• Attempt any five questions.
• All questions carry equal marks.
Prove that a rectangular solid of maximum volume within a sphere is a cube.
If u=tan⁻¹((x³+y³)/(x-y)) then prove that x ∂u/∂x + y ∂u/∂y = sin 2u.
Show that the surface area of solid generated by revolution of the loop of curve x=t², y=t-t³/3 about the x-axis is 3π.
Change the order of integration in the following integral and then evaluate ∫₀^∞∫ₓ^∞ e^(-y)/y dy dx.
Find the Fourier Series for f(x)=x+x² in (-π,π).
Find the half range sine series for f(x)=x(π-x) in (0,π). Hence Deduce that 1-1/3³+1/5³-...=π³/32.
Let W₁ and W₂ be subspaces of a vector space V and assume that W₁ ∩ W₂ = {0}. Let w₁∈W₁ and w₂∈W₂ be such that w₁≠0 and w₂≠0. Prove that {w₁, w₂} is linearly independent.
i) Prove that W₁ ∩ W₂ is a subspace of V.
ii) Give an example to show that W₁ ∪ W₂ need not be a subspace of V.
iii) Is W₁ ∪ W₂ a subspace of V?
Find Eigen Value and Eigen vectors of the matrix.
Diagonalizable the matrix.
Reduce the matrix to the normal form, hence find its rank.
Find the first 3 terms in the Maclaurin series for
i) sin⁻¹x
ii) xe^(-x).
Investigate for what values of λ and μ the simultaneous equations have
i) no solution
ii) a unique solution
iii) an infinite number of solutions.
Find the Taylor series for the function x⁴+x-2 centered at a=1.
Evaluate ∫₀^∞∫ₓ^∞ e^(-y)/y dy dx.
If xˣ yʸ zᶻ = C then show that ∂²z/∂x∂y = -(x log ex)⁻¹.
