Implicit Differentiation

Q.

What is the derivative of $y=\tan \left(x\right)$?

Q.

If $y=e^x\log x$, then $\frac{dy}{dx}$is

1. $\frac{e^x}{x}$

2. $e^x\left(\log x+\frac{1}{x}\right)$

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

4. $\frac{e^x}{\log x}$

Q. The value of $\int \frac{\sin 7x}{\cos x}dx$ is
(where $C$ is constant of integration)

1. $-\frac{\cos 6x}{3}+\frac{2\cos 4x}{3}-\cos 2x+\cos x+C$
2. $-\frac{\cos 6x}{3}+\frac{\cos 4x}{2}-\cos 2x+\ln |\cos x|+C$
3. $-\frac{\sin 6x}{3}+\frac{2\sin 3x}{3}-\sin 2x+\ln |\sin x|+C$
4. $-\frac{\sin 6x}{3}+2\sin 4x-\sin 2x+\sin x+C$

Q. Prove that:- 1) Sin (A+B) =sinAcosB+cosAsinB 2) Sin(A-B) =sinAcosB-cosAsinB 3) Cos(A-B) =cosAcosB+sinAsinB 4) Cos(A+B) =cosAcosB-sinAsinB

Q.

If $\alpha$ and $\beta$are two different complex numbers with $\left|\beta \right|=1$, then $\left|\frac{(\beta -\alpha )}{(1-\overline{\alpha }\beta }\right|$ is equal to

Q. If $y=\sqrt{\sin x+y}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

1. $\frac{\cos x}{1-2y}$
2. $\frac{\cos x}{2y-1}$
3. $\frac{\sin x}{1-2y}$
4. $\frac{\sin x}{2y-1}$

Q.

If $x^2+y^2+z^2=r^2$, then ${\tan }^{-1}\frac{xy}{zr}+{\tan }^{-1}\frac{yz}{xr}+{\tan }^{-1}\frac{zx}{yr}=$

1. $\mathrm{\pi }$

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

3. $0$

4. None of these

Q.

Prove

cotAcosA/cotA+cosA = cot A-cosA/cotAcosA

Q. If $y=e^{ax}\cos bx$, then prove that

$\frac{d^{2}y}{d{x}^2}-2a\frac{dy}{dx}+(a^2+b^2)y=0$

Q.

If $y=\left(1+x^2\right)\left({\tan }^{-1}x\right)-x$ , then $\frac{dy}{dx}=$

1. ${\tan }^{-1}x$

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

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

4. $2x\left({\tan }^{-1}x\right)-1$

Q. The value of the determinant is
$\left|\begin{array}{ccc}cos(A-P)& cos(A-Q)& cos(A-R)\\ cos(B-P)& cos(B-Q)& cos(B-R)\\ cos(C-P)& cos(C-Q)& cos(C-R)\end{array}\right|$

1. zero for all values of P, Q, R, A, B, and C.
2. one for all values of P, Q, R, A, B, and C.
3. zero only if P= Q= R= A= B= C.
4. one only if P= Q= R= A= B= C.

Q. If the inverse trigonometric functions take principal values, then ${\cos }^{-1}(\frac{3}{10}\cos \left({\tan }^{-1}\left(\frac{4}{3}\right)\right)+\frac{2}{5}\sin \left({\tan }^{-1}\left(\frac{4}{3}\right)\right))$ is equal to:

1. $\frac{\pi }{3}$
2. $0$
3. $\frac{\pi }{6}$
4. $\frac{\pi }{4}$

Q.

Fill in the blanks:

$1{\mathrm{m}}^3={\mathrm{\_\_\_\_\_\_\_\_cm}}^3$

Q.

Explain the relation between $litre,d{m}^3$ and $ml.$

Q.

The value of ${\cos }^{-1}\left(\cos \left(12\right)\right)-{\sin }^{-1}\left(\sin \left(14\right)\right)$

1. $-2$

2. $8\mathrm{\pi }-26$

3. $4\pi +2$

4. None of these

Q.

