The angle between the two circles, each passing through the centre of the other is

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

• $\mathrm{\pi }$

$\mathrm{If} {\log }_{\frac{1}{\sqrt{3}}}\left\{\frac{{\left|\mathrm{z}\right|}^2 - \left|\mathrm{z}\right| + 1}{2 + \left|\mathrm{z}\right|}\right\} > - 2, \mathrm{then} \mathrm{z} \mathrm{lies} \mathrm{inside}$

• a triangle

• an ellipse

• a circle

• a square

A circle having centre at the origin passes through the three vertices of an equilateral triangle the length of its median being 9 units. Then the equation of that circle is

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 9$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 18$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 36$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 81$

$1 + \cos \left(10°\right) \hspace{0.17em}+ \cos \left(20°\right) + \cos \left(30°\right) = ?$

• $4\sin \left(10°\right)\sin \left(20°\right)\sin \left(30°\right)$

• $4\cos \left(5°\right)\cos \left(10°\right)\cos \left(15°\right)$

• $4\cos \left(10°\right)\cos \left(20°\right)\cos \left(30°\right)$

• $4\sin \left(5°\right)\sin \left(10°\right)\sin \left(15°\right)$

The set of all values of a such that both the points (1, 2) and (3, 4) lie on the same side of the line 3x - 5y + a = 0

• $\left\{\mathrm{x} \in \mathrm{IR} \mathrm{x} > 11\right\}$

• $\left\{\mathrm{x} \in \mathrm{IR} \mathrm{x} > 11\right\} \cup \left\{\mathrm{x} \in \mathrm{IR} \mathrm{x} < 7\right\}$

• $\left\{\mathrm{x} \in \mathrm{IR} \mathrm{x} < 7\right\}$

• $\mathrm{\varphi }$

$\mathrm{If} \mathrm{x} = \frac{1 . 3}{3 . 6} + \frac{1 . 3 . 5}{3 . 6. 9} + \frac{1 . 3 . 5 . 7}{3 . 6. 9 . 12} + ... \mathrm{to} \mathrm{infinite} \mathrm{terms}, \mathrm{then} 9{\mathrm{x}}^2 + 24\mathrm{x} = ?$

• 31

• 11

• 41

• 21

${}^{37}\mathrm{C}_4 + \sum _{\mathrm{r} = 1}^5{}^{\left(42 - \mathrm{r}\right)}\mathrm{C}_{\mathrm{r}} = ?$

• ${}^{41}\mathrm{C}_4$

• ${}^{39}\mathrm{C}_4$

• ${}^{38}\mathrm{C}_4$

• ${}^{42}\mathrm{C}_4$

48.

A container s the shape of an inverted cone. Its height is 6 m and radius is 4m at the top. If it is filled with water at the rate of 3m/min then the rate of change of height of water(in mt/min) when the water level is 3 m is

• $\frac{3}{4\mathrm{\pi }}$

• $\frac{2}{9\mathrm{\pi }}$

• $16\mathrm{\pi }$

• $2\mathrm{\pi }$

$\mathrm{If} \mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{are} \mathrm{the} \mathrm{lengths} \mathrm{of} \mathrm{the} \mathrm{tangents} \mathrm{from} \mathrm{the} \mathrm{vertices} \mathrm{of} \mathrm{a} \mathrm{triangle} \mathrm{to} \mathrm{its} \mathrm{incircle}. \mathrm{Then}$

• $\mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma } = \frac{1}{{\mathrm{r}}^2}\left(\mathrm{\alpha \beta \gamma }\right)$

• $\mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma } = \frac{1}{\mathrm{r}}\left(\mathrm{\alpha \beta \gamma }\right)$

• $\frac{1}{\mathrm{\alpha }} + \frac{1}{\mathrm{\beta }} + \frac{1}{\mathrm{\gamma }} = \mathrm{r}\left(\mathrm{\alpha \beta \gamma }\right)$

• ${\mathrm{\alpha }}^2 + {\mathrm{\beta }}^2 + {\mathrm{\gamma }}^2 = \frac{2}{\mathrm{r}}\left(\mathrm{\alpha \beta \gamma }\right)$

The angle between the tangents drawn from the point (1, 2) to the ellipse 3x² + 2y² = 5 is

• ${\tan }^{-1}\left(\frac{12\sqrt{5}}{5}\right)$

• ${\tan }^{-1}\left(\frac{12\sqrt{5}}{13}\right)$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$