11.

${\mathrm{sech}}^{ - 1}\left(\frac{1}{\sqrt{2}}\right) + {\mathrm{csch}}^{ - 1}\left( - 1\right) = ?$

• 0

• $\sqrt{2} + 1$

• $\sqrt{2}$

• $\sqrt{2} - 1$

The lines y = 2x + $\sqrt{76}$ and 2y + x = 8 touch the ellipse x² + y² = 1. If the point of $\frac{{\mathrm{x}}^2}{16} + \frac{{\mathrm{y}}^2}{12} = 1$ intersection of these two lines lie on a circle, whose centre coincides with the centre of that ellipse, then the equation of that circle is

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 28$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 12$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 12$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = {\left(4 + \sqrt{8}\right)}^2$

The equation of the pair of lines through the point (2, 1) and perpendicular to the pair of lines 4xy + 2x + 6y + 3 = 0 is

• xy - x - 2y + 2 = 0

• xy + x - 2y - 2 = 0

• xy + x + 2y - 6 = 0

• xy - x + 2y - 2 = 0

The harmonic mean of two numbers is $- \frac{8}{5}$ and their geometric mean is 2. The quadratic equation whose roots are twice those numbers is

• ${\mathrm{x}}^2 + 5\mathrm{x} + 4 = 0$

• ${\mathrm{x}}^2 + 10\mathrm{x} + 16 = 0$

• ${\mathrm{x}}^2 - 10\mathrm{x} + 16 = 0$

• ${\mathrm{x}}^2 - 5\mathrm{x} + 4 = 0$

$\mathrm{If} \mathrm{z} \mathrm{is} \mathrm{a} \mathrm{complex} \mathrm{number} \mathrm{with} \left|\mathrm{z}\right| \ge 5. \mathrm{Then} \mathrm{the} \mathrm{least} \mathrm{value} \mathrm{of} \left|\mathrm{z} + \frac{2}{\mathrm{z}}\right| \mathrm{is}$

• $\frac{24}{5}$

• $\frac{26}{5}$

• $\frac{23}{5}$

• $\frac{29}{5}$

For the function f(x) = (x - 1)(x - 2) defined on [0, ½] the value of c satisfying Lagrange's mean value theorem is

• $\frac{1}{5}$

• $\frac{1}{3}$

• $\frac{1}{7}$

• $\frac{1}{4}$

If the roots of the equation x² -7x² + 14x - 8 = 0 are in geometric progression, then the difference between the largest and the smallest roots is

• 4

• 2

• $\frac{1}{2}$

• 3

If the mean and variance of a binomial variate X are $\frac{4}{3}, \frac{8}{9}$respectively, then P(X = 2) =

• $\frac{4}{27}$

• $\frac{16}{81}$

• $\frac{8}{27}$

• $\frac{8}{81}$

$\mathrm{If} \mathrm{\alpha } \mathrm{is} \mathrm{a} \mathrm{non}-\mathrm{real} \mathrm{root} \mathrm{of} {\mathrm{x}}^7 = 1, \mathrm{then} \mathrm{\alpha }(1 + \mathrm{\alpha }) (1 + {\mathrm{\alpha }}^2 + {\mathrm{\alpha }}^4) =$

• 1

• 2

• - 1

• - 2

$$\begin{gathered} \mathrm{In} ∆\mathrm{ABC}, \mathrm{L}, \mathrm{M}, \mathrm{N} \mathrm{are} \mathrm{points} \mathrm{on} \mathrm{BC}, \mathrm{CA}, \mathrm{AB} \\ \mathrm{respectivley}, \mathrm{dividing} \mathrm{them} \mathrm{in} \mathrm{the} \mathrm{ratio} 1 : 2, \\ 2 : 3, 3: 5. \mathrm{If} \mathrm{the} \mathrm{point} \mathrm{K} \mathrm{divides} \mathrm{AB} \mathrm{in} \mathrm{the} \mathrm{ratio} 5: 3, \mathrm{then}\frac{\left|\overline{\mathrm{AL}} + \overline{\mathrm{BM}} + \overline{\mathrm{CN}}\right|}{\left|\overline{\mathrm{CK}}\right|} = ? \end{gathered}$$

• $\frac{5}{8}$

• $\frac{2}{5}$

• $\frac{3}{5}$

• $\frac{1}{15}$