If both ` (x-2) and (x - (1)/(2))` are fractors of `p x ^(2) 5x +r,` then show that p=r.

If both (x-2) and ( x- (1)/(2)) are the factors of ax^(2) + 5x + b , then show that a =b .

If the ratio of the roots of a x ^2+2b x+c=0 is same as the ratios of roots of p x^2+2q x+r=0, then a. (2b)/(a c)=(q^2)/(p r) b. b/(a c)=(q^)/(p r) c. (b^2)/(a c)=(q^2)/(p r) d. none of these

If the roots of the equation (q-r)x^(2)+(r-p)x+p-q=0 are equal, then show that p, q and r are in A.P.

If (x+2) is a common factor of (px^2+qx+r) and (qx^2+px+r) then a) p=q or p+q+r=0 b) p=r or p+q+r=0 c) q=r or p+q+r=0 d) p=q=-1/2r

If alpha,beta are the roots of a x^2+b x+c=0a n dalpha+h ,beta+h are the roots of p x^2+q x+r=0then h= a. -1/2(a/b-p/q) b. (b/a-q/p) c. 1/2(b/a-q/p) d. none of these

Three points P (h, k), Q(x_(1) , y_(1))" and " R (x_(2) , Y_(2)) lie on a line. Show that (h - x_(1)) (y_(2) - y_(1)) = (k - y_(1)) (x_(2) - x_(1)) .

If the roots of equation x^3+px^2+qx+r=0 are in A.P then show that 2p^3-9pq+27r = 0 .