Parametric Differentiation

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 $x=2cost-cot2t,y=2sint-sin2t$, then $\frac{d^{2}y}{d{x}^2}$ at t = $\frac{\pi }{2}$ is

1. $\frac{-3}{2}$
2. $\frac{3}{2}$
3. $\frac{-5}{2}$
4. $\frac{5}{2}$

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

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

Q. If $y=1+t^4$ and $x=3t^3+t$ then what is $\frac{dy}{dx}$

1. 
2. 
3. 
4. 

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

1. 2
2. $\frac{1}{2}$
3. 0
4. 1

Q. If $x=2\cos 2t,y=\sin 2t$ then -

1. ${\frac{dy}{dx}|}_{t=\frac{3\pi }{8}}=\frac{1}{2}$
2. ${\frac{dy}{dx}|}_{t=\frac{\pi }{4}}=\frac{1}{2}$
3. ${\frac{d^{2}y}{d{x}^2}|}_{t=\frac{\pi }{4}}=\frac{-1}{4}$
4. ${\frac{d^{2}y}{d{x}^2}|}_{t=\frac{\pi }{6}}=-2$

Q. If $x=a{\sec }^3\theta \mathrm{and}y=a{\tan }^3\theta ,\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}\frac{\mathrm{dy}}{\mathrm{dx}}\mathrm{at}\theta =\frac{\pi }{3}\mathrm{is}$

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

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