What is $\frac{1}{{\log }_2\left(\mathrm{N}\right)} + \frac{2}{{\log }_3\left(\mathrm{N}\right)} + \frac{1}{{\log }_4\left(\mathrm{N}\right)} + ... + \frac{1}{{\log }_{100}\left(\mathrm{N}\right)} = \left(\mathrm{N} \ne 1\right)?$

• $\frac{1}{{\log }_{100!}\left(\mathrm{N}\right)}$

• $\frac{1}{{\log }_{99!}\left(\mathrm{N}\right)}$

• $\frac{99}{{\log }_{100!}\left(\mathrm{N}\right)}$

• $\frac{99}{{\log }_{99!}\left(\mathrm{N}\right)}$

What is the greatest integer among the following by which the number 5⁵ + 7⁵ is divisible ?

• 6

• 8

• 11

• 12

$$\begin{gathered} \mathrm{What} \mathrm{is} \mathrm{the} \mathrm{number} \mathrm{of} \mathrm{non}-\mathrm{zero} \mathrm{terms} \mathrm{in} \mathrm{the} \mathrm{expansion} \mathrm{of} \\ (1 + 2\sqrt{3}{\mathrm{\}}}^{11} + (1 - 2\sqrt{3}{)}^{11} (\mathrm{after} \mathrm{simplification}) ? \end{gathered}$$

• 4

• 5

• 6

• 11

$$\begin{gathered} \mathrm{If} \mathrm{x} = 1 - \mathrm{y} + {\mathrm{y}}^2 - {\mathrm{y}}^3 + ... \mathrm{up} \mathrm{to} \mathrm{infinite} \mathrm{terms}, \mathrm{where} \mathrm{IyI} < 1, \\ \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct}? \end{gathered}$$

• $\mathrm{x} = \frac{1}{1 + \mathrm{y}}$

• $\mathrm{x} = \frac{1}{1 - \mathrm{y}}$

• $\mathrm{x} = \frac{\mathrm{y}}{1 + \mathrm{y}}$

• $\mathrm{x} = \frac{\mathrm{y}}{1 - \mathrm{y}}$

What is the sum of all two-digit numbers which when divided by 3 leave 2 as the remainder ?

• 1565

• 1585

• 1635

• 1655

If 0 < a < 1, the value of log₁₀(a) is negative. This is justified by:

• Negative power of 10 is less than 1

• Negative power of 10 is between 0 and 1

• Negative power of 10 is positive

• Negative power of 10 is negative

The third term of a GP is 3. What is the product of the first five terms ?

• 216

• 226

• 243

• Cannot be determined due to insufficient data

$\mathrm{fx}, 3/2, \mathrm{z} \mathrm{If} \mathrm{x}, \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right., \mathrm{z} \mathrm{are} \mathrm{in} \mathrm{AP}; \mathrm{x}, 3, \mathrm{z} \mathrm{are} \mathrm{in} \mathrm{GP}; \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{will} \mathrm{be} \mathrm{in} \mathrm{HP} ?$

• x, 6, z

• x, 4, z

• x, 2, z

• x, 1, z

$\mathrm{What} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{the} \mathrm{sum} ({\mathrm{i}}^{\mathrm{n}} + {\mathrm{i}}^{\mathrm{n} + 1}), \mathrm{where}$ $\mathrm{i} = \sqrt{ - 1} ?$

• i

• 2i

•  - 2i

• 1 + i

20.

If x, x - y and x + y are the angles of a triangle (not an equilateral triangle) such that tan(x - y), tan x and tan (x + y) are in GP, then what is x equal to ?

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$