The set of all real values of $$\lambda$$ for which the quadratic equations, ($$\lambda$$2 + 1)x2 – 4$$\lambda$$x + 2 = 0 always have exactly one root in the interval (0, 1) is :

Let [t] denote the greatest integer $$\le$$ t. Then the equation in x, [x]2 + 2[x+2] - 7 = 0 has :

Let $$\lambda \ne 0$$ be in R. If $$\alpha$$ and $$\beta$$ are the roots of the equation, x2 - x + 2$$\lambda$$ = 0 and $$\alpha$$ and $$\gamma$$ are the roots of the equation, $$3x^2-10x+27\lambda =0$$, then {{\beta \gamma } \over \lambda } is equal to:

The product of the roots of the equation 9x2 - 18|x| + 5 = 0 is :

If $$\alpha$$ and $$\beta$$ are the roots of the equation, 7x2 – 3x – 2 = 0, then the value of {\alpha \over {1 - {\alpha ^2}}} + {\beta \over {1 - {\beta ^2}}} is equal to :

If $$\alpha$$ and $$\beta$$ be two roots of the equation x2 – 64x + 256 = 0. Then the value of {\left( {{{{\alpha ^3}} \over {{\beta ^5}}}} \right)^{1/8}} + {\left( {{{{\beta ^3}} \over {{\alpha ^5}}}} \right)^{1/8}} is :

If $$\alpha$$ and $$\beta$$ are the roots of the equation 2x(2x + 1) = 1, then $$\beta$$ is equal to :

Let $$\alpha$$ and $$\beta$$ be the roots of the equation 5x2 + 6x – 2 = 0. If Sn = $$\alpha$$n + $$\beta$$n, n = 1, 2, 3...., then :

Let a, b $$\in$$ R, a $$\ne$$ 0 be such that the equation, ax2 – 2bx + 5 = 0 has a repeated root $$\alpha$$, which is also a root of the equation, x2 – 2bx – 10 = 0. If $$\beta$$ is the other root of this equation, then $$\alpha$$2 + $$\beta$$2 is equal to :

The number of real roots of the equation, e4x + e3x – 4e2x + ex + 1 = 0 is :

Let \alpha = {{ - 1 + i\sqrt 3 } \over 2}. If $$a=\left(1+\alpha \right)\sum _{k=0}^{100}{\alpha }^{2k}$$ and $$b=\sum _{k=0}^{100}{\alpha }^{3k}$$, then a and b are the roots of the quadratic equation :

Let S be the set of all real roots of the equation, 3x(3x – 1) + 2 = |3x – 1| + |3x – 2|. Then S :

Let $$\alpha$$ and $$\beta$$ be the roots of the equation x2 - x - 1 = 0. If pk = $${\left(\alpha \right)}^k+{\left(\beta \right)}^k$$ , k $$\ge$$ 1, then which one of the following statements is not true?

Let $$\alpha$$ and $$\beta$$ be two real roots of the equation (k + 1)tan2x - $$\sqrt{2}$$ . $$\lambda$$tanx = (1 - k), where k($$\ne$$ - 1) and $$\lambda$$ are real numbers. if tan2 ($$\alpha$$ + $$\beta$$) = 50, then a value of $$\lambda$$ is:

Let f(x) be a quadratic polynomial such that f(–1) + f(2) = 0. If one of the roots of f(x) = 0 is 3, then its other root lies in :

Let P(3, 3) be a point on the hyperbola, {{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1. If the normal to it at P intersects the x-axis at (9, 0) and e is its eccentricity, then the ordered pair (a2, e2) is equal to :

If the line y = mx + c is a common tangent to the hyperbola {{{x^2}} \over {100}} - {{{y^2}} \over {64}} = 1 and the circle x2 + y2 = 36, then which one of the following is true?

Let e1 and e2 be the eccentricities of the ellipse, {{{x^2}} \over {25}} + {{{y^2}} \over {{b^2}}} = 1(b < 5) and the hyperbola, {{{x^2}} \over {16}} - {{{y^2}} \over {{b^2}}} = 1 respectively satisfying e1e2 = 1. If $$\alpha$$ and $$\beta$$ are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair ($$\alpha$$, $$\beta$$) is equal to :

A hyperbola having the transverse axis of length $$\sqrt{2}$$ has the same foci as that of the ellipse 3x2 + 4y2 = 12, then this hyperbola does not pass through which of the following points?

For some \theta \in \left( {0,{\pi \over 2}} \right), if the eccentricity of the hyperbola, x2–y2sec2$$\theta$$ = 10 is $$\sqrt{5}$$ times the eccentricity of the ellipse, x2sec2$$\theta$$ + y2 = 5, then the length of the latus rectum of the ellipse, is :