Using remainder theorem, find the value of k if on dividing $2 {\mathrm{x}}^3 + 3 {\mathrm{x}}^2 - \mathrm{k} \mathrm{x} + 5 \mathrm{by} \mathrm{x} - 2$, leaves a remainder 7.

If ( 3 a + 2 b ) : ( 5 a + 3 b ) = 18 : 29.  Find  a : b.

Given  $\mathrm{A} = \left[ \begin{array}{cc} 2& 0\\ - 1& 7\end{array} \right] \mathrm{and} \mathrm{I} = \left[ \begin{array}{cc}1& 0\\ 0& 1\end{array} \right]$  and  A² = 9 A + mI. find m.

The mean of following numbers is 68. Find the value of  ‘x’.  45,  52,  60,  x,  69,  70,  26,  81  and  94  Hence estimate the median. 

The slope of a line joining  P ( 6, k )  and  Q (1- 3k, 3 ) is $\frac{1}{2}$. Find

(i) k

(ii) Midpoint of  PQ, using the value of  ‘k’  found in  (i).

Without using trigonometrical tables, evaluate:

${\mathrm{cosec}}^2 57° - {\tan }^2 33° + \cos 44° \mathrm{cosec} 46° - \sqrt{2} \cos 45° - {\tan }^2 60°$

A certain number of metallic cones, each of radius  2 cm  and height  3 cm  are melted and recast into a solid sphere of radius  6 cm. Find the number of cones.

Solve the following inequation, write the solution set and represent it on the number line.

$- 3 ( \mathrm{x} - 7 ) \ge 15 - 7 \mathrm{x} > \frac{\mathrm{x} + 1}{3}, \mathrm{x} \notin \mathrm{R}$

In the figure given below,  AD  is a diameter.  O  is the centre of the circle.
AD  is parallel to  BC  and  $\angle$CBD = $30°$. Find:

(i)   $\angle$OBD 

(ii)  $\angle$AOB

(iii)  $\angle$BED

A game of numbers has cards marked with 11, 12, 13, … 40. A card is drawn at random. Find the probability that the number on the card drawn is :

(i) A perfect square

(ii) Divisible by 7