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 factors of (ax^(2)+5x+b) show that a=b

If x^(2)-3 x+2 is a factor of x^(4)-p x^(2)+q , then the values of p and q are

If p(x) = 5x^(2)-3x + 7 , then p(1) equals to :

. If P Q and P R are the tangents from P to the ellipse (x^(2))/(2)+y^(2)=1 , and the equation of Q R is x+3 y=1 , then P =

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 coefficients of r^(th) , (r+1)^(t h) and (r+2)^(th ) terms in the expansion of (1+x)^(14) are in A.P, then the value of r is

Use the Factor Theorem to determine whether g (x) is a factor of p (x) in each of the following cases: (i) p(x) = 2x^3+x^2-2x-1,g(x) =x+1 (ii) p(x)=x^3+3x+1, g (x) =x+2 (iii) p(x) = x^3-4x^2+x+6,g(x)=x-3

Let for a != a_1 != 0 , f(x)=ax^2+bx+c , g(x)=a_1x^2+b_1x+c_1 and p(x) = f(x) - g(x) . If p(x) = 0 only for x = -1 and p(-2) = 2 then the value of p(2) .

If X_(1) = (x_(1), y_(1)) and X_(2) = (x_(2),y_(2)) are two optimal solutions of a L.P.P., then

Area of the triangle with vertices (a, b ), (x_(1), y_(1)) and (x_(2), y_(2)) , where a, x_(1), x_(2) are in G.P. with common ratio r and b, y_(1), y_(2) are in G.P. with common ratio s is :