a) Verify Rolle's theorem for $f(x)=x^4-1$ in $[-1, 1]$ (Unit 1)

b) If $u=\log (x^3+y^3 z^3-3xyz)$, Show that $$\left(\frac{\partial}{\partial x} + \frac{\partial}{\partial y} + \frac{\partial}{\partial z}\right)^2 u = -\frac{9}{(x+y+z)^2}$$ (Unit 1)

a) Change the order of integration in $\int_0^a \int_y^a f(x,y)dxdy$ (Unit 2)

b) Find the area of the cardioid $r=a(1+\cos\theta)$ (Unit 2)

a) Test the convergence of the series $$\frac{1}{1.2.3} + \frac{2}{2.3.4} + \frac{3}{3.4.5} + \dots$$ (Unit 3)

b) Find the Fourier series for the function $f(x)=x^3$ in $(-\pi, \pi)$. (Unit 3)

a) Determine whether the following vectors in $R^4$ are linearly dependent or independent: (i) (1,2,-3,1), (3,7,1,-2), (1,3,7,-4) (ii) (1,3,1,-2), (2,5,-1,3), (1,3,7,-2) (Unit 4)

b) Find a basis and the dimension of the subspace W of P(t) spanned by: $U=t^3+t^2-3t+2$, $V=2t^3+t^2+t-4$, $W=4t^3+3t^2-5t+2$ (Unit 4)

a) Find eigenvalues and eigenvectors of matrix $$A = \begin{bmatrix} 8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{bmatrix}$$ (Unit 5)

b) Find the rank of the matrix $\begin{bmatrix} 2 & 1 & 3 \\ 4 & 7 & 13 \\ 4 & -3 & -1 \end{bmatrix}$ (Unit 5)

a) Prove that $\beta(m,n)=\beta(m+1, n)+\beta(m, n+1)$ where m, n > 0 (Unit 2)

b) Test the convergence of the series $$\frac{x}{1.2} + \frac{x^2}{2.3} + \frac{x^3}{3.4} + \frac{x^4}{4.5} \dots$$ (Unit 3)

a) Show that the given system of equations $x+y+z=6, \ x+2y+3z=14, \ x+4y+7z=30$ are consistent and solve them. (Unit 5)

b) Evaluate $\int_0^1 x^4(1-\sqrt{x})^5dx$ (Unit 2)

a) Obtain Taylor's series expansion of the function $f(x,y)=e^{xy}$ about (1,1) up to third degree terms. (Unit 1)

b) Discuss the extreme values (maxima and minima) of the function $x^3+y^3-3axy$. (Unit 1)