Implicit Differentiation

Q.

If $x,y\mathrm{and}z$ in AP, then $\frac{1}{\sqrt{x}+\sqrt{y}},\frac{1}{\sqrt{z}+\sqrt{x}},\frac{1}{\sqrt{y}+\sqrt{z}}$ are in:

1. AP

2. GP

3. HP

4. AP and HP

Q. Let $x^k+y^k=a^k,(a,k>0)$ and $\frac{dy}{dx}+{\left(\frac{y}{x}\right)}^{\frac{1}{3}}=0$, then $k$ is

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

Q.

$\left(\frac{d}{dx}\right)\left[\frac{(1+cotx)}{(1-cotx)}\right]=$

Q. $Ifsin(x+y)=lo{g}_e(x+y),then\frac{dy}{dx}isequalto$

1. 2
2. - 2
3. 1
4. - 1

Q. If $x{e}^{xy}-y-{\sin }^{2}x=0$ then $\frac{dy}{dx}$ at $x=0$ is

Q. Let $x^k+y^k=a^k,(a,k>0)$ and $\frac{dy}{dx}+{\left(\frac{y}{x}\right)}^{\frac{1}{3}}=0$, then $k$ is

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

Q. $If{x}^2+y^2=t-\frac{1}{t}and{x}^4+y^4=t^2+\frac{1}{t^2},then{x}^{3}y\frac{dy}{dx}equals$

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

Q. Which of the following is true for $y\left(x\right)$ that satisfies the differential equation $\frac{dy}{dx}=xy-1+x-y;y\left(0\right)=0$

1. $y\left(1\right)=1$
2. $y\left(1\right)=e^{-\frac{1}{2}}-1$
3. $y\left(1\right)=e^{\frac{1}{2}}-e^{-\frac{1}{2}}$
4. $y\left(1\right)=e^{\frac{1}{2}}-1$

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

1. 2
2. cos a
3. sin a
4. None of these

Q. Which of the following is true for $y\left(x\right)$ that satisfies the differential equation $\frac{dy}{dx}=xy-1+x-y;y\left(0\right)=0$

1. $y\left(1\right)=1$
2. $y\left(1\right)=e^{\frac{1}{2}}-1$
3. $y\left(1\right)=e^{\frac{1}{2}}-e^{-\frac{1}{2}}$
4. $y\left(1\right)=e^{-\frac{1}{2}}-1$

Q. Consider the curve $\sin x+\sin y=1$, lying in the first quadrant , then
$\begin{array}{cccc}& \text{List- I}& & \text{List-II}\\ \text{(I)}& \underset{x\to \pi /2}{lim}\frac{d^{2}y}{d{x}^2}=& \text{(P)}& 0\\ \text{(II)}& \underset{x\to 0^{+}}{lim}{x}^{3/2}\frac{d^{2}y}{d{x}^2}=& \text{(Q)}& 1\\ \text{(III)}& \underset{x\to 0^{+}}{lim}{x}^2\frac{d^{2}y}{d{x}^2}=& \text{(R)}& \frac{1}{\sqrt{2}}\\ \text{(IV)}& \underset{x\to \pi /2}{lim}\frac{dy}{dx}=& \text{(S)}& \frac{1}{2\sqrt{2}}\\ & & \text{(T)}& \sqrt{2}\\ & & \text{(U)}& 3\end{array}$

Which of the following is the only INCORRECT combination?

1. $I\to T$
2. $II\to S$
3. $III\to P$
4. $IV\to P$