If $\left(\mathrm{\alpha } + \sqrt{\mathrm{\beta }}\right) \mathrm{and} \left(\mathrm{\alpha } - \sqrt{\mathrm{\beta }}\right)$ are the roots of the equation x² + px + q = 0, where $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{p} \mathrm{and} \mathrm{q}$ are real, then the roots of the equation (p² - 4q)(p²x² + 4px) - 16q = 0 are

• $\left(\frac{1}{\mathrm{\alpha }} + \frac{1}{\sqrt{\mathrm{\beta }}}\right) \mathrm{and} \left(\frac{1}{\mathrm{\alpha }} - \frac{1}{\sqrt{\mathrm{\beta }}}\right)$

• $\left(\frac{1}{\sqrt{\mathrm{\alpha }}} + \frac{1}{\mathrm{\beta }}\right) \mathrm{and} \left(\frac{1}{\sqrt{\mathrm{\alpha }}} - \frac{1}{\mathrm{\beta }}\right)$

• $\left(\frac{1}{\sqrt{\mathrm{\alpha }}} + \frac{1}{\sqrt{\mathrm{\beta }}}\right) \mathrm{and} \left(\frac{1}{\sqrt{\mathrm{\alpha }}} - \frac{1}{\sqrt{\mathrm{\beta }}}\right)$

• $\left(\sqrt{\mathrm{\alpha }} + \sqrt{\mathrm{\beta }}\right) \mathrm{and} \left(\sqrt{\mathrm{\alpha }} - \sqrt{\mathrm{\beta }}\right)$

The number of solutions of the equation ${\log }_2\left({\mathrm{x}}^2 + 2\mathrm{x} - 1\right) = 1$ is

• 0

• 1

• 2

• 3

The sum of the series $1 + \frac{1}{2}{}^{\mathrm{n}}\mathrm{C}_1 + \frac{1}{3}{}^{\mathrm{n}}\mathrm{C}_2 + ... + \frac{1}{\mathrm{n} + 1}{}^{\mathrm{n}}\mathrm{C}_{\mathrm{n}}$ is equal to

• $\frac{2^{\mathrm{n} + 1} - 1}{\mathrm{n} + 1}$

• $\frac{3\left(2^{\mathrm{n}} - 1\right)}{2\mathrm{n}}$

• $\frac{2^{\mathrm{n}} + 1}{\mathrm{n} + 1}$

• $\frac{2^{\mathrm{n}} + 1}{2\mathrm{n}}$

The value of $\sum _{\mathrm{r} = 2}^{\infty }\frac{1 + 2 + ... + \left(\mathrm{r} - 1\right)}{\mathrm{r}!}$

• e

• 2e

• $\frac{\mathrm{e}}{2}$

• $\frac{3\mathrm{e}}{2}$

5.

The remainder obtained when 1! + 2! + ... + 95! is divided by 15 is

• 14

• 3

• 1

• 0

Let ${\left(1 + \mathrm{x}\right)}^{10} = \sum _{\mathrm{r} = 0}^{10}{\mathrm{c}}_{\mathrm{r}}{\mathrm{x}}^{\mathrm{T}} \mathrm{and} \sum _{\mathrm{r} = 0}^7{\left(1 + \mathrm{x}\right)}^7 = {\mathrm{d}}_{\mathrm{r}}{\mathrm{x}}^{\mathrm{T}}$. If $\mathrm{P} = \sum _{\mathrm{r} = 0}^5{\mathrm{c}}_{\mathrm{r}}{\mathrm{x}}^{\mathrm{T}} \mathrm{and} \mathrm{Q} = \sum _{\mathrm{r} = 0}^3{\mathrm{d}}_{2\mathrm{r} + 1 }, \mathrm{then} \frac{\mathrm{P}}{\mathrm{Q}}$ is equal to

• 4

• 8

• 16

• 32

Two decks of playing cards are well shuffled and 26 cards are randomly distributed to a player. Then, the probability that the player gets all distinct cards is

• ${}^{52}\mathrm{C}_{\mathrm{r}}/{}^{104}\mathrm{C}_{26}$

• $2 \times {}^{52}\mathrm{C}_{\mathrm{r}}/{}^{104}\mathrm{C}_{26}$

• $2^{13} \times {}^{52}\mathrm{C}_{\mathrm{r}}/{}^{104}\mathrm{C}_{26}$

• $2^{26} \times {}^{52}\mathrm{C}_{\mathrm{r}}/{}^{104}\mathrm{C}_{26}$

An um contains 8 red and 5 white balls. Three balls are drawn at random. Then, the probability that balls of both colours are drawn is

• $\frac{40}{143}$

• $\frac{70}{143}$

• $\frac{3}{13}$

• $\frac{10}{13}$

Let R be the set of real numbers and the functions f : R ➔ R and g : R ➔ R be defined by f(x) = x² + 2x - 3 and g(x) = x + 1. Then, the value of x for which f(g(x)) = g(f(x)) is

• - 1

• 0

• 1

• 2

If a, b and c are in arithmetic progression, then the roots of the equation ax - 2bx + c = 0 are

• $1 \mathrm{and} \frac{\mathrm{c}}{\mathrm{a}}$

• $- \frac{1}{\mathrm{a}} \mathrm{and} - \mathrm{c}$

• $- 1 \mathrm{and} - \frac{\mathrm{c}}{\mathrm{a}}$

• $- 2 \mathrm{and} - \frac{\mathrm{c}}{2\mathrm{a}}$