• Attempt any five questions.
• All questions carry equal marks.
Verify Rolle's theorem for f(x)=x⁴-1 in [-1, 1]
If u=log(x³+y³z³-3xyz), Show that (∂/∂x + ∂/∂y + ∂/∂z)² u = -9/(x+y+z)²
Change the order of integration in ∫₀ᵃ ∫_yᵃ f(x,y)dxdy
Find the area of the cardioid r=a(1+cosθ)
Test the convergence of the series 1/(1·2·3) + 2/(2·3·4) + 3/(3·4·5) + ...
Find the Fourier series for the function f(x)=x³ in (-π, π).
Determine whether the following vectors in R⁴ 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)
Find a basis and the dimension of the subspace W of P(t) spanned by: U=t³+t²-3t+2, V=2t³+t²+t-4, W=4t³+3t²-5t+2
Find eigenvalues and eigenvectors of matrix A
Find the rank of the matrix
Prove that β(m,n)=β(m+1, n)+β(m, n+1) where m, n > 0
Test the convergence of the series x/(1·2) + x²/(2·3) + x³/(3·4) + x⁴/(4·5) ...
Show that the given system of equations x+y+z=6, x+2y+3z=14, x+4y+7z=30 are consistent and solve them.
Evaluate ∫₀¹ x⁴(1-√x)⁵ dx
Obtain Taylor's series expansion of the function f(x,y)=e^(xy) about (1,1) up to third degree terms.
Discuss the extreme values (maxima and minima) of the function x³+y³-3axy.
