The possible values of `a` such that the equation `x^2+2a x+a=sqrt(a^2+x-1/(16))-1/(16),xgeq-a ,` has two distinct real roots are given by: `[0,1]` (b) `[-oo,0]` `[0,oo]` (d) `(3/4,oo)`

If the equation sin ^(2) x - k sin x - 3 = 0 has exactly two distinct real roots in [0, pi] , then find the values of k .

If the roots of the equation x^2-8x+a^2-6a=0 are real and distinct, then find all possible value of adot

If a

Let S be the set of all non-zero real numbers such that the quadratic equation alphax^2-x+alpha=0 has two distinct real roots x_1a n dx_2 satisfying the inequality |x_1-x_2|<1. Which of the following intervals is (are) a subset (s) of S ? (1/2,1/(sqrt(5))) b. (1/(sqrt(5)),0) c. (0,1/(sqrt(5))) d. (1/(sqrt(5)),1/2)

The equation |sqrt(x^2+(y-1)^2)-sqrt(x^2+(y+1)^2)|=K will represent a hyperbola for (a) K in (0,2) (b) K in (-2,1) (c) K in (1,oo) (d) K in (0,oo)

The real number k for which the equation, 2x^3+""3x""+""k""=""0 has two distinct real roots in [0, 1] (1) lies between 2 and 3 (2) lies between -1 and 0 (3) does not exist (4) lies between 1 and 2

Solve the equation 3x^3 - 16x^2 + 23x -6 = 0 if the product of two roots is 1.

The exhaustive set of values of a for which inequation (a -1)x^2- (a+1)x+ a -1>=0 is true AA x >2 (a) (-oo,1) (b)[7/3,oo) (c) [3/7,oo) (d) none of these

The range of a for which the equation x^2+ax-4=0 has its smaller root in the interval (-1,2)i s a. (-oo,-3) b. (0,3) c. (0,oo) d. (-oo,-3)uu(0,oo)

If f(x)=int_(x^2)^(x^2+1)e^-t^2dt , then f(x) increases in (0,2) (b) no value of x (0,oo) (d) (-oo,0)