What is the value of ${\log }_7\left(\sqrt{7\sqrt{7\sqrt{7}}}\right) = ?$

• $3{\log }_2\left(7\right)$

• $1 - 3{\log }_2\left(7\right)$

• $1 - 3{\log }_7\left(2\right)$

• $\frac{7}{8}$

If an infinite GP has the first term x and the sum 5, then which of the following is correct ?

• $\mathrm{x} < - 10$

• $- 10 < \mathrm{x} < 0$

• $0 < \mathrm{x} < 10$

• $\mathrm{x} > 10$

Consider the following expressions :

$$\begin{gathered} 1. \mathrm{x} + {\mathrm{x}}^2 - \frac{1}{\mathrm{x}} \\ 2. \sqrt{{\mathrm{ax}}^2 + \mathrm{bx} + \mathrm{x} - \mathrm{c} + \frac{\mathrm{d}}{\mathrm{x}} - \frac{\mathrm{e}}{{\mathrm{x}}^2}} \\ 3. 3{\mathrm{x}}^2 - 5\mathrm{x} +\hspace{0.17em}\mathrm{ab} \\ 4. \frac{2}{{\mathrm{x}}^2 - \mathrm{ax} + {\mathrm{b}}^3} \\ 5. \frac{1}{\mathrm{x}} - \frac{2}{\mathrm{x}\hspace{0.17em}+ 5} \\ \mathrm{Which} \mathrm{of} \mathrm{the} \mathrm{above} \mathrm{are} \mathrm{rational} \mathrm{expression} ? \end{gathered}$$

• 1, 4 and 5 only

• $1, 3, 4 \mathrm{and} 5 \mathrm{only}$

• 2, 4 and 5 only

• 1 and 2 only

4.

A square matrix A is called orthogonal if

• A = A²

• $\mathrm{A}\text{'} = {\mathrm{A}}^{ - 1}$

• $\mathrm{A} = {\mathrm{A}}^{ - 1}$

• $\mathrm{A} = \mathrm{A}\text{'}$

If A, B and C are subsets of a Universal set, then which one of the following is not correct ?

• $\mathrm{A} \cup \left(\mathrm{B} \cap \mathrm{C}\right) = \left(\mathrm{A} \cup \mathrm{B}\right) \cap \left(\mathrm{A} \cup \mathrm{C}\right)$

• $\mathrm{A}\text{'} \cup \left(\mathrm{A} \cup \mathrm{B}\right) = \left(\mathrm{B}\text{'} \cap \mathrm{A}\right)\text{'} \cup \mathrm{A}$

• $\mathrm{A}\text{'} \cup \left(B \cup C\right) = \left(C\text{'} \cap B\right)\text{'} \cup \mathrm{A}\text{'}$

• $\left(\mathrm{A} \cap \mathrm{B}\right) \cup \mathrm{C} = \left(\mathrm{A} \cup \mathrm{C}\right) \cap \left(\mathrm{B} \cup \mathrm{C}\right)$

Let x be the number of integers lying between 2999 and 8001 which have at least two digits equal. Then x is equal to

• 2480

• 2481

• 2482

• 2483

The sum of the series $3 - 1 + \frac{1}{3} - \frac{1}{9} + ... =$

• $\frac{20}{9}$

• $\frac{9}{20}$

• $\frac{9}{4}$

• $\frac{4}{9}$

A survey was conducted among 300 students. It was found that 125 students like to play cricket, 145 students like toplay football and 90 students like to play tennis. 32 students like to play exactly two games out of the three games.

How many students like to play all the three games ?

• 14

• 21

• 28

• 35

A survey was conducted among 300 students. It was found that 125 students like to play cricket, 145 students like toplay football and 90 students like to play tennis. 32 students like to play exactly two games out of the three games. How many students like to play exactly only one game?

How many students like to play all the three games ?

• 196

• 228

• 254

• 268

$$\begin{gathered} \mathrm{If} \mathrm{\alpha } \mathrm{and} \mathrm{\beta }( \ne 0) \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{quadratic} \mathrm{equation} {\mathrm{x}}^2 + \mathrm{\alpha x} - \mathrm{B} = 0, \\ \mathrm{then} \mathrm{the} \mathrm{quadratic} \mathrm{expression} -{\mathrm{x}}^2 + \mathrm{\alpha x} + \mathrm{\beta } \mathrm{where} \mathrm{x} \in \mathrm{R} \mathrm{has} \end{gathered}$$

• $\mathrm{Least} \mathrm{value} - \frac{1}{4}$

• $\mathrm{Least} \mathrm{value} - \frac{9}{4}$

• $\mathrm{Greatest} \mathrm{value} \frac{1}{4}$

• $\mathrm{Greatest} \mathrm{value} \frac{9}{4}$