11.

The area (in sq units) of the equilateral triangle formed by the tangent at ($\sqrt{3}$, 0) to the hyperbola x² - 3y² = 3 with the pair of asymptotes of the hyperbola is

• $\sqrt{2}$

• $\sqrt{3}$

• $\frac{1}{\sqrt{3}}$

• $2\sqrt{3}$

The radius of the circle r = $12\cos \left(\mathrm{\theta }\right) + 5\sin \left(\mathrm{\theta }\right) \mathrm{IS}$

• $\frac{5}{12}$

• $\frac{17}{2}$

• $\frac{15}{2}$

• None of these

If x-coordinate of a point P on the line joining the points Q(2, 2, 1) and R(5, 1, - 2) is 4, then the z-coordinate of P is

• - 2

• - 1

• 1

• 2

A straight line is equally inclined to all the three coordinate axes. Then, an angle made by the line with the y-axis is

• ${\cos }^{-1}\left(\frac{1}{3}\right)$

• ${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

• ${\cos }^{-1}\left(\frac{2}{\sqrt{3}}\right)$

• $\frac{\mathrm{\pi }}{4}$

The equation of the sphere through the points (1, 0, 0), (0, 1, 0) and (1, 1, 1)and having the smallest radius

• $3\left({\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2\right) - 4\mathrm{x} - 4\mathrm{y} - 2\mathrm{z} + 1 = 0$

• $2\left({\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2\right) - 3\mathrm{x} - 3\mathrm{y} - \mathrm{z} + 1 = 0$

• $\left({\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2\right) - \mathrm{x} - \mathrm{y} - \mathrm{z} + 1 = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2 - 2\mathrm{x} - 2\mathrm{y} + 4\mathrm{z} + 1 = 0$

$\underset{\mathrm{x}\to \infty }{\lim }{\left(\frac{\mathrm{x} + 6}{\mathrm{x} + 1}\right)}^{\mathrm{x} + 4} = ?$

• e⁴

• e⁶

• e⁵

• e

$$\begin{gathered} \mathrm{Let} \mathrm{f} :\hspace{0.17em}\mathrm{R} \to \mathrm{R} \mathrm{be} \mathrm{defined} \mathrm{by} \\ \mathrm{f}\left(\mathrm{x}\right) = \left\{\mathrm{\alpha } + \frac{\sin \left(\mathrm{x}\right)}{\mathrm{x}}, \mathrm{if} \mathrm{x}> 0 \\ 2, \mathrm{if} \mathrm{x} = 0 \\ \mathrm{\beta } + \left[\frac{\sin \left(\mathrm{x}\right) - \mathrm{x}}{{\mathrm{x}}^3}\right], \mathrm{if} \mathrm{x} < 0\right\} \\ \mathrm{where}, \left[\mathrm{x}\right] \mathrm{denotes} \mathrm{the} \mathrm{integral} \mathrm{part} \mathrm{of} \mathrm{x}. \\ \mathrm{If} \mathrm{f} \mathrm{continuous} \mathrm{at} \mathrm{x} = 0, \mathrm{then} \mathrm{\beta } - \mathrm{\alpha } \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• - 1

• 1

• 0

• 2

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = {\left({\mathrm{x}}^2 - 1\right)}^7, \mathrm{then} {\mathrm{f}}^{\left(14\right)}\left(\mathrm{x}\right) = ?$

• 0

• 2!

• 7!

• 14!

The coordinates of the point P on the curve $\mathrm{x} = \mathrm{a}\left(\mathrm{\theta } + \sin \left(\mathrm{\theta }\right)\right), \mathrm{y} = \mathrm{a}\left(1 - \cos \left(\mathrm{\theta }\right)\right),$ where the tangent is inclined at an angle $\frac{\mathrm{\pi }}{4}$ to x-axis, are

• $\left[\mathrm{a}\left(\frac{\mathrm{\pi }}{4} - 1\right), \mathrm{a}\right]$

• $\left[\mathrm{a}\left(\frac{\mathrm{\pi }}{2} + 1\right), \mathrm{a}\right]$

• $\left[\mathrm{a}\left(\frac{\mathrm{\pi }}{2}\right), \mathrm{a}\right]$

• (a, a)

If $∆$ is the area of the triangle formed by the positive x-axis and the normal and tangent to the circle x² + y² = 4 at (1, $\sqrt{3}$) then $∆$ is equal to

• $\frac{\sqrt{3}}{2}$

• $\sqrt{3}$

• $2\sqrt{3}$

• 6