a) Expand $(1+x)^x$ by Maclaurin's Theorem.

b) Find the Maximum value of $u=\sin x \sin y \sin(x+y)$.

a) Find the volume of the solid generated by the revolution of the Cardioid $r=a(1+\cos\theta)$ about the initial line.

b) Prove that $\int_{0}^{\infty}\cos(x^2)dx=\frac{1}{2}\sqrt{\frac{\pi}{2}}$.

a) Show that Sequence ($x_n$) where $|x|<1$ converge to 0.

b) Find the Fourier Series for the function $f(x)=x\sin x, (-\pi

a) Show that the following equations are consistent and solve them.

$x-y+2z=4$

$3x+y+4z=6$

$x+y+z=1$

b) If $w_1$ and $w_2$ be two subspace of V(F) then Show that $w_1 \cap w_2$ also subspace of V(F).

a) Find the Characteristic equation of the matrix $\begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & -4 \\ 1 & 0 & -1 \end{pmatrix}$ and hence find the Eigen values and Eigen vectors.

b) Show that the following matrix A is Diagonalizable. $A = \begin{pmatrix} 1 & 0 & -1 \\ 1 & 2 & 1 \\ 2 & 2 & 3 \end{pmatrix}$.

a) Find the Maximum and Minimum value of $u=a^2x^2+b^2y^2+c^2z^2$, where $x^2+y^2+z^2=1$ and $lx+my+nz=0$.

b) Find the Fourier Series for the function $f(x)=x+x^2, (-\pi

a) Show that the surfaces area of the solid generated by revolution of the loop of the curve $x=t^2, y=t-\frac{1}{3}t^3$ about the x axis is $3\pi$.

b) Investigate for what values of $\lambda$ and $\mu$ the simultaneous equations.

$X+Y+Z=6$

$X+2Y+3Z=10$

$X+2Y+\lambda Z=\mu$

a) Expand $\log x$ in power $(x-1)$ by Taylor's theorem and hence find the value $\log 1.1$.

b) Evaluate $\iint xy\ dx\ dy$ where the region of integration is $x+y<1$ in the positive quadrant.