A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If x denotes the percentage of them, who like both coffee and tea, then x cannot be :

• 36

• 63

• 38

• 54

$$\begin{gathered} \mathrm{If} \mathrm{S} \mathrm{is} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{the} \mathrm{first} 10 \mathrm{terms} \mathrm{of} \mathrm{the} \mathrm{series} {\tan }^{-1}\left(\frac{1}{3}\right) + {\tan }^{-1}\left(\frac{1}{7}\right) + {\tan }^{-1}\left(\frac{1}{13}\right) + {\tan }^{-1}\left(\frac{1}{3}\right) \\ + {\tan }^{-1}\left(\frac{1}{21}\right) + ... , \mathrm{then} \tan \left(\mathrm{S}\right) = ? \end{gathered}$$

• $- \frac{6}{5}$

• $\frac{5}{11}$

• $\frac{5}{6}$

• $\frac{10}{11}$

3.

$\mathrm{If} 2^{10} + 2^9 . 3^1 +\hspace{0.17em}{2}^8 . 3^2 +\hspace{0.17em}... + 2 . 3^9 +3^{10} = 3^{10}= \mathrm{S} - 2^{11} \mathrm{Then} \mathrm{S} = ?$

• $3^{11}$

• $2 . 3^{11}$

• $\frac{3^{11}}{2} + 2^{10}$

• $3^{11} - 2^{11}$

If the common tangent to the parabolas, y² = 4x and x² = 4y also touches the circle, x² + y² = c², then c is equal to :

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{2\sqrt{2}}$

• $\frac{1}{2}$

• $\frac{1}{4}$

$$\begin{gathered} \mathrm{If} \mathrm{\alpha } \mathrm{is} \mathrm{the} \mathrm{positive} \mathrm{root} \mathrm{of} \mathrm{the} \mathrm{equation}, \mathrm{p}\left(\mathrm{x}\right) = {\mathrm{x}}^2 – \mathrm{x} – 2 = 0, \mathrm{then} \\ \underset{\mathrm{x}\to \mathrm{\alpha }}{\lim }\frac{\sqrt{1 - \cos \left(\mathrm{p}\left(\mathrm{x}\right)\right)}}{\mathrm{x} +\hspace{0.17em}\mathrm{\alpha } - 4} = ? \end{gathered}$$

• $\frac{3}{2}$

• $\frac{3}{\sqrt{2}}$

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{2}$

If the co–ordinates of two points A and B are $\left(\sqrt{7}, 0\right) \mathrm{and} \left( - \sqrt{7}, 0\right)$ respectively and P is any point on the conic, 9x² + 16y² = 144, then PA + PB is equal to :

• 8

• 16

• 9

• 6

The product of the roots of the equation 9x² – 18|x| + 5 = 0 is

• $\frac{25}{81}$

• $\frac{5}{27}$

• $\frac{25}{9}$

• $\frac{5}{9}$

If the point P on the curve, 4x² + 5y² = 20 is farthest from the point Q(0, – 4), then PQ² is equal to :

• 29

• 48

• 21

• 36

If (a, b, c) is the image of the point (1, 2, – 3) in the line,

$\frac{\mathrm{x} + 1}{2} = \frac{\mathrm{y} - 3}{ - 2} = \frac{\mathrm{z}}{ - 1}$, then a + b + c is equal to :

• 3

• - 1

• 2

• 1

If the four complex numbers $\mathrm{z}, \overline{\mathrm{z}}, \overline{\mathrm{z}} - 2\mathrm{Re}\left(\overline{\mathrm{z}}\right) \mathrm{and} \mathrm{z} - 2\mathrm{Re}\left(\mathrm{z}\right)$ represent the vertices of a square of side 4 units in the Argand plane, then |z| is equal to :

• 4

• 2

• $4\sqrt{2}$

• $2\sqrt{2}$