Q.1. Attempt any five of the following :

(a) Show that the vectors \((1,0,-1),(0,-3,2)\) and \((1,2\),
1) form a basis for the vector space \(R^3(R)\).

(b) If \(\lambda\) is a characteristic root of a non-singular matrix
A, then prove that \(\frac{|A|}{\lambda}\) is a characteristic root of Adj. A.

(c) Let \(f\) be defined on IR by setting \(f(x)=x\), if \(x\) is rational, and \(f(x)=1-x\), if \(x\) is irrational. Show that \(f\) is continuous at \(x=\frac{1}{2}\) but is discontinuous at every other point.