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

If x nearly equal to 1 show that (px^q - qx^p)/(x^q - x^p) = (1)/(1 -x) ( 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 ~(p harr q) -= ( p ^^ ~q) vv(~p ^^q) .

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

Show that (p^^q)vv(~p)vv(p^^~q) is a tautology

(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) .

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))

If the roots of the equation qx^(2)+2px+2q=0 are real and unequal, prove that the roots of the equation (p+q)x^(2)+2qx+(p-q)=0 are imaginary.

If the equations x^2+p x+q=0 and x^2+p^(prime)x+q^(prime)=0 have a common root, then it must be equal to a. (p^(prime)-p ^(prime) q)/(q-q^(prime)) b. (q-q ')/(p^(prime)-p) c. (p^(prime)-p)/(q-q^(prime)) d. (p q^(prime)-p^(prime) q)/(p-p^(prime))