31.

$$\begin{gathered} \mathrm{If} \mathrm{\omega } \mathrm{is} \mathrm{a} \mathrm{complex} \mathrm{root} \mathrm{of} \mathrm{unity}, \mathrm{then} \mathrm{for} \mathrm{any} \\ \mathrm{n} > 1, \sum _{\mathrm{r} = 1}^{\mathrm{n} - 1}\mathrm{r}\left(\mathrm{r} + 1 - \mathrm{\omega }\right)\left(\mathrm{r} +\hspace{0.17em}1 - {\mathrm{\omega }}^2\right) = \end{gathered}$$

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

• $\frac{\mathrm{n}\left(\mathrm{n} +\hspace{0.17em}1\right)\left(2\mathrm{n} +\hspace{0.17em}1\right)}{6}$

• $\frac{\mathrm{n}\left(\mathrm{n} - 1\right)}{4}\left({\mathrm{n}}^2 + 3\mathrm{n} +\hspace{0.17em}4\right)$

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

Let N be the set of all natural numbers, Z be
the set of all integers and $\mathrm{\sigma } : \mathrm{N} \to \mathrm{Z}$ be defined by

$$\begin{gathered} \mathrm{\sigma }\left(\mathrm{n}\right) = \left\{\frac{\mathrm{n}}{2}, \mathrm{if} \mathrm{n} \mathrm{is} \mathrm{even} \\ - \frac{\mathrm{n} - 1}{2}, \mathrm{if} \mathrm{n} \mathrm{is} \mathrm{odd}\right\}. \mathrm{Then}, \end{gathered}$$

• $\sigma$ is onto but not one-one

• $\mathrm{\sigma } \mathrm{is} \mathrm{one}-\mathrm{one} \mathrm{but} \mathrm{not} \mathrm{onto}$

• $\mathrm{\sigma } \mathrm{is} \mathrm{neither} \mathrm{one}-\mathrm{one} \mathrm{nor} \mathrm{onto}$

• $\mathrm{\sigma } \mathrm{is} \mathrm{one}-\mathrm{one} \mathrm{and} \mathrm{onto}$

The variance of the following data is

xi	6	10	14	1	24	28	30
fi	2	4	7	12	8	4	3

• 33.4

• 34.3

• 43.4

• 44.3

$\mathrm{p}, {\mathrm{x}}_1, {\mathrm{x}}_2 , ..., {\mathrm{x}}_{\mathrm{n}} \mathrm{and} \mathrm{q}, {\mathrm{y}}_1, {\mathrm{y}}_2, ..., {\mathrm{y}}_{\mathrm{n}}$ are two arithmetic progressions with common differences a and b respectively. If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the arithmetic means of ${\mathrm{x}}_1, {\mathrm{x}}_2, ..., {\mathrm{x}}_{\mathrm{n}} \mathrm{and} {\mathrm{y}}_1, {\mathrm{y}}_2, ..., {\mathrm{y}}_{\mathrm{n}}$ respectively. Then the locus of P $\left(\mathrm{\alpha }, \mathrm{\beta }\right)$ is

• $\mathrm{a}\left(\mathrm{x} - \mathrm{p}\right) = \mathrm{b}\left(\mathrm{y} - \mathrm{q}\right)$

• $\mathrm{b}(\mathrm{x} - \mathrm{p}) = \mathrm{a}(\mathrm{y} - \mathrm{q})$

• $\mathrm{\alpha }\left(\mathrm{x} - \mathrm{p}\right) = \mathrm{\beta }\left(\mathrm{y} - \mathrm{q}\right)$

• $\mathrm{p}\left(\mathrm{x} - \mathrm{\alpha }\right) = \mathrm{q}\left(\mathrm{y} - \mathrm{\beta }\right)$

$$\begin{gathered} \mathrm{If} \mathrm{\alpha } \ne 0, \mathrm{and} \mathrm{the} \mathrm{mean} \mathrm{deviation} \mathrm{of} \mathrm{the} \mathrm{observations} \\ \left\{\mathrm{k\alpha }\right\}, \mathrm{for} \mathrm{k} = 1, 2, ..., 50 \mathrm{about} \mathrm{its} \mathrm{median} \mathrm{is} 50, \mathrm{then} \left|\mathrm{\alpha }\right| = ? \end{gathered}$$

• 4

• 3

• 2

• 5

If 2kx + 3y - 1 = 0, 2x + y + 5 = 0 are conjugate lines with respect to the circle x² + y² - 2x - 4y - 4 = 0, then k =

• 3

• 4

• 1

• 2

$\mathrm{If} \frac{5{\mathrm{x}}^2 + 2}{{\mathrm{x}}^3 + \mathrm{x}} = \frac{{\mathrm{A}}_1}{\mathrm{x}} + \frac{{\mathrm{A}}_2\mathrm{x} + {\mathrm{A}}_3}{{\mathrm{x}}^2 + 1}, \mathrm{then} \left({\mathrm{A}}_1, {\mathrm{A}}_2, {\mathrm{A}}_3\right) = ?$

• (0, 2, 3)

• $\left(3, 0, 2\right)$

• $\left(2, 3, 0\right)$

• $\left(2, 0, 3\right)$

$\mathrm{The} \mathrm{sum} \mathrm{of} \mathrm{the} \mathrm{n} \mathrm{terms} \mathrm{of} \frac{1}{2 . 5} + \frac{1}{5 . 8} +\hspace{0.17em}\frac{1}{8 . 11} + ... \mathrm{is}$

• $\frac{3\mathrm{n}}{2\left(3\mathrm{n} + 2\right)}$

• $\frac{3\mathrm{n}}{3\mathrm{n} + 2}$

• $\frac{\mathrm{n}}{2\left(3\mathrm{n} + 2\right)}$

• $\frac{\mathrm{n}}{3\mathrm{n} + 2}$

The equations of the parabola whose axis is parallel to the X-axis and which passes through the points (- 2, 1), (1, 2)(- 1, 3) is

• $18{\mathrm{y}}^2 - 12\mathrm{x} - 21\mathrm{y} - 21 = 0$

• $5{\mathrm{y}}^2 + 2\mathrm{x} - 21\mathrm{y} + 20 = 0$

• $15{\mathrm{y}}^2 + 12\mathrm{x} - 11\mathrm{y} + 20 = 0$

• $25{\mathrm{y}}^2 - 2\mathrm{x} - 65\mathrm{y} + 36 = 0$

$$\begin{gathered} \mathrm{If} \mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} {\mathrm{x}}^3 + {\mathrm{px}}^2 + \mathrm{qx} +\hspace{0.17em}\mathrm{r} = 0 \\ \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \left(1 + {\mathrm{\alpha }}^2\right)\left(1 + {\mathrm{\beta }}^2\right)\left(1 + {\mathrm{\gamma }}^2\right) \mathrm{is} \end{gathered}$$

• ${\left(\mathrm{r} - \mathrm{p}\right)}^2 + {\left(\mathrm{r} - \mathrm{q}\right)}^2$

• ${\left(1 + \mathrm{p}\right)}^2 + {\left(1 + \mathrm{q}\right)}^2$

• ${\left(\mathrm{r}\hspace{0.17em}+ \mathrm{p}\right)}^2 + {\left(\mathrm{q} + 1\right)}^2$

• ${\left(\mathrm{r} - \mathrm{p}\right)}^2 + {\left(\mathrm{q} - 1\right)}^2$