If x nearly equal to 1 show that `(px^q - qx^p)/(x^q - x^p) = (1)/(1 -x)` ( nearly)

If x nearly equal to 1 show that (px^p - qx^q)/(p-q) = x^(p+q) (nearly)

If p is nearly equal to q and n gt 1 , such that ((n+1) p+(n-1)q)/((n-1)p+(n +1)q) = ((p)/(q))^(k) , then the value of k, is

Show that: x p q p x q q q xx-p)(x^2+p x-2q^2) .

(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) .

The value of lim_(x to 1) ((p)/(1-x^(p))-(q)/(1-xq)),p,q,inN, equals

If each pair of the three equations x^(2) - p_(1)x + q_(1) =0, x^(2) -p_(2)c + q_(2)=0, x^(2)-p_(3)x + q_(3)=0 have common root, prove that, p_(1)^(2)+ p_(2)^(2) + p_(3)^(2) + 4(q_(1)+q_(2)+q_(3)) =2(p_(2)p_(3) + p_(3)p_(1) + p_(1)p_(2))

Consider the statements : P : There exists some x IR such that f(x) + 2x = 2(1+x2) Q : There exists some x IR such that 2f(x) +1 = 2x(1+x) f(x)=(1-x)^(2) sin^(2) x+x^(2)" "AA x in R Then (A) both P and Q are true (B) P is true and Q is false (C) P is false and Q is true (D) both P and Q are false.

int (px^(p+2q-1)-qx^(q-1))/(x^(2p+2q)+2x^(p+q)+1) dx is equal to (1) -x^p/(x^(p+q)+1)+C (2) x^q/(x^(p+q)+1)+C (3) -x^q/(x^(p+q)+1)+C (4) x^p/(x^(p+q)+1)+C

Using properties of determinants, show that |{:(x, p, q), ( p, x, q), (q,q,x):}|= (x-p) (x ^(2) + px - 2q ^(2))

Let P(x) =x^(2)+ 1/2x + b and Q(x) = x^(2) + cx + d be two polynomials with real coefficients such that P(x) Q(x) = Q(P(x)) for all real x. Find all the real roots of P(Q(x)) =0.