Parametric Differentiation

Q. the value of theta for which sin theta = cos theta where , 180<theta<360

Q.

Prove that:

$\text{sin 3x+ sin 2x - sin x = 4 sin x cos}\frac{x}{2}\text{cos}\frac{3x}{2}$

Q.

$y={\left(\tan x\right)}^{{\left(\tan x\right)}^{\left(\tan x\right)}}$ , then at $x=\frac{\mathrm{\pi }}{4}$ , the value of $\frac{dy}{dx}=$

1. $0$

2. $1$

3. $2$

4. None of these

Q. Find the derivative of the following function:
f(x)= $\frac{ax+b}{(p{x}^2+qx+r)}$

Q. Express cos 6 theta in terms of cos theta

Q. If $y=\sqrt{(a-x)(x-b)}-(a-b)ta{n}^{-1}\sqrt{\left(\frac{a-x}{x-b}\right)}$, then $\frac{dy}{dx}$ is equal to

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

Q. If $x=sec\theta -cos\theta ,y=se{c}^{10}\theta -co{s}^{10}\theta$ and $(x^2+4){\left(\frac{dy}{dx}\right)}^2=k(y^2+4)$, then k is equal to

1. $\frac{1}{100}$
2. $1$
3. $10$
4. $100$

Q. If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of $\left|z\right|$.

Q. If $\cos \mathrm{\theta }=\frac{1}{2}\left(a+\frac{1}{a}\right),$ and $\cos 3\mathrm{\theta }=\lambda \left(a^3+\frac{1}{a^3}\right)$, then $\lambda =$
(a) $\frac{1}{4}$

(b) $\frac{1}{2}$

(c) 1

(d) none of these

Q. If tan θ = $x-\frac{1}{4x}$, then sec θ − tan θ is equal to
(a) $-2x,\frac{1}{2x}$
(b) $-\frac{1}{2x},2x$
(c) 2x
(d) $2x,\frac{1}{2x}$

Q. Prove that:
$\cos 6°\cos 42°\cos 66°\cos 78°=\frac{1}{16}$

Q. Write the interval in which the value of 5 cos θ + 3 cos $\left(\mathrm{\theta }+\frac{\pi }{3}\right)+3$ lies.

Q. If $x=e^{\theta }(\sin \theta +\cos \theta )$ and $y=e^{\theta }(\sin \theta –\cos \theta ),$ where $\theta$ is a real parameter, then $\frac{d^{2}y}{d{x}^2}$ at $\theta =\frac{\pi }{6}$ is

1. $\frac{\sqrt{3}}{2}$
2. $\frac{4}{3\sqrt{3}}{e}^{-\pi /6}$
3. $\frac{8}{3\sqrt{3}}{e}^{-\pi /6}$
4. $\frac{4}{3}$

Q. If $x=2\sin \theta -\sin 2\theta$ and $y=2\cos \theta -\cos 2\theta$, $\theta \in [0,2\pi ],$ then $\frac{d^{2}y}{d{x}^2}$ at $\theta =\pi$ is :

1. $-\frac{3}{8}$
2. $\frac{3}{4}$
3. $\frac{3}{2}$
4. $-\frac{3}{4}$
5. None of these

Q. If x = a $cos\theta ,y=bsin\theta ,then\frac{d^{3}y}{d{x}^3}$ is equal to

1. $\left(\frac{-3b}{a^3}\right)cose{c}^4\theta co{t}^4\theta$
2. $\left(\frac{3b}{a^3}\right)cose{c}^4\theta cot\theta$
3. $\left(\frac{-3b}{a^3}\right)cose{c}^4\theta cot\theta$
4. None of the above

Q. If $f\left(x\right)=\frac{2x}{1+x^2}$, show that f(tan θ) = sin 2θ.

Q. In $∆ABC,\mathrm{if}\angle B=60°,$ prove that $\left(a+b+c\right)\left(a-b+c\right)=3ca$.

Q. If $\mathrm{cosec\theta }+\mathrm{cot\theta }=\frac{11}{2}$, then tan θ =
(a) $\frac{21}{22}$

(b) $\frac{15}{16}$

(c) $\frac{44}{117}$

(d) $\frac{117}{44}$

Q.

what is the value of cot 660 degrees?

Q. If ${\tan }^2\mathrm{\theta }=1-e^2,$ prove that $\sec \mathrm{\theta }+{\tan }^3\mathrm{\theta }\mathrm{cosec}\mathrm{\theta }=2{\left(2-e^2\right)}^{3/2}$.

Q. If $t=\frac{\left(}{1-\mathrm{cos\theta }}\mathrm{and}s=2\tan \left(\frac{\theta }{2}\right)$, then find $\frac{\mathrm{dt}}{\mathrm{ds}}\mathrm{at}\theta =\frac{\pi }{2}$

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

Q. Given the parametric equations $x=f\left(t\right),y=g\left(t\right)$, then $\frac{d^{2}y}{d{x}^2}$ equals

1. $\frac{\frac{d^{2}y}{d{t}^2}.\frac{dx}{dt}-\frac{dy}{dt}\frac{d^{2}x}{d{t}^2}}{{\left(\frac{dx}{dt}\right)}^2}$
2. $\frac{\frac{dx}{dt}\frac{d^{2}y}{d{t}^2}-\frac{d^{2}x}{d{t}^2}\frac{dy}{dt}}{{\left(\frac{dx}{dt}\right)}^3}$
3. $\frac{\frac{d^{2}y}{d{t}^2}}{\frac{d^{2}x}{d{t}^2}}$
4. $\frac{\frac{d^{2}y}{d{t}^2}.\frac{dx}{dt}-\frac{dy}{dt}\frac{d^{2}x}{d{t}^2}}{\left(\frac{dx}{dt}\right)}$

Q. If sec $\mathrm{\theta }=\mathrm{x}+\frac{1}{4\mathrm{x}}$, then sec θ + tan θ =
(a) $x,\frac{1}{x}$
(b) $2x,\frac{1}{2x}$
(c) $-2x,\frac{1}{2x}$
(d) $-\frac{1}{x},x$

Q. Prove that:

$\frac{\sin \mathrm{\theta }+\sin 2\mathrm{\theta }}{1+\cos \mathrm{\theta }+\cos 2\mathrm{\theta }}=\tan \mathrm{\theta }$

Q. If $e^x=\frac{\sqrt{1+t}-\sqrt{1-t}}{\sqrt{1+t}+\sqrt{1-t}}$ and $\tan \frac{y}{2}=\sqrt{\frac{1-t}{1+t}}$ then $\frac{dy}{dx}$ at $t=\frac{1}{2}$ is

1. $\frac{1}{2}$
2. $-\frac{1}{2}$
3. $0$
4. None of these

Q. $\underset{x\to 0}{\lim }\frac{x^2}{\sin x^2}$