Observe the following statements :

I. The circle x² + y² - 6x - 4y - 7 = 0 touches y-axis.

II. The circle x² + y² + 6x + 4y - 7 = 0 touches

x-axis. Which of the following is a correct statement ?

• Both I and II are true

• Neither I nor II is true

• I is true, II is false

• I is false, II is true

The length of the tangent drawn to the circle x² + y² - 2x + 4y - 11 = 0

• 1

• 2

• 3

• 4

43.

If b and c are the lengths of the segments of any focal chord of a parabola y² = 4ax, then the length of the semi-latus rectum is

• $\frac{\mathrm{bc}}{\mathrm{b} + \mathrm{c}}$

• $\sqrt{\mathrm{bc}}$

• $\frac{\mathrm{b} + \mathrm{c}}{2}$

• $\frac{2\mathrm{bc}}{\mathrm{b} + \mathrm{c}}$

Equations of the latus rectum of the ellipse

9x² + 4y² - 18x - 8y - 23 = 0 are : 

• $\mathrm{y} = \pm \sqrt{5}$

• $\mathrm{x} = \pm \sqrt{5}$

• $\mathrm{y} = 1 \pm \sqrt{5}$

• $\mathrm{x} = - 1 \pm \sqrt{5}$

If the eccentricity of a hyperbola is $\sqrt{3}$; then the eccentricity of its conjugate hyperbola is :

• $\sqrt{2}$

• $\sqrt{3}$

• $\sqrt{\frac{3}{2}}$

• $2\sqrt{3}$

$\mathrm{If} 0 < \mathrm{p} <\mathrm{q}, \mathrm{then} \underset{\mathrm{n}\to \infty }{\lim }{\left({\mathrm{q}}^{\mathrm{n}} + {\mathrm{p}}^{\mathrm{n}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{n}$}\right.} = ?$

• e

• p

• q

• 0

$\underset{\mathrm{x}\to \infty }{\lim }\left[\sqrt{{\mathrm{x}}^2 + 2\mathrm{x} - 1} - \mathrm{x}\right] = ?$

• $\infty$

• $\frac{1}{2}$

• 4

• 1

$$\begin{gathered} \mathrm{If} {\mathrm{l}}_1 = \underset{\mathrm{x}\to 2^{ + }}{\lim }\left(\mathrm{x} + \left[\mathrm{x}\right]\right), {\mathrm{l}}_2 = \underset{\mathrm{x}\to 2^{ - }}{\lim }\left(2\mathrm{x} - \left[\mathrm{x}\right]\right) \\ \mathrm{and} {\mathrm{l}}_3 = \underset{\mathrm{x}\to \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\lim }\frac{\cos \left(\mathrm{x}\right)}{\left(\mathrm{x} - {\displaystyle \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}, \mathrm{then} : \end{gathered}$$

• ${\mathrm{l}}_1 < {\mathrm{l}}_2 < {\mathrm{l}}_3$

• ${\mathrm{l}}_2 < {\mathrm{l}}_3 < {\mathrm{l}}_1$

• ${\mathrm{l}}_3 < {\mathrm{l}}_2 < {\mathrm{l}}_1$

• ${\mathrm{l}}_1 < {\mathrm{l}}_3 < {\mathrm{l}}_2$

The sides of the rectangle of greatest area that can be inscribed in the ellipse x² + 4y² = 64 are :

• $\left(6\sqrt{2}, 4\sqrt{2}\right)$

• $\left(8\sqrt{2}, 4\sqrt{2}\right)$

• $\left(8\sqrt{2}, 8\sqrt{2}\right)$

• $\left(16\sqrt{2}, 4\sqrt{2}\right)$

The polar equation of the circle with centre $\left(2, \frac{\mathrm{\pi }}{2}\right)$ are radius 3 units is:

• ${\mathrm{r}}^2 + 4\mathrm{rcos}\left(\mathrm{\theta }\right) = 5$

• ${\mathrm{r}}^2 + 4\mathrm{rsin}\left(\mathrm{\theta }\right) = 5$

• ${\mathrm{r}}^2 - 4\mathrm{rsin}\left(\mathrm{\theta }\right) = 5$

• ${\mathrm{r}}^2 - 4\mathrm{rcos}\left(\mathrm{\theta }\right) = 5$