Section A
Q1.
(a) Let A be a square matrix of order 3 such that each of its diagonal elements is 'a' and each of its off-diagonal elements is 1. If B = bA is orthogonal, determine the values of a and b.
(b)
Let V be the vector space of all 2 × 2 matrices over the field R. Show that W is not a subspace of V, where
(i) W contains all 2 × 2 matrices with zero determinant.
(ii) W consists of all 2 × 2 matrices A such that A2 = A.
(c)
Using the Mean Value Theorem, show that
(i) f(x) is constant in [a, b], if f'(x) = 0 in [a, b].
(ii) f(x) is a decreasing function in (a, b), if f'(x) exists and is < 0 everywhere in (a, b).
(d) Let u(x, y) = ax2 + 2hxy + by2 and v(x, y) = Ax2 + 2Hxy + By2. Find the Jacobian J = (\partial(u, v))/(\partial(x, y)), and hence show that u, v are independent unless (a)/(A) = (b)/(B) = (h)/(H).
(e) Find the equations of the planes parallel to the plane 3x - 2y + 6z + 8 = 0 and at a distance 2 from it.
