The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in

• $\left(- \infty , - 3\right) \cup \left(9, \infty \right)$

• $\left( - \infty , - 9\right) \cup \left(3, \infty \right)$

• $\left(- 3, \infty \right)$

• $\left( - \infty , 9\right)$

Box I contains 30 cards numbered I to 30 and Box II contains 20 cards numbered 31 to 50. A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box I is :

• $\frac{2}{3}$

• $\frac{2}{5}$

• $\frac{8}{17}$

• $\frac{4}{17}$

$\mathrm{The} \mathrm{domain} \mathrm{of} \mathrm{the} \mathrm{function} \mathrm{f}\left(\mathrm{x}\right) = {\sin }^{-1}\left(\frac{\left|\mathrm{x}\right| + 5}{{\mathrm{x}}^2 + 1}\right) \mathrm{is} \left( - \infty , - \mathrm{a}\right) \cup \left(\mathrm{a}, \infty \right), \mathrm{Then} \mathrm{a} =?$

• $\frac{1 + \sqrt{17}}{2}$

• $\frac{\sqrt{17} - 1}{2}$

• $\frac{\sqrt{17}}{2}$

• $\frac{\sqrt{17}}{2} + 1$

If |x| < 1, |y| < 1 and x $\ne$ y, then the sum to infinity of the following series (x + y) + (x² + xy + y²) + (x³ + x²y + xy² + y³) + ..... is :

• $\frac{\mathrm{x} +\hspace{0.17em}\mathrm{y} + \mathrm{xy}}{\left(1 - \mathrm{x}\right)\left(1 - \mathrm{y}\right)}$

• $\frac{\mathrm{x} + \mathrm{y} - \mathrm{xy}}{\left(1 - \mathrm{x}\right)\left(1 - \mathrm{y}\right)}$

• $\frac{\mathrm{x} + \mathrm{y} - \mathrm{xy}}{\left(1 - \mathrm{x}\right)\left(1 + \mathrm{y}\right)}$

• $\frac{\mathrm{x} + \mathrm{y} + \mathrm{xy}}{\left(1 + \mathrm{x}\right)\left(1 + \mathrm{y}\right)}$

$\mathrm{The} \mathrm{value} \mathrm{of} {\left(\frac{1 + \sin \left({\displaystyle \frac{2\mathrm{\pi }}{9}}\right) + \mathrm{icos}\left(\frac{2\mathrm{\pi }}{9}\right)}{1 + \sin \left(\frac{2\mathrm{\pi }}{9}\right) - \mathrm{icos}\left(\frac{2\mathrm{\pi }}{9}\right)}\right)}^3 \mathrm{is} :$

• $- \frac{1}{2}\left(\sqrt{3} - \mathrm{i}\right)$

• $\frac{1}{2}\left(\sqrt{3} - \mathrm{i}\right)$

• $\frac{1}{2}\left(1 - \mathrm{i}\sqrt{3}\right)$

• $- \frac{1}{2}\left(1 - \mathrm{i}\sqrt{3}\right)$

$$\begin{gathered} \mathrm{Let} \mathrm{\alpha } > 0, \mathrm{\beta } > 0 \mathrm{be} \mathrm{such} \mathrm{that} {\mathrm{\alpha }}^3 + {\mathrm{\beta }}^2 = 4. \mathrm{If} \mathrm{the} \mathrm{maximum} \mathrm{value} \mathrm{of} \mathrm{the} \mathrm{term} \\ \mathrm{independent} \mathrm{of} \mathrm{x} \mathrm{in} \mathrm{the} \mathrm{binomial} \mathrm{expansion} \mathrm{of} {\left({\mathrm{\alpha x}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$9$}\right.} + {\mathrm{\beta x}}^{- \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\right)}^{10} \mathrm{is} 10\mathrm{k}, \mathrm{then} \mathrm{k} \mathrm{is} \mathrm{equal} \mathrm{to} : \end{gathered}$$

• 176

• 336

• 352

• 84

$$\begin{gathered} \mathrm{If} \mathrm{R} = \left\{\left(\mathrm{x}, \mathrm{y}\right) :\hspace{0.17em}\mathrm{x}, \mathrm{y} \in \mathrm{Z}, {\mathrm{x}}^2 + 3{\mathrm{y}}^2 \le 8\right\} \mathrm{is} \mathrm{a} \mathrm{relation} \mathrm{on} \mathrm{the} \mathrm{set} \mathrm{of} \mathrm{integers} \mathrm{Z}, \\ \mathrm{then} \mathrm{the} \mathrm{domain} \mathrm{R}{ }^{- 1} \mathrm{is} ; \end{gathered}$$

• $\left\{ - 1, 0, 1\right\}$

• $\left\{ - 2, - 1, 1, 2\right\}$

• $\left\{0, 1\right\}$

• $\left\{ - 2, - 1, 0, 1, 2\right\}$

If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0)is equal to

•  - 24

•  - 12

• 6

• 12

A line parallel to the straight line 2x – y = 0 is tangent to the hyperbola $\frac{{\mathrm{x}}^2}{4} - \frac{{\mathrm{y}}^2}{2} = 1$  at the point (x₁, y₁). Then ${{\mathrm{x}}_1}^2 + 5{{\mathrm{y}}_1}^2$ is equal to :

• 10

• 5

• 8

• 6

The contrapositive of the statement “If I reach the station in time, then I will catch the train” is:

• If I will catch the train, then I reach the station in time.

• If do not reach the station in time, then I will not catch the train

• If I do not reach the station in time,then I will catch the train.

• If I will not catch the train, then I do not reach the station in time.