• Attempt any five questions.
• All questions carry equal marks.
State Rolle's theorem hence verify for f(x)=x²+2x defined in the interval [-2, 0].
Find the first six terms of the expansions of the function eˣ log(1+y) in a Taylor series in the neighbourhood of the point (0, 0).
The temperature u(x,y,z) at any point in space is u=400xz² find the highest temperature on surface of the sphere x²+y²+z²=1.
If u=x²tan⁻¹(y/x)-y²tan⁻¹(x/y) find the value of ∂²u/∂x∂y.
Find shortest distance from the origin to the curve x²+4xy+6y²=140.
Find du/dt if u=x²+y², x=a cos t, y=b sin t.
Change the order of integration in ∫₀¹∫ₓ²²⁻ˣ xy dy dx and hence evaluate.
Evaluate ∬ e²ˣ⁺³ʸ dxdy over the triangle bounded by x=0, y=0 and x+y=1.
Test the convergence of the series 1 + x/2 + x²/5 + x³/10 + ... + xⁿ/(n²+1) + ...
Expand as a half range f(x)=x sin x series and cosine series for the interval 0<x<2.
Expand f(x)=x sin x, 0<x<2π as a Fourier series.
Find the a₀ and aₙ if the function f(x)=x+x² is expanded in Fourier series defined in (-1, 1).
i) If A is a skew symmetric matrix then show that A² is a symmetric matrix.
ii) Find eigen values of the matrix [[5,4],[1,2]].
Find the inverse of the matrix by using elementary row transformations.
Verify Cayley Hamilton theorem for the matrix A and hence find A⁻¹.
Test the consistency and hence, solve the following set of equations: x+2y-z=3, 3x-y+2z=1, 2x-2y+3z=2, x-y+z=-1.