If $Sin(x+y)=\log (x+y)$ then $\frac{dy}{dx}$is equal to

1. $2$

2. $-2$

3. $1$

4. $-1$

Q.

If y= cos(1-x) find dy by dx

Q. Let $y=y\left(x\right)$ be a function of $x$ satisfying $y\sqrt{1-x^2}=k-x\sqrt{1-y^2}$ where $k$ is a constant and $y\left(\frac{1}{2}\right)=-\frac{1}{4}$. Then $\frac{dy}{dx}$ at $x=\frac{1}{2}$, is equal to :

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

Q.

Prove that cos² 2x – cos² 6x = sin 4x sin 8x

Q.

If $y=\log \left(x\right)e^{\left(\tan x+x^2\right)}$ , then $\frac{dy}{dx}$ is equal to?

1. $e^{\left(\tan x+{\left(x\right)}^2\right)}\left(\frac{1}{x}+\left(se{c}^{2}x+x\right)\log \left(x\right)\right)$

2. $e^{\left(\tan x+{\left(x\right)}^2\right)}\left(\frac{1}{x}+\left(se{c}^{2}x-x\right)\log \left(x\right)\right)$

3. $e^{\left(\tan x+{\left(x\right)}^2\right)}\left(\frac{1}{x}+\left(se{c}^{2}x+2x\right)\log \left(x\right)\right)$

4. $e^{\left(\tan x+{\left(x\right)}^2\right)}\left(\frac{1}{x}+\left(se{c}^{2}x-2x\right)\log \left(x\right)\right)$

Q. If $2^x+2^y=2^{x+y}$ then the value of $\frac{dy}{dx}atx=y=1is$

1. - 1
2. 0
3. 1
4. 2

Q. Sin(A+B)+sin(A+B)/Cos(A+B)+cos(A-B)=TanA

Q.

Prove that:

1+tan(A)tan(A/2) = tan(A)cot(A/2)-1 = sec(A)

Q.

If $\sqrt{(1-x^2)}+\sqrt{(1-y^2)}=a(x-y)$, then $\frac{dy}{dx}=$

1. $\frac{\sqrt{1-x^2}}{\sqrt{1-y^2}}$

2. $\frac{\sqrt{1-y^2}}{\sqrt{1-x^2}}$

3. $\frac{\sqrt{x^2-1}}{\sqrt{y^2-1}}$

4. none of these

Q.

The value of $\left(\frac{50}{6}\right)-\left(\frac{5}{1}\right)\left(\frac{40}{6}\right)+\left(\frac{5}{2}\right)\left(\frac{30}{6}\right)-\left(\frac{5}{3}\right)\left(\frac{20}{6}\right)+\left(\frac{5}{4}\right)\left(\frac{10}{6}\right)where\left(\frac{n}{r}\right)denotes{}^n{C}_r$ is

1. 15625

2. 0

3. 1000000

4. 2250000

Q.

The resultant of the two vectors having magnitude $2$ and $3$ is $1$. What is their cross product?

1. $6$

2. $3$

3. $1$

4. $0$

Q.

What is the order of convergence of Newton Raphson Method?

Q.

The general solution of $\left|\sin x=\cos x\right|$ is (when $n\in \mathrm{\mathbb{Z}}$) given by

1. $n\pi +\frac{\pi }{4}$

2. $2n\pi \pm \frac{\pi }{4}$

3. $n\pi \pm \frac{\pi }{4}$

4. $n\pi -\frac{\pi }{4}$

Q.

If $f\left(x\right)={\log }_e\left({\log }_{e}x\right)$, then $f\left(e\right)$ is equal to

1. $e^{-1}$

2. $e$

3. $1$

4. $0$

Q.

Suppose $f\left(x\right)$ is a polynomial of degree four, having critical points at $\left(-1,0,1\right)$. If $T=\left\{x\in \mathrm{ℝ}:f\left(x\right)=f\left(0\right)\right\}$, then the sum of squares of all the elements of $T$ is:

1. $2$

2. $6$

3. $8$

4. $4$