If the pair of straight lines given by Ax² + 2Hxy + By² = 0 (H² > AB) forms an equilateral triangle with line ax + by + c = 0, then (A + 3B)(3A + B) is equal to :

• H²

• - H²

• 2H²

• 4H²

The area (in sq units) of the quadrilateral formed by two pairs of lines ${\mathrm{\lambda }}^2{\mathrm{x}}^2 - {\mathrm{m}}^2{\mathrm{y}}^2 - \mathrm{n}\left(\mathrm{\lambda x} + \mathrm{my}\right) = 0$ and ${\mathrm{\lambda }}^2{\mathrm{x}}^2 - {\mathrm{m}}^2{\mathrm{y}}^2 + \mathrm{n}\left(\mathrm{\lambda x} + \mathrm{my}\right) = 0$ is :

• $\frac{{\mathrm{n}}^2}{2\left|\mathrm{\lambda m}\right|}$

• $\frac{{\mathrm{n}}^2}{\left|\mathrm{\lambda m}\right|}$

• $\frac{\mathrm{n}}{2\left|\mathrm{\lambda m}\right|}$

• $\frac{{\mathrm{n}}^2}{4\left|\mathrm{\lambda m}\right|}$

If the circle x² + y² + 6x - 2y + k = 0 bisects the circumference of the circle x² + y² + 2x - 6y - 15 = 0, then k is equal to:

• 21

• - 21

• 23

• - 23

If P is a point such that the ratio of the square of the lengths of the tangents from P to the circles x² + y² + 2x - 4y - 20 = 0 and x² + y² - 4x + 2y - 2y - 44 = 0 is 2 : 3, then the locus of P is a circle with centre :

• (7, - 8)

• (- 7, 8)

• (7, 8)

• (- 7, - 8)

If 5x - 12y + 10 = 0 and 12y - 5x + 16 = 0 are two tangents to a circle, then the radius of the circle is

• 1

• 2

• 4

• 6

The eccentricity of the ellipse 9x² + 5y² - 18x - 20y - 16 = 0, is:

• $\frac{1}{2}$

• $\frac{2}{3}$

• $\frac{3}{2}$

• $2$

The product of the lengths of perpendiculars drawn from any point on the hyperbola x² - 2y² - 2 = O to its asymptotes is

• $\frac{1}{2}$

• $\frac{2}{3}$

• $\frac{3}{2}$

• 2

The equation of the parabola with focus (0, 0)and directrix x + y = 4 is

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 - 2\mathrm{xy} + 8\mathrm{x} +8\mathrm{y} -16 = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 - 2\mathrm{xy} + 8\mathrm{x} + 8\mathrm{y} = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 + 8\mathrm{x} + 8\mathrm{y} - 16= 0$

• ${\mathrm{x}}^2 - {\mathrm{y}}^2 + 8\mathrm{x} +8\mathrm{y} - 16= 0$

$\underset{\mathrm{x}\to \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}{\lim } \left[\frac{3\sin \left(\mathrm{x}\right) - \sqrt{3}\cos \left(\mathrm{x}\right)}{6\mathrm{x} - \mathrm{\pi }}\right] \mathrm{is} \mathrm{equal} \mathrm{to} :$

• $\sqrt{3}$

• $\frac{1}{\sqrt{3}}$

• $\frac{1}{3}$

• $- \frac{1}{3}$

20.

$\mathrm{If} \mathrm{a} > 0, \underset{\mathrm{x}\to \mathrm{a}}{\lim } \frac{{\mathrm{a}}^{\mathrm{x}} - {\mathrm{x}}^{\mathrm{a}}}{{\mathrm{x}}^{\mathrm{x}} - {\mathrm{a}}^{\mathrm{a}}} = - 1, \mathrm{then} \mathrm{a} \mathrm{is} \mathrm{equal} \mathrm{to} :$

• 0

• 1

• e

• 2e