Section A
Q1.
(a)
Consider the following quadratic form :
q(x, y, z) = 2x2 + 2y2 + 6z2 + 2xy - 6yz - 6zx, where (x, y, z) are the coordinates of the vector X with respect to the standard basis (1, 0, 0), (0, 1, 0), (0, 0, 1) of R3. Find the expression of q(x, y, z) with respect to the basis B = (1)/(√(6)),(1)/(√(6)),(-2)/(√(6))),(1)/(√(2)),(-1)/(√(2)),0),(1)/(√(3)),(1)/(√(3)),(1)/(√(3)))}.
Is q positive definite ? Justify your answer.
(b) Prove that the product of two Hermitian matrices A, B is Hermitian if and only if A and B commute. Give an example of a pair of 3 × 3 symmetric matrices such that their product is again symmetric (do not consider only diagonal matrices) and also check whether they commute or not.
(c)
Using Beta and Gamma functions, evaluate the following integrals :
(i) ∫02 x(8-x3)^1/3 dx
(ii) ∫01 (x2 dx)/(√(1-x5))
(d)
Evaluate ∬R x2 dx dy,
where R is the region in the first quadrant bounded by the hyperbola xy = 16 and the lines y = x, y = 0 and x = 8.
(e) Find the equation of the plane passing through the points (1, -1, 1) and (-2, 1, -1) and perpendicular to the plane 2x + y + z + 5 = 0.
