Let S be the sum of the first 9 term of the series :(x + ka} + (x² + (k + 2)a} + {x³ + (k + 4)a} + {x⁴ + (k + 6)a} + ........ where a $\ne$ 0 and x $\ne$ 1. If $\mathrm{S} = \frac{{\mathrm{x}}^{10} - \mathrm{x} +\hspace{0.17em}45\mathrm{a}\left(\mathrm{x} - 1\right)}{\mathrm{x} - 1}$, then k is equal to

•  - 3

• 1

•  - 5

• 3

The imaginary part of

${\left(3 +\hspace{0.17em}2\sqrt{ - 54}\right)}^{\frac{1}{2}} - {\left(3 - 2\sqrt{ - 54}\right)}^{\frac{1}{2}} \mathrm{can} \mathrm{be}$

• $\sqrt{ - 6}$

• $\sqrt{6}$

• $- 2\sqrt{6}$

• 6

$\underset{\mathrm{x}\to 0}{\lim }{\left(\tan \left(\frac{\mathrm{\pi }}{4} + \mathrm{x}\right)\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.} = ?$

• 2

• 1

• e

• e²

For some $\mathrm{\theta } \in \left(0, \frac{\mathrm{\pi }}{2}\right)$, if the eccentricity of the hyperbola, ${\mathrm{x}}^2 - {\mathrm{y}}^2{\sec }^2\left(\mathrm{\theta }\right) = 10$ is 5 times the eccentricity of the ellipse , ${\mathrm{x}}^2{\sec }^2\left(\mathrm{\theta }\right) + {\mathrm{y}}^2 = 5$, then the length of the latus rectum of the ellipse, is

• $\frac{2\sqrt{5}}{3}$

• $2\sqrt{6}$

• $\frac{4\sqrt{5}}{3}$

• $\sqrt{30}$

5.

Which of the following is a tautology ?

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

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

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

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

Let E^c denote the complement of an event E. Let E₁, E₂ and E₃ be any pairwise independent events withP(E₁) > 0 and $\mathrm{P}\left({\mathrm{E}}_1 \cap {\mathrm{E}}_2\cap {\mathrm{E}}_3\right) = 0 \mathrm{then} \left({{\mathrm{E}}_3}^{\mathrm{c}} \cap \raisebox{1ex}{${{\mathrm{E}}_3}^{\mathrm{c}}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{E}}_1$}\right.\right)$ is equal to

• $\mathrm{P}\left({{\mathrm{E}}_3}^{\mathrm{c}}\right) - \mathrm{P}\left({\mathrm{E}}_2\right)$

• $\mathrm{P}\left({{\mathrm{E}}_3}^{\mathrm{c}}\right) - \mathrm{P}\left({{\mathrm{E}}_2}^{\mathrm{c}}\right)$

• $\mathrm{P}\left({\mathrm{E}}_3\right) - \mathrm{P}\left({\mathrm{E}}^{\mathrm{c}}\right)$

• $\mathrm{P}\left({{\mathrm{E}}_2}^{\mathrm{C}}\right) + \mathrm{P}\left({\mathrm{E}}_3\right)$

If the equation ${\cos }^4\left(\mathrm{\theta }\right) + {\sin }^4\left(\mathrm{\theta }\right) + \mathrm{\lambda } = 0$ has real solutions for $\mathrm{\theta }, \mathrm{then} \mathrm{\lambda }$ lies in interval :

• $\left( - \frac{5}{4}, - 1\right)$

• $\left[ - 1, - \frac{1}{2}\right]$

• $\left[- \frac{1}{2}, - \frac{1}{4}\right]$

• $\left[ - \frac{3}{2}, - \frac{5}{4}\right]$

The set of all possible values of $\mathrm{\theta }$ in the interval (0, $\mathrm{\pi }$) for which the points(1, 2) and (sin($\mathrm{\theta )}$, cos($\mathrm{\theta )}$) lie on the same side of the line x + y = 1 is

• $\left(0, \frac{\mathrm{\pi }}{2}\right)$

• $\left(0, \frac{\mathrm{\pi }}{4}\right)$

• $\left(\frac{\mathrm{\pi }}{4}, \frac{3\mathrm{\pi }}{4}\right)$

• $\left(0, \frac{3\mathrm{\pi }}{4}\right)$

Let f(x) be a quadratic polynomial such that f( – 1) + f(2) = 0. If one of the roots of f(x) = 0 is 3, then its other root lies in :

• (1, 3)

• ( - 1, 0)

• ( - 3, - 1)

• (0, 1)

The area (in sq. units) of an equilateral triangle inscribed in the parabola y² = 8x, with one of its vertices on the vertex of this parabola is

• $64\sqrt{3}$

• $192\sqrt{3}$

• $128\sqrt{3}$

• $256\sqrt{3}$