$\mathrm{If} \mathrm{n} \in \mathrm{N}, \mathrm{then} {121}^{\mathrm{n}} - {25}^{\mathrm{n}} + {1900}^{\mathrm{n}} - {\left( - 4\right)}^{\mathrm{n}} \mathrm{is} \mathrm{divisible} \mathrm{by} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} ?$

• 1904

• 2000

• 2002

• 2006

$\mathrm{If} \mathrm{n} = \left(2007\right)!, \mathrm{then} \mathrm{what} \mathrm{is} \frac{1}{{\log }_2\left(\mathrm{n}\right)} + \frac{1}{{\log }_3\left(\mathrm{n}\right)} + \frac{1}{{\log }_4\left(\mathrm{n}\right)} + ... + \frac{1}{{\log }_{2017}\left(\mathrm{n}\right)} = ?$

• 0

• 1

• $\frac{\mathrm{n}}{2}$

• n

In the expansion of (1 + x)⁴³, if the coefficients of (2r + 1)^th and (r + 2)^th terms are equal, then what is the value of r (r $\ne$ 1) ?

• 5

• 14

• 21

• 22

4.

What is the principal argument of (- 1 - i), where i = $\sqrt{ - 1} ?$

• $\frac{\mathrm{\pi }}{4}$

• $- \frac{\mathrm{\pi }}{4}$

• $- \frac{3\mathrm{\pi }}{4}$

• $\frac{3\mathrm{\pi }}{4}$

Let $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ be real numbers and z be a complex number. If ${\mathrm{z}}^2 + \mathrm{\alpha z} + \mathrm{\beta } = 0$ has two distinct non-real roots with Re(z) = 1, then it is necessary that :

• $\mathrm{\beta } \in \left( - 1, 0\right)$

• $\left|\mathrm{\beta }\right| = 1$

• $\mathrm{\beta } \in \left(1, \infty \right)$

• $\mathrm{\beta } \in \left(0, 1\right)$

$$\begin{gathered} \mathrm{Let} \mathrm{A} \mathrm{and} \mathrm{B} \mathrm{be} \mathrm{subsets} \mathrm{of} \mathrm{X} \mathrm{and} \mathrm{C} = (\mathrm{A} \cap \mathrm{B}\text{'}) \cup (\mathrm{A}\text{'} \cap \mathrm{B}), \mathrm{where} \mathrm{A}\text{'} \mathrm{and} \mathrm{B}\text{'} \mathrm{are} \\ \mathrm{complements} \mathrm{of} \mathrm{A} \mathrm{and} \mathrm{B} \mathrm{respectively} \mathrm{in} \mathrm{X}. \mathrm{What} \mathrm{is} \mathrm{C} \mathrm{equal} \mathrm{to}? \end{gathered}$$

• $(A \cup B\text{'}) - (A \cap B\text{'})$

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

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

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

How many numbers between 100 and 1000 can beformed with the digits 5, 6, 7, 8, 9, if the repetition of digits is not allowed ?

• 3⁵

• 5³

• 120

• 60

The number of non-zero integral solutions of the equation   ${\left|1 - 2\mathrm{i}\right|}^{\mathrm{x}} = 5^{\mathrm{x}}$ is :

• Zero (No solution)

• One

• Two

• Three

If the ratio of AM to GM of two positive numbers a and b is 5 : 3, then a : b is equal to :

• 3 : 5

• 2 : 9

• 9 : 1

• 5 : 3

If the coefficients of am and aⁿ in the expansion of (1+ a)^{m + n} are $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$, then which one of the following is correct ?

• $\mathrm{\alpha } = 2\mathrm{\beta }$

• $\mathrm{\alpha } = \mathrm{\beta }$

• $2\mathrm{\alpha } = \mathrm{\beta }$

• $\mathrm{\alpha } = \left(\mathrm{m} + \mathrm{n}\right)\mathrm{\beta }$