• Attempt any five questions.
• All questions carry equal marks.
Prove that π/3 > (1/√53)cos⁻¹(3/5) > π/3 - 1/8 using Lagrange's mean value theorem.
Find the minimum and maximum value of f(x,y)=x³+3xy²-3x²-3y²+4.
Find C of Cauchy's Mean value theorem on [a, b] for the function f(x)=eˣ and g(x)=e⁻ˣ, (a,b>0).
Prove that Γ(n)Γ(1-n)=π/sin nπ.
By Changing the order of integration, evaluate ∫₀¹∫₀^√(1-x²) y² dy dx.
Find the area of a plane in the form of a quadrant of the ellipse x²/a²+y²/b²=1.
Let W be a subspace of a finite dimensional vector space V(F). Then dim(V/W) = dim V - dim W.
Obtain the Fourier series to represent f(x)=x sin x, 0<x<2π.
Show that T: V₂(R) → V₃(R) is defined as T(a, b) = (a-b, b-a, -a) is linear transformation.
Test the convergence of the series Σ(√(n+1)-√(n-1)).
Verify Cayley-Hamilton theorem for the matrix A. Hence find A⁻¹.
Examine the consistency of the system of equations. If consistent, solve: x+y+z=3, x+2y+3z=4, x+4y+9z=6.
Diagonalize the matrix A.
