Prove without expansion that `|a h+bgga b+c h bf+b afh b+b c af+b cc bg+fc|=a|a h+bga h bf+b a h b af+b cgf|`

If a : b = c : d, and e : f = g : h, then (ae + bf ) : (ae – bf ) =?

(i) a , b, c are in H.P. , show that (b + a)/(b -a) + (b + c)/(b - c) = 2 (ii) If a^(2), b^(2), c^(2) are A.P. then b + c , c + a , a + b are in H.P. .

(Associativity ) Let f: A to B , g : B to C " and " h: C to . Then prove that ( h o g) o f = h o ( g o f )

If a,b,c are in H.P, then

If distinct and positive quantities a ,b ,c are in H.P. then (a) b/c= (a-b)/(b-c) (b) b^2>ac (c) b^2< ac (d) a/c=(a-b)/(b-c)