If `B ,C` are square matrices of order `na n difA=B+C ,B C=C B ,C^2=O ,` then without using mathematical induction, show that for any positive integer `p ,A^(p+1)=B^p[B+(p+1)C]` .

If B, C are square matrices of order n and if A = B + C , BC = CB , C^2=0 then which of the following is true for any positive integer N

If B ,\ C are n rowed square matrices and if A=B+C , B C=C B , C^2=O , then show that for every n in N , A^(n+1)=B^n(B+(n+1)C) .

If a,b and c are in A.P. and also in G.P., show that : a=b=c .

If a,b and c are in A.P, show that (b+c), (c+a) and (a+b) are also in A.P.

A B C\ a n d\ D B C are two isosceles triangles on the same bas B C and vertices A\ a n d\ D are on the same side of B C . If A D is extended to intersect B C at P , show that A B D\ ~= A C D (ii) A B P\ ~= A C P

If a, b, c, d are in G. P., show that a+b,b+c,c+d are also in G. P.

If A , B , C are three real numbers and p=[A+B+C] and q=[A]+[B]+[C]dot (where [.] represents greatest integer function). Then maximum value of [p-q] is: 1 (b) 0 (c) 2 (d) 3

If a, b, c are in A.P, then show that: b+c-a ,\ c+a-b ,\ a+b-c are in A.P.

In A B C , P divides the side A B such that A P : P B=1:2 . Q is a point in A C such that P Q B C . Find the ratio of the areas of A P Q and trapezium B P Q C .

If a, b, c are in A.P, then show that: \ b c-a^2,\ c a-b^2,\ a b-c^2 are in A.P.