If `P(x)=x^2+ax+b and Q(x)=x^2+a_1x+b_1,a,b,a_1,b_1 epsilon R` and equation `P(x).Q(x)=0` has at most one real root, then
(A) `(1+a+b)(1+a_1+b_1)gt0` (B) `(1+a+b)(1+a_1+b_1)lt0`
(C) `(1+a+b)/(1+a_1_b_1)gt0` (D) `1+a+bgt0`

Prove that the points (a ,\ b),\ (a_1,\ b_1) and (a-a_1,\ b-b_1) are collinear if a b_1=a_1b .

Show that one of the roots of equation ax^2+bx+c=0 may be reciprocal of one of the roots of a_1x^2+b_1x+c_1=0 if (aa_1-c c_1)^2=(bc_1-ab_1)(b_1c-a_1b)

If a_1x^2 + b_1 x + c_1 = 0 and a_2x^2 + b_2 x + c_2 = 0 has a common root, then the common root is

The equation A/(x-a_1)+A_2/(x-a_2)+A_3/(x-a_3)=0 ,where A_1,A_2,A_3gt0 and a_1lta_2lta_3 has two real roots lying in the invervals. (A) (a_1,a_2) and (a_2,a_3) (B) (-oo,a_1) and (a_3,oo) (C) (A_1,A_3) and (A_2,A_3) (D) none of these

If a ,\ b ,\ c are the roots of the equation x^3+p x+q=0 , then find the value of the determinant |(1+a,1 ,1), (1, 1+b,1),( 1, 1 ,1+c)| .

If one of the roots of the equation ax^2 + bx + c = 0 be reciprocal of one of the a_1 x^2 + b_1 x + c_1 = 0 , then prove that (a a_1-c c_1)^2 =(bc_1-ab_1) (b_1c-a_1 b) .

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

If the equation of the locus of a point equidistant from the points (a_1, b_1) and (a_2, b_2) is (a_1-a_2)x+(b_1-b_2)y+c=0 , then find the value of c .

The equation [1x y][1 3 4 0 2-1 0 0 1]=[0] has fo ry=0 b. rational roots for y=-1 d. integral roots Then (ii) a. (p) (r) b. (q) (p) c. (p) (q) d. (r) (p)

Let f (x) =ax ^(2) +bx + c,a ne 0, such the f (-1-x)=f (-1+ x) AA x in R. Also given that f (x) =0 has no real roots and 4a + b gt 0. Let p =b-4a, q=2a +b, then pq is: