Let `S_p` and `S_q` be the coefficients of `x^p` and `x^q` respectively in `(1 + x)^(p+q)`, then :

Let C_1 and C_2 be parabolas x^2 = y - 1 and y^2 = x-1 respectively. Let P be any point on C_1 and Q be any point C_2 . Let P_1 and Q_1 be the reflection of P and Q, respectively w.r.t the line y = x then prove that P_1 lies on C_2 and Q_1 lies on C_1 and PQ >= [P P_1, Q Q_1] . Hence or otherwise , determine points P_0 and Q_0 on the parabolas C_1 and C_2 respectively such that P_0 Q_0 <= PQ for all pairs of points (P,Q) with P on C_1 and Q on C_2

Let C_1 and C_2 be respectively, the parabolas x^2=y-1 and y^2=x-1 Let P be any point on C_1 and Q be any point on C_2 . Let P_1 and Q_1 be the refelections of P and Q, respectively with respect to the line y=x. Arithemetic mean of PP_1 and Q Q_1 is always less than

In the expansion of (1+x)^n , the coefficient of x^(p-1) and of x^(q-1) are equal. Show that p+q=n+2, p ne q .

The 5th, 8th and 11th terms of a G.P. are p, q and x, respectively. Show that q^2=ps .

The 5th, 8th, and 11th terms of a GP are p, q and s respectively. Find relation between p,q and s.

If x^p . y^q = (x + y)^(p + q) , show that (dy)/(dx) = y/x

Let alpha and beta be the roots of x^2-x+p=0 and gamma and delta be the root of x^2-4x+q=0. If alpha,beta,and gamma,delta are in G.P., then the integral values of p and q , respectively, are

If x^(p) y^(q) = (x + y)^((p + q)) " then " (dy)/(dx)= ?

If p and q are the roots of the equation x^2-p x+q=0 , then (a) p=1,\ q=-2 (b) p=1,\ q=0 (c) p=-2,\ q=0 (d) p=-2,\ q=1

If the pth , qth , rth , terms of a GP . Are x,y,z respectively prove that : x^(q-r).y^(r-p).z^(p-q)=1