31.

The value of  is

• $\frac{2}{3}\left(\sqrt{2} - 1\right)$

• $\frac{2}{3}\left(\sqrt{2} + 1\right)$

• $\frac{2}{3}\left(2\sqrt{2} + 1\right)$

The sum of n terms of the following series 1³ + 3³ + 5³ + 7³ + ... is

• n²(2n² - 1)

• n³(n - 1)

• n³ + 8n + 4

• 2n⁴ + 3n²

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are roots of ax + bx + c = 0, then the equation whose roots are ${\mathrm{\alpha }}^2 \mathrm{and} {\mathrm{\beta }}^2$ is

• a²x² - (b² - 2ac)x + c² = 0

• a²x² + (b² - ac)x + c² = 0

• a²x² + (b² + ac)x + c² = 0

• a²x² + (b² + 2ac)x + c² = 0

If w is an imginary cube root of unity, then the value of (2 - w)(2 - w²) + 2(3 - w)(3 - w²) + ... + (n - 1)(n - w)(n - w²) is

• $\frac{{\mathrm{n}}^2}{4}{\left(\mathrm{n} +\hspace{0.17em}1\right)}^2 - \mathrm{n}$

• $\frac{{\mathrm{n}}^2}{4}{\left(\mathrm{n} + 1\right)}^2 + \mathrm{n}$

• $\frac{{\mathrm{n}}^2}{4}{\left(\mathrm{n} + 1\right)}^2$

• $\frac{{\mathrm{n}}^2}{4}\left(\mathrm{n} + 1\right) - \mathrm{n}$

If ${}^{\mathrm{n}}\mathrm{C}_{\mathrm{r} - 1} = 36, {}^{\mathrm{n}}\mathrm{C}_{\mathrm{r}} = 84 \mathrm{and} {}^{\mathrm{n}}\mathrm{C}_{\mathrm{r} + 1} = 126, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} {}^{\mathrm{n}}\mathrm{C}_8 \mathrm{is}$

• 10

• 7

• 9

• 8

In a group of 14 males and 6 females. 8 and 3 of the males and females, respectively are aged above 40 yr. The probability that a person selected at random from the group is aged above 40 yr given that the selected person is a female, is

• $\frac{2}{7}$

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{5}{6}$

The equation x³ - yx² + x - y = 0 represents

• a hyperbola and two straight lines

• a straight line

• a parabola and two straight lines

• a straight line and a circle

The locus of the mid-points of chords of the circle x² + y² = 1, which subtends a right angle at the origin, is

• x² + y² = $\frac{1}{4}$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = \frac{1}{2}$

• xy = 0

• x² - y² = 0

• x = - a

• x = a

• x = 0

• x = - $\frac{\mathrm{a}}{2}$

The points of the ellipse 16x² + 9y² = 400 at which the ordinate decreases at the same rate at which the abscissa increases is/are given by

• $\left(3, \frac{16}{3}\right) \mathrm{and} \left(- 3, - \frac{16}{3}\right)$

• $\left(3, - \frac{16}{3}\right) \mathrm{and} \left(- 3, \frac{16}{3}\right)$

• $\left(\frac{1}{16}, \frac{1}{9}\right) \mathrm{and} \left(- \frac{1}{16}, - \frac{1}{9}\right)$

• $\left(\frac{1}{16}, - \frac{1}{9}\right) \mathrm{and} \left(- \frac{1}{16}, \frac{1}{9}\right)$