Section A
Q1. Let V = mathbbR4. Find a basis and dimension of the subspace W = (a, b, c, d) in V : a = b + c, c = b + d Describe explicitly a linear transformation from \mathbbR3 to \mathbbR3, which has its range spanned by (1, 0, -1) and (1, 2, 2). Find the relation between the radii of a right circular cylinder and a cone if the former with maximum possible curved surface area is inscribed in the latter. Find the limit of (\cot x - tan x)loge x, when x → 0. Show that if ax2 + 2hxy + by2 + 2gx + 1 = 0 represents two straight lines, then b < 0 and bg2 + h2 = ab.
(a) Let V = \mathbbR4. Find a basis and dimension of the subspace W = \(a, b, c, d) ∈ V : a = b + c, c = b + d\ 8 Describe explicitly a linear transformation from \mathbbR3 to \mathbbR3, which has its range
(b) spanned by (1, 0, -1) and (1, 2, 2). 8 Find the relation between the radii of a right circular cylinder and a cone if the
(c) former with maximum possible curved surface area is inscribed in the latter. 8 Find the limit of (\cot x - tan x)loge x, when x → 0. 8
(d) Show that if ax2 + 2hxy + by2 + 2gx + 1 = 0 represents two straight lines, then
(e) b < 0 and bq^{2} + h^{2} = ab. 8 2.
(i) form and show that it represents a parabola. Find the latus rectum of the parabola. 6 (ii) A variable sphere passes through the points (0,0, ± c) and cuts the lines u - x tan θ = 0 = z - c y + x tan θ = 0 = z + c in the points P and Q. If |PQ| = 2a (where a is a + ve number), then show that the centre of all such spheres lies on the circle x2 + y2 = (a2 - c2)\csc22θ, z = 0. 9 JBNV-U-MATH/47 2
