The line y = x + $\mathrm{\lambda }$ is tangent to the ellipse 2x² + 3y² = 1. Then, $\mathrm{\lambda }$ is

• - 2

• 1

• $\sqrt{\frac{5}{6}}$

• $\sqrt{\frac{2}{3}}$

Let S be the set of points, whose abscissae and ordinates are natural numbers. Let p e S, such that the sum of the distance of P from (8, 0) and (0, 12) is minimum among all elemants in S. Then, the number of such points P in S is

• 1

• 3

• 5

• 11

The cosine of the angle between any two diagonals of a cube is

• $\frac{1}{3}$

• $\frac{1}{2}$

• $\frac{2}{3}$

• $\frac{1}{\sqrt{3}}$

If x is a positive real number different from 1 such that ${\log }_{\mathrm{a}}\left(\mathrm{x}\right), {\log }_{\mathrm{b}}\left(\mathrm{x}\right), {\log }_{\mathrm{c}}\left(\mathrm{x}\right)$ are in AP, then

• $\mathrm{b} = \frac{\mathrm{a} + \mathrm{c}}{2}$

• $\sqrt{\mathrm{ac}}$

• ${\mathrm{c}}^2 = {\left(\mathrm{ac}\right)}^{{\log }_{\mathrm{a}}\left(\mathrm{b}\right)}$

• None of these

If a, x are real numbers and $\left|\mathrm{a}\right|$ < 1, $\left|\mathrm{x}\right|$ < 1, then 1 + (1+ a) x + (1+ a + a²)x² + ...$\infty$ is equal to

• $\frac{1}{\left(1 - \mathrm{a}\right)\left(1 - \mathrm{ax}\right)}$

• $\frac{1}{\left(1 - \mathrm{a}\right)\left(1 - \mathrm{x}\right)}$

• $\frac{1}{\left(1 - \mathrm{x}\right)\left(1 - \mathrm{ax}\right)}$

• $\frac{1}{\left(1 + \mathrm{ax}\right)\left(1 - \mathrm{a}\right)}$

If ${\log }_{0.3}\left(\mathrm{x} - 1\right) < {\log }_{0.09}\left(\mathrm{x} - 1\right)$, then x lies in the interval

• $\left(2, \infty \right)$

• (1, 2)

• (- 2, - 1)

• None of these

The value of $\sum _{\mathrm{n} = 1}^{13}\left({\mathrm{i}}^{\mathrm{n}} + {\mathrm{i}}^{\mathrm{n} + 1}\right), \mathrm{i} = \sqrt{- 1}$ is

• i

• i - 1

• 1

• 0

8.

If ${\mathrm{z}}_{1 }, {\mathrm{z}}_2, {\mathrm{z}}_3$ are imaginary numbers such that $\left|{\mathrm{z}}_1\right| = \left|{\mathrm{z}}_2\right| = \left|{\mathrm{z}}_3\right| = \left|\frac{1}{{\mathrm{z}}_1} + \frac{1}{{\mathrm{z}}_2} + \frac{1}{{\mathrm{z}}_3}\right|$ = 1, then $\left|z_1 + z_2 + z_3\right|$ is

• equal to 1

• less than

• greater than 1

• equal to 3

If p.q are the · roots of the equation x² + px + q =0, then

• p = 1, q = - 2

• p = 0, q = 1

• p = - 2, q = 0

• p = - 2, q = 1

The number of ways in which the letters of the word ARRANGE can be permuted such that the R's occur together, is

• $\frac{7!}{2! 2!}$

• $\frac{7!}{2!}$

• $\frac{6!}{2!}$

• $5! \times 2!$