99}{400}$

3. $\frac{25}{101}$

4. $\frac{101}{408}$

Q.

If $y={\log }_{10}x+{\log }_{x}10+{\log }_{x}x+{\log }_{10}10$, then $\frac{dy}{dx}$

1. $\frac{1}{x\log 10}-\frac{\log 10}{x(\log x{)}^2}$

2. $\frac{1}{x{\log }_{2}10}-\frac{1}{x{\log }_{10}\mathrm{e}}$

3. $\frac{1}{x\log 10}+\frac{\log 10}{x(\log x{)}^2}$

4. none of these

Q.

For every pair of continuous function $\mathrm{f},\mathrm{g}:\left[0,1\right]\to \mathrm{ℝ}$ such that $\max \mathrm{f}\left(x\right):x\in \left[0,1\right]=\max \mathrm{g}\left(x\right):x\in \left[0,1\right]$, the correct statement(s) is (are)

1. ${\left[\mathrm{f}\left(c\right)\right]}^2+3\mathrm{f}\left(c\right)={\left[\mathrm{g}\left(\mathrm{c}\right)\right]}^2+3\mathrm{g}\left(\mathrm{c}\right)$ for some $\mathrm{c}\in \left[0,1\right]$

2. ${\left[\mathrm{f}\left(c\right)\right]}^2+\mathrm{f}\left(c\right)={\left[\mathrm{g}\left(c\right)\right]}^2+3\mathrm{g}\left(c\right)$ for some $\mathrm{c}\in \left[0,1\right]$

3. ${\left[\mathrm{f}\left(c\right)\right]}^2+3\mathrm{f}\left(c\right)={\left[\mathrm{g}\left(\mathrm{c}\right)\right]}^2+\mathrm{g}\left(\mathrm{c}\right)$ for some $\mathrm{c}\in \left[0,1\right]$

4. ${\left[\mathrm{f}\left(c\right)\right]}^2={\left[\mathrm{g}\left(c\right)\right]}^2$ for some $\mathrm{c}\in \left[0,1\right]$

Q. If $x=3\tan t$ and $y=3\sec t$, then the value of $\frac{d^{2}y}{d{x}^2}\text{at}t=\frac{\pi }{4}$, is :

1. $\frac{1}{6}$
2. $\frac{1}{6\sqrt{2}}$
3. $\frac{1}{3\sqrt{2}}$
4. $\frac{3}{2\sqrt{2}}$

Q. Let $f:R\to R$ where $f\left(x\right)=\frac{x^2+4x+7}{x^2+x+1},$ then $f\left(x\right)$ is

1. one-one function
2. many-one function
3. not a function itself
4. a constant function

Q.

Evaluate:${\int }_0^{\frac{\mathrm{\pi }}{2}}x\sin xdx$

1. $0$

2. $1$

3. $-1$

4. $2$

Q. If $\alpha ,\beta$ and $\gamma$ are three consecutive terms of a non-constant G.P. such that the equations $\alpha x^2+2\beta x+\gamma =0$ and $x^2+x–1=0$ have a common root, then $\alpha (\beta +\gamma )$ is equal to :

1. $\beta \gamma$
2. $\alpha \beta$
3. $\alpha \gamma$
4. $0$

Q. Let $f\left(x\right)=15-|x-10|;x\in R$. Then the set of all values of $x$, at which the function, $g\left(x\right)=f\left(f\right(x\left)\right)$ is not differentiable is:

1. $\{5,10,15\}$
2. $\{5,10,15,20\}$
3. $\{10,15\}$
4. $\left\{10\right\}$

Q.

Prove the identity $\frac{cot\theta +\cos ec\theta -1}{cot\theta -\cos ec\theta +1}=\frac{1+\cos \theta }{\sin \theta }$.

Q.

The value of ${\int }_0^1\frac{1}{e^x+e}dx$ is

1. $\left(\frac{1}{e}\right)\log \left[\left(\frac{1+e}{2}\right)\right]$

2. $\log \left[\left(\frac{1+e}{2}\right)\right]$

3. $\left(\frac{1}{e}\right)\log (1+e)$

4. $\log \left[\frac{2}{\left(1+e\right)}\right]$

5. $\left(\frac{1}{e}\right)\log \left[\frac{2}{\left(1+e\right)}\right]$

Q.

If $x=sec\Phi -\tan \Phi ,y=\cos ec\Phi +cot\Phi$, then

1. $x=\frac{y+1}{y-1}$

2. $x=\frac{y-1}{y+1}$

3. $y=\frac{1+x}{1-x}$

4. None of these

Q.

$\sin x=\frac{1}{4},x$ in Quadrant $II$. Find the value of $\sin \frac{x}{2}$

Q.

One of the values of ${\left(\frac{(1+i)}{\sqrt{2}}\right)}^{\frac{2}{3}}$ is

1. $\sqrt{3}+i$

2. $-\mathrm{i}$

3. $\mathrm{i}$

4. $-\sqrt{3}+i$

Q.

If $C_{35}^{189}+C_X^{189}=C_X^{190}$, then $X$ is equal to

1. $34$

2. $35$

3. $36$

4. $37$