• Attempt any five questions.
• All questions carry equal marks.
State Lagrange'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 cos y in a Taylor series in the neighbourhood of the point (0, 0).
Estimate the extreme values of the function x³+y³-63(x+y)+12xy.
If u=((y-x)/xy)((z-x)/xz) find the value of x²uₓ+y²u_y+z²u_z.
Show that the rectangular solid of maximum volume that can be inscribed in a given sphere is a cube.
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.
i) Find the value of Γ(3/2).
ii) Evaluate ∫₀¹ x³(1-√x)² dx.
Test the series 1 + x/2 + x²/5 + x³/10 + ... + xⁿ/(n²+1) + ...
Show that β(l,m)=Γ(l)Γ(m)/Γ(l+m).
Expand as a half range f(x)=x sin x series and cosine series for the interval 0<x<2.
Transform the matrix into normal form and hence find its rank.
Find the inverse of the matrix by using elementary row transformations.
Find the eigen values and eigen vectors of the matrix.
Test the consistency and hence, solve the following set of equations.
