Parametric Differentiation

Q.

If $2y\cos \theta =x\sin \theta$ and $2xsec\theta -y\cos ec\theta =3$, then $x^2+4y^2=$

1. $4$

2. $-4$

3. $\pm 4$

4. None of these

Q.

The imaginary part of ${\tan }^{-1}\left(\cos \theta +i\sin \theta \right)$

1. $\tan h^{-1}\left(\sin \theta \right)$

2. $\tan h^{-1}\left(\infty \right)$

3. $\frac{1}{2}\tan h^{-1}\left(\sin \theta \right)$

4. None of these

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 $y=\sqrt{(a-x)(x-b)}-(a-b)ta{n}^{-1}\sqrt{\left(\frac{a-x}{x-b}\right)}$, 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. $\left(\frac{-3b}{a^3}\right)cose{c}^4\theta co{t}^4\theta$
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 $x=3\tan t$ 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. A curve is represented parametrically by the equations $x=f\left(t\right)=a^{\ln \left(b^t\right)}$ and $y=g\left(t\right)=b^{-\ln \left(a^t\right)};a,b>0$ and $a\ne 1,b\ne 1$ where $t\in R$.

Which of the following is not a correct expression for $\frac{dy}{dx}?$

1. $\frac{-1}{f(t{)}^2}$
2. $-\left(g\right(t){)}^2$
3. $\frac{-g\left(t\right)}{f\left(t\right)}$
4. $\frac{-f\left(t\right)}{g\left(t\right)}$

Q. A function is represented parametrically by the equations $x=\log |2t|,y=|{\tan }^{-1}|t||,t<0.$ Then the value of $\left|\right(1+t^2{)}^2[\frac{1}{t}\frac{d^{2}y}{d{x}^2}-{\left(\frac{dy}{dx}\right)}^2]|$ is

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

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

1. $\frac{4t^3}{9t^2+1}$
2. $\frac{9t^2+1}{4t^3}$
3. $4t^3(9t^2+1)$
4. $9t^2-4t^3$

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 $x=sec\theta -cos\theta ,y=se{c}^{10}\theta -co{s}^{10}\theta$ 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. The value of $\cos (2{\cos }^{-1}x+{\sin }^{-1}x)$ when $x=\frac{1}{5}$ is

1. $\frac{-\sqrt{24}}{5}$
2. $\frac{\sqrt{16}}{5}$
3. $\frac{\sqrt{24}}{5}$
4. $\frac{-\sqrt{16}}{5}$