If `5^(-p)=4^(-q)=20^(r )`. Show that `(1)/(p)+(1)/(q)+(1)/(r )=0`

(a) If r^(2) = pq , show that p : q is the duplicate ratio of (p + r) : (q + r) . (b) If (p - x) : (q - x) be the duplicate ratio of p : q then show that : (1)/(p) + (1)/(q) = (1)/(r) .

if q is the mean proportional between p and r , prove that : p^(2) - q^(2) + r^(2) = q^(4) ((1)/(p^(2))-(1)/(q^(2)) + (1)/(r^(2))) .

The roots of the equation (q-r)x^2+(r-p)x+p-q=0 are (A) (r-p)/(q-r),1 (B) (p-q)/(q-r),1 (C) (q-r)/(p-q),1 (D) (r-p)/(p-q),1

If p > q > 0a n dp r < -1 < q r , then find the value of tan^(-1)((p-q)/(1+p q))+tan^(-1)((q-r)/(1+q r))+tan^(-1)((r-p)/(1+r p)) .

If r^[th] and (r+1)^[th] term in the expansion of (p+q)^n are equal, then [(n+1)q]/[r(p+q)] is (a) 1/2 (b) 1/4 (c) 1 (d) 0

If A=((p,q),(0,1)) , then show that A^(8)=((p^(8),q((p^(8)-1)/(p-1))),(0,1))

If p : q = 1(1)/(3) : 1(1)/(2) and q : r = (1)/(2) : (1)/(3) , find p : r

Show that (pvvq) to r -= (p to r) ^^ (q to r)

If P -= ((1)/(x_(p)), p ), Q = ((1)/(x_(q)),q), R = ((1)/(x_(r)), r) where x _(k) ne 0 , denotes the k^(th) terms of a H.P. for k in N , then :

If a=x y^(p-1),\ b=x y^(q-1) and c=x y^(r-1) , prove that a^(q-r)b^(r-p)\ c^(p-q)=1