Derivative of Some Standard Functions

Q. Let $A$ be a $2\times 2$ real matrix with entries from $\{0,1\}$ and $|A|\ne 0$ . Consider the following two statements:

(P) If $A\ne I_2,$ then $|A|=-1$

(Q) If $|A|=1$, then $\text{tr}\left(A\right)=2$,
where $I_2$ denotes $2\times 2$ identity matrix and $\text{tr}\left(A\right)$ denotes the sum of the diagonal entries of $A$. Then :

1. Both (P) and (Q) are false
2. (P) is true and (Q) is false
3. Both (P) and (Q) are true
4. (P) is false and (Q) is true

Q. find domain of f(x) = √log_3(cos(sinx))

Q.

If $y=\sqrt{\frac{1+e^x}{1-e^x}}$ , then $\frac{dy}{dx}$ is equal to?

1. $\frac{e^x}{(1-e^x)\sqrt{(1-e^{2x})}}$

2. $\frac{e^x}{(1-e^x)\sqrt{(1-e^x)}}$

3. $\frac{e^x}{(1-e^x)\sqrt{(1+e^{2x})}}$

4. $\frac{e^x}{(1-e^x)\sqrt{(1+e^x)}}$

Q.

Write the following cubes in expanded form: ${\left(\frac{3}{2}x+1\right)}^3$

Q.

If ${\tan }^{-1}x+{\tan }^{-1}y+{\tan }^{-1}z=\frac{\pi }{2}$, then

1. $x+y+z-xyz=0$

2. $x+y+z+xyz=0$

3. $xy+xz+zy+1=0$

4. $xy+xz+zy-1=0$

Q.

Write the following cube In expanded form: ${\left(x-\frac{2}{3}y\right)}^3$.

Q. If f(x) = |cos x|, then the value of $f\text{'}\left(\frac{3\pi }{4}\right)$ is

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

Q. If x is equal to P sec theta + Q tan theta and Y is equal to p tan theta + Q sec theta then prove that x square minus y square is equal to p square minus qsquare

Q.

If $\tan \left(A\right)+cot\left(A\right)=4$, then ${\tan }^4\left(A\right)+co{t}^4\left(A\right)$ is equal to

1. $110$

2. $191$

3. $80$

4. $194$

Q.

Prove : cos^2 π/8+cos^2 3π/8+cos^2 5π/8+cos^2 7π/8=2

Q. If $y=x\tan \frac{x}{2},$ then $(1+\cos x)\frac{dy}{dx}-\sin x=$

1. $xy$
2. $y$
3. $0$
4. $x$

Q. Integral 1+cos^2x/1-cos2x.dx

Q. Find the maximum value of cos(cos(cos x))

Q.

cos{3π/2 + x } cos(2π+x) [ cot {3π/2 - z} + cot (2π+x) ] =1

Q. let A=[1 0 1 1] and P=[ cos pi/6 sin pi/6 -sinpi/6 cospi/6] and Q=PAP^T then P^TQ^2013P

Q.

Find the second derivative of e^xcosx

1. -e^xsinx

2. -2e^xcosx

3. e^x(sinx + cosx)

4. -2e^xsinx

Q. Range of function f(x) = cos(ksinx) is [ -1, 1], then the least positive integral value of k will be?

Q. Let $f\left(x\right)$ be a polynomial of degree $3$ such that $f(-1)=10,f\left(1\right)=6,f\left(x\right)$ has a critical point at $x=-1$ and $f^{\prime }\left(x\right)$ has a critical point at $x=1$. Then $f\left(x\right)$ has a local minima at $x$ equals to

Q.

Show that the local maximum value of $x+\frac{1}{x}$ is less than the local minimum value.

Q. Find the domain of f (x) = 1/( | sinx | - sinx )

Q. ntif cot theta+cos theta=p, cot theta-cos theta=q then the value of p square- q square isn nt(1)2pqn nt(2)4pqn nt(3)2pqn nt(4)4pqn

Q. If cos x - sin x=2 sinx Then prove that cosx + sinx=2cosx

Q. If f(x) = sin2x - cos2x, Find f^I (π/6)

Q. ntx= cos inverse (8t-8t+1), y=sin inverse (3t-4t) [0