a) Find the points where the function $x^3+y^3-3axy$ has maximum or minimum value.

b) Find the Taylor's expansion of $y=\sin x$ about point $x=\pi/2$.

a) The part of the parabola $y^2=4ax$ cut off by the latus rectum revolves about the tangent at the vertex. Find the volume of the reel thus generated.

b) Prove that: $\int_{0}^{1}\frac{dx}{\sqrt{1-x^4}} = \frac{(\Gamma(1/4))^2}{6\sqrt{2\pi}}$.

a) Show that the following series is Convergent. $\frac{1}{4} - \frac{1}{4^2} + \frac{1}{4^3} - \frac{1}{4^4} + \dots$

b) Obtain the Half-Range Sine Series for \( f(x) = e^x \) in \( 0 < x < 1 \).

a) Show that the set $w=\{(a,b,0):a,b \in R\}$ is subspace of $R^3$.

b) Are the following vectors LD? If so express one of these as a LC of other two.

$X_1=(1,3,4,2)$, $X_2=(3,-5,2,2)$, $X_3=(-2,1,-3,2)$.

a) Find a similarity transformation that diagonalise the matrix. $A = \begin{pmatrix} -2 & 2 & -3 \\ 2 & 1 & -6 \\ -1 & -2 & 0 \end{pmatrix}$.

b) Find the Eigen value and Corresponding Eigen Vectors of the following Matrix. $\begin{pmatrix} 8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{pmatrix}$.

a) Define Beta and Gamma Function and show that relation $B(m,n)=\frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$.

b) Evaluate:

i) $\int_{0}^{2}x^2(1-x)^3dx$

ii) $\int_{0}^{1}\sqrt{x(1-x)}dx$

a) Prove that the surface area of the solid generated by the revolution of the ellipse $x^2/a^2+y^2/b^2=1$ about the major axis is : $2(\pi ab) \cdot \{\sqrt{1-e^2} + \frac{\sin^{-1}e}{e}\}$.

b) Show that the sequence $(n^{1/n})$ converge to 1.

a) Prove that a rectangular solid of maximum volume within a sphere is a cube.

b) Verify Rolle's Theorem for the function $y=x^2+2, a=-2$ and $b=2$.