Let $$\alpha$$, $$\beta$$ be two roots of the equation x2 + (20)1/4x + (5)1/2 = 0. Then $$\alpha$$8 + $$\beta$$8 is equal to

The set of all values of K > $$-$$1, for which the equation $$(3x^2+4x+3{)}^2-(k+1)(3x^2+4x+3)(3x^2+4x+2)+k(3x^2+4x+2{)}^2=0$$ has real roots, is :

cosec18$${}^{\circ }$$ is a root of the equation :

The sum of the roots of the equation $$x+1-2{\log }_2(3+2^x)+2{\log }_4(10-2^{-x})=0$$, is :

The numbers of pairs (a, b) of real numbers, such that whenever $$\alpha$$ is a root of the equation x2 + ax + b = 0, $$\alpha$$2 $$-$$ 2 is also a root of this equation, is :

A hyperbola passes through the foci of the ellipse {{{x^2}} \over {25}} + {{{y^2}} \over {16}} = 1 and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is :

The locus of the midpoints of the chord of the circle, x2 + y2 = 25 which is tangent to the hyperbola, {{{x^2}} \over 9} - {{{y^2}} \over {16}} = 1 is :

Consider a hyperbola H : x2 $$-$$ 2y2 = 4. Let the tangent at a point P(4, $$\sqrt{6}$$) meet the x-axis at Q and latus rectum at R(x1, y1), x1 > 0. If F is a focus of H which is nearer to the point P, then the area of $$\Delta$$QFR is equal to :

Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x2 $$-$$ y2 = 3. If L is also a tangent to the parabola y2 = $$\alpha$$x, then $$\alpha$$ is equal to :

The locus of the centroid of the triangle formed by any point P on the hyperbola $$16x^2-9y^2+32x+36y-164=0$$, and its foci is :

The point $$P\left(-2\sqrt{6},\sqrt{3}\right)$$ lies on the hyperbola {{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1 having eccentricity {{\sqrt 5 } \over 2}. If the tangent and normal at P to the hyperbola intersect its conjugate axis at the point Q and R respectively, then QR is equal to :

The locus of the mid points of the chords of the hyperbola x2 $$-$$ y2 = 4, which touch the parabola y2 = 8x, is :

Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices (a, c), (2, b) and (a, b) be \left( {{{10} \over 3},{7 \over 3}} \right). If $$\alpha$$, $$\beta$$ are the roots of the equation $$a{x}^2+bx+1=0$$, then the value of $${\alpha }^2+{\beta }^2-\alpha \beta$$ is :

The triangle of maximum area that can be inscribed in a given circle of radius 'r' is :

If in a triangle ABC, AB = 5 units, \angle B = {\cos ^{ - 1}}\left( {{3 \over 5}} \right) and radius of circumcircle of $$\Delta$$ABC is 5 units, then the area (in sq. units) of $$\Delta$$ABC is :

Let {{\sin A} \over {\sin B}} = {{\sin (A - C)} \over {\sin (C - B)}}, where A, B, C are angles of triangle ABC. If the lengths of the sides opposite these angles are a, b, c respectively, then :

The equation of the plane passing through the point (1, 2, -3) and perpendicular to the planes 3x + y - 2z = 5 and 2x - 5y - z = 7, is :

The distance of the point (1, 1, 9) from the point of intersection of the line {{x - 3} \over 1} = {{y - 4} \over 2} = {{z - 5} \over 2} and the plane x + y + z = 17 is :

Let a, b$$\in$$R. If the mirror image of the point P(a, 6, 9) with respect to the line {{x - 3} \over 7} = {{y - 2} \over 5} = {{z - 1} \over { - 9}} is (20, b, $$-$$a$$-$$9), then | a + b |, is equal to :

The vector equation of the plane passing through the intersection of the planes $$\overrightarrow{r}.\left(\hat{i}+\hat{j}+\hat{k}\right)=1$$ and $$\overrightarrow{r}.\left(\hat{i}-2\hat{j}\right)=-2$$, and the point (1, 0, 2) is :