If the integer $\mathrm{\lambda } \mathrm{and} \mathrm{\mu }$ are chosen at random between 1 and 100, then the probability that a number of the form $7^{\mathrm{\lambda }} + 7^{\mathrm{\mu }}$ is divisible by 5 is

• 1/4

• 1/7

• 1/8

• 1/49

If 2^x + 2^y =2^{x + y}, then the value of $\frac{\mathrm{dy}}{\mathrm{dx}}$ at x = y = 1 is

• 0

• - 1

• 1

• 2

The Newton Raphson method converges fast, if f'($\mathrm{\alpha }$) is

• small

• large

• 0

• None of these

If n is positive integer, then (1 + i)ⁿ + (1 - i)ⁿ is equal to

• ${\left(\sqrt{2}\right)}^{\mathrm{n} - 2}\cos \left(\frac{\mathrm{n\pi }}{4}\right)$

• ${\left(\sqrt{2}\right)}^{\mathrm{n} - 2}\sin \left(\frac{\mathrm{n\pi }}{4}\right)$

• ${\left(\sqrt{2}\right)}^{\mathrm{n} + 2}\cos \left(\frac{\mathrm{n\pi }}{4}\right)$

• ${\left(\sqrt{2}\right)}^{\mathrm{n} + 2}\sin \left(\frac{\mathrm{n\pi }}{4}\right)$

If $\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{x}\left(1 + \mathrm{acos}\left(\mathrm{x}\right)\right) - \mathrm{bsin}\left(\mathrm{x}\right)}{{\mathrm{x}}^3} = 1$, then

• $\mathrm{a} = - \frac{5}{2}, \mathrm{b} = - \frac{1}{2}$

• $\mathrm{a} = - \frac{3}{2}, \mathrm{b} = - \frac{1}{2}$

• $\mathrm{a} = - \frac{3}{2}, \mathrm{b} = - \frac{5}{2}$

• $\mathrm{a} = - \frac{5}{2}, \mathrm{b} = - \frac{3}{2}$

6.

If ${\log }_{\mathrm{x}}\left(\mathrm{a}\right), {\mathrm{a}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \mathrm{and} {\log }_{\mathrm{b}}\left(\mathrm{x}\right)$ are in GP, then x is equal to

• ${\log }_{\mathrm{b}}\left({\log }_{\mathrm{a}}\left(\mathrm{b}\right)\right)$

• $\log \left(\log \left(\mathrm{a}\right)\right)$

• $\log \left(\log \left(\mathrm{a}\right)\right) + \log \left(\log \left(\mathrm{b}\right)\right)$

• ${\log }_{\mathrm{a}}\left({\log }_{\mathrm{b}}\left(\mathrm{a}\right)\right)$

At the foot of a mountain the elevation of its summit is 45°. After ascending 2 km towards the mountain up an incline of 30°, the elevation changes to 60°. The height ofmountain will be

• $\sqrt{3} - 1$

• $1 - \sqrt{3}$

• $1 + \sqrt{3}$

• $\sqrt{3}$

The arithmetic calculations in a computer are performed by

• input unit

• ALU

• output unit

• memory

If $\frac{1 + 4\mathrm{p}}{4}, \frac{1 - \mathrm{p}}{4}, \frac{1 - 2\mathrm{p}}{2}$ are the probabilities of three mutually exclusive events, then

• $\frac{1}{3} \le \mathrm{p} \le \frac{1}{2}$

• $\frac{1}{3} \le \mathrm{p} \le \frac{2}{3}$

• $\frac{1}{6} \le \mathrm{p} \le \frac{1}{2}$

• None of these

If $\frac{\mathrm{a} + \mathrm{b}}{1 - \mathrm{ab}}, \frac{\mathrm{b} + \mathrm{c}}{1 - \mathrm{bc}}$ are in AP, then $\mathrm{a}, \frac{1}{\mathrm{b}}$ and c are in

• AP

• GP

• HP

• None of these