If both the roots of the quadratic equation x2 $$-$$ mx + 4 = 0 are real and distinct and they lie in the interval [1, 5], then m lies in the interval :

The sum of the solutions of the equation $$\left|\sqrt{x}-2\right|+\sqrt{x}\left(\sqrt{x}-4\right)+2=0$$ (x > 0) is equal to:

If a directrix of a hyperbola centred at the origin and passing through the point (4, –2$$\sqrt{3}$$ ) is 5x = 4$$\sqrt{5}$$ and its eccentricity is e, then :

If 5x + 9 = 0 is the directrix of the hyperbola 16x2 – 9y2 = 144, then its corresponding focus is :

If the line y = mx + 7$$\sqrt{3}$$ is normal to the hyperbola {{{x^2}} \over {24}} - {{{y^2}} \over {18}} = 1 , then a value of m is :

If the eccentricity of the standard hyperbola passing through the point (4,6) is 2, then the equation of the tangent to the hyperbola at (4,6) is :

If the vertices of a hyperbola be at (–2, 0) and (2, 0) and one of its foci be at (–3, 0), then which one of the following points does not lie on this hyperbola?

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is :

Equation of a common tangent to the parabola y2 = 4x and the hyperbola xy = 2 is :

The equation of a tangent to the hyperbola 4x2 – 5y2 = 20 parallel to the line x – y = 2 is :

A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length 4 along the x-axis. Then the eccentricity of the hyperbola is :

Let 0 < \theta < {\pi \over 2}. If the eccentricity of the hyperbola {{{x^2}} \over {{{\cos }^2}\theta }} - {{{y^2}} \over {{{\sin }^2}\theta }} = 1 is greater than 2, then the length of its latus rectum lies in the interval :

A circle cuts a chord of length 4a on the x-axis and passes through a point on the y-axis, distant 2b from the origin. Then the locus of the centre of this circle, is :

With the usual notation, in $$\Delta$$ABC, if $$\angle A+\angle B$$ = 120o, a = $$\sqrt{3}$$ $$+$$ 1, b = $$\sqrt{3}$$ $$-$$ 1 then the ratio $$\angle A:\angle B,$$ is :

In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. If x2 – c2 = y, where c is the length of the third side of the triangle, then the circumradius of the triangle is :

Given {{b + c} \over {11}} = {{c + a} \over {12}} = {{a + b} \over {13}} for a $$\Delta$$ABC with usual notation. If {{\cos A} \over \alpha } = {{\cos B} \over \beta } = {{\cos C} \over \gamma }, then the ordered triad ($$\alpha$$, $$\beta$$, $$\gamma$$) has a value :

If the lengths of the sides of a triangle are in A.P. and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is :

The angles A, B and C of a triangle ABC are in A.P. and a : b = 1 : $$\sqrt{3}$$. If c = 4 cm, then the area (in sq. cm) of this triangle is :

A triangle has a vertex at (1, 2) and the mid points of the two sides through it are (–1, 1) and (2, 3). Then the centroid of this triangle is :

If an angle between the line, {{x + 1} \over 2} = {{y - 2} \over 1} = {{z - 3} \over { - 2}} and the plane, $$x-2y-kz=3$$ is {\cos ^{ - 1}}\left( {{{2\sqrt 2 } \over 3}} \right), then a value of k is :