30 let r- ((a.a2, ay): a e r be avector space over r. then which of following is not a subapace of r' 7 (a) w-{(0,a2, @y) : dg, dy e r) b) w,-t(.@,0) : q 20) d)

30 Let R- ((a.a2, ay): a e R be avector space over R. Then which of following is not a subapace of R' 7 (A) W-{(0,a2, @y) : dg, dy e R) B) W,-T(.@,0) : q 20) D) None of thesg 31. Let W, and Wa be subspaces of a vector space V. Then which one 1s not always a subspace of V ? LB) W Wa (D) None of these (A) WinWa (C)Wi+ W 32 Let V be a vector space over a ficld F and W be a subset ot V. Then W is a Subspace of V iff W is closed under: (A) vector addition (B) calar multiplication () Both (A) and (6) (D) None of these 33 The characteristic roots of a Hermitian matrix are (A) either pure imaginary or zero 8) all real (C) all complex all pure imagınary 34. The characteristic roots of a square matrix are given by (A) 2,-1,1 (B) 2,-1+ 3,-1-3 ) 1,-2,1 (D) None of these