$\mathrm{If} \mathrm{\alpha } \mathrm{and} \mathrm{\beta } \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation}, 7{\mathrm{x}}^2 – 3\mathrm{x} – 2 = 0, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \frac{\mathrm{\alpha }}{1 - {\mathrm{\alpha }}^2} + \frac{\mathrm{\beta }}{1 - {\mathrm{\beta }}^2} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{1}{24}$

• $\frac{27}{36}$

• $\frac{27}{16}$

• $\frac{3}{8}$

The statement

$\left(\mathrm{p} \to \left(\mathrm{q} \to \mathrm{p}\right)\right) \to \left(\mathrm{p}\to \left(\mathrm{p} \vee \mathrm{q}\right)\right) \mathrm{is}$

• equivalent to $\left(\mathrm{p} \vee \mathrm{q}\right) \wedge \left(~ \mathrm{p}\right)$

• a contradiction

• a tautology

• equivalent to $\left(\mathrm{p} \wedge \mathrm{q}\right) \vee \left(~ \mathrm{q}\right)$

If the line y = mx + c is a common tangent to the hyperbola $\frac{{\mathrm{x}}^2}{100} - \frac{{\mathrm{y}}^2}{64} = 1 \mathrm{and} \mathrm{the} \mathrm{circle} {\mathrm{x}}^2 + {\mathrm{y}}^2 = 36, \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{true}$

• 4c² = 369

• 5m = 4

• c² = 369

• 8m + 5 = 0

If the length of the chord of the circle, x² + y² = r²(r > 0) along the line, y – 2x = 3 is r, then r² is equal to :

• $\frac{9}{5}$

• 12

• $\frac{12}{5}$

• $\frac{24}{5}$

If the sum of the second,third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eighth terms is 243, then the sum of the first 50 terms of this G.P. is :

• $\frac{2}{13}\left(3^{50} - 1\right)$

• $\frac{1}{26}\left(3^{49} - 1\right)$

• $\frac{1}{13}\left(3^{50} - 1\right)$

• $\frac{1}{26}\left(3^{50} - 1\right)$

If the mean and the standard deviation of the data 3, 5, 7, a, b are 5 and 2 respectively, then a and b are the roots of the equation : 

• x² - 20x + 18 = 0

• 2x² - 20x + 19 = 0

• x² - 10x + 19 = 0

• x² - 10x + 18 = 0

7.

$\mathrm{The} \mathrm{value} \mathrm{of} {\left(\frac{ - 1 + \mathrm{i}\sqrt{3}}{1 - \mathrm{i}}\right)}^{30} \mathrm{is} :$

• 6⁵

• 2¹⁵i

• - 2¹⁵

•  - 2¹⁵i

$\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{x}\left({\mathrm{e}}^{\left({}^{\sqrt{1 + {\mathrm{x}}^2 + {\mathrm{x}}^4} - 1}\right)/\mathrm{x}} - 1\right)}{\sqrt{1 + {\mathrm{x}}^2 + {\mathrm{x}}^4} - 1}$

• does not exist

• is equal to 1

• is equal to $\sqrt{\mathrm{e}}$

• is equal to 0

$$\begin{gathered} \mathrm{Solve} \mathrm{L} = {\sin }^2\left(\frac{\mathrm{\pi }}{16}\right) - {\sin }^2\left(\frac{\mathrm{\pi }}{8}\right) \mathrm{and} \\ \mathrm{M} = {\cos }^2\left(\frac{\mathrm{\pi }}{16}\right) - {\cos }^2\left(\frac{\mathrm{\pi }}{8}\right) \end{gathered}$$

• $\mathrm{M} = \frac{1}{4\sqrt{2}} + \frac{1}{4}\cos \left(\frac{\mathrm{\pi }}{8}\right)$

• $\mathrm{M} = \frac{1}{2\sqrt{2}} + \frac{1}{2}\cos \left(\frac{\mathrm{\pi }}{8}\right)$

• L =  - $\mathrm{L} = - \frac{1}{2\sqrt{2}} + \frac{1}{2}\cos \left(\frac{{\displaystyle \mathrm{\pi }}}{8}\right)$

• $\mathrm{L} = \frac{1}{4\sqrt{2}} - \frac{1}{4}\cos \left(\frac{\mathrm{\pi }}{8}\right)$

$$\begin{gathered} \mathrm{If} \mathrm{for} \mathrm{some} \mathrm{\alpha } \in \mathrm{R}, \mathrm{the} \mathrm{lines} \\ {\mathrm{L}}_1 : \frac{\mathrm{x} + 1}{2} = \frac{\mathrm{y} - 2}{ - 1} = \frac{\mathrm{z} - 1}{1} \mathrm{and} \\ {\mathrm{L}}_2 :\hspace{0.17em}\frac{\mathrm{x} + 2}{\mathrm{\alpha }} = \frac{\mathrm{y} + 1}{5 - \mathrm{\alpha }} = \frac{\mathrm{z} + 1}{1} \mathrm{are} \mathrm{coplanar}, \mathrm{then} \mathrm{the} \mathrm{line} {\mathrm{L}}_2 \mathrm{passes} \mathrm{through} \mathrm{point} : \end{gathered}$$

• (2, - 10, - 2)

• (10, 2, 2)

• ( - 2, 10, 2)

• (10, - 2, - 2)