Show that the function f : R \(\rightarrow\)R defined by f(x) = \({x\over{x^2+1}}{\forall \ x\in R}\) is neither one-one nor onto.

4

(a)

Which is greater tan 1 or tan^-1 1? 

4

(a)

Find the value of the expression \(sin\left( 2{ tan }^{ -1 }\frac { 1 }{ 3 } \right) +cos\left( { tan }^{ -1 }2\sqrt { 2 } \right) \)

4

(a)

A spherical ball of salt is dissolving in water such a manner that the rate of decrease of the volume at any instant is proportional to the surface .Prove that the radius is decreasing at a constant rate.

4

(a)

Let \(A=\begin{bmatrix} 2 & 3 \\ -1 & 2 \end{bmatrix}\), then show that \({ A }^{ 2 }-4A+7l=0\). Using this result, calculate \({ A }^{ 5 }\) also.

4

(a)

Find the matrix A, satisfying the matrix equation

\(\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix}A\begin{bmatrix} -3 & 2 \\ 5 & -3 \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)

4

(a)

Find the value of \(\theta\) satisfying \(\begin{vmatrix} 1 & 1 & sin3\theta \\ -4 & 3 & cos2\theta \\ 7 & -7 & -2 \end{vmatrix}=0\)

4

(a)

Evaluate the integral: \(\int tan^8\ x\ sec^4\ x\ dx\)

4

(a)

Evaluate the integral: \(\int {x^3\over x^4+3x^2+2}dx.\)

4

(a)

Evaluate the integral: \(\int{x^2\over x^4+x^2-2}dx\)

4

(a)

Evaluate the integral: \({\sqrt x\over \sqrt{a^3-x^3}}dx\)

4

(a)

If \(\overrightarrow { a } =\hat { i } -\hat { j } +7\hat { k } \) and \(\overrightarrow { b } =5\hat { i } -\hat { j } +\lambda \hat { k } \) then find the value of \(\lambda\)  so that \(\overrightarrow { a } +\overrightarrow { b } \ and\ \overrightarrow { a } -\overrightarrow { b } \) are perpendicular vectors.

4

(a)

If \(f(x)=\frac { \sqrt { 2 } cos\quad x-1 }{ cot\quad x-1 } ,\quad x\neq \frac { \pi }{ 4 } \), find the value of  \(f\left( \frac { \pi }{ 4 } \right) \), so that f (x) becauses continuous at x =\(\frac { \pi }{ 4 } \). 

4

(a)

Find the area of greatest rectangle can be inscribed in an ellipse \({x^2\over a^2}+{y^2\over b^2}=1\).

4

(a)

Integrate:\(\int^{1}_{0}{dx\over e^x+e^{-x}}\)

4

(a)

If x^py^q = (x + y)^p+q, prove that \((i)\frac { dy }{ dx } =\frac { y }{ x } and\quad (ii)\frac { { d }^{ 2 }y }{ { d }x^{ 2 } } =0.\)

4

(a)

Verify MVT for the following functions:
f(x)= sin x-sin 2x in [0,\(\pi\)]

4

(a)

Differentiate w.r.t. x or find \(\frac { dy }{ dx } \) : \({ 2 }^{ cos^{ 2 }x }\)

4

(a)

Differentiate w.r.t. x or find \(\frac { dy }{ dx } \) : \(x={ e }^{ \theta }\left( \theta +\frac { 1 }{ \theta } \right) ,y={ e }^{ -\theta }\left( \theta -\frac { 1 }{ \theta } \right) \)

4

(a)

A swimming pool is to be drained for cleaning. If L represents-the number of litres of water in the pool t seconds after the pool has been plugged off to drain and L = 200(10 - t)². How fast is the water running out at the end of 5 s and what is the average rate at which the water flows out during the first 5 s?

4

(a)

A biased die is such that \(P(4)=\frac{1}{10}\) and other scores-being equally likely. The die is tossed twice. If X is the 'number of four seen', then find the variance of the random variable X.

4

(a)

Prove that the curves y² = 4x and x² = 4y divide the area of the square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts. 

4

(a)