A die has 6 faces marked by the given numbers as shown below:

The die is thrown once. What is the probability of getting
(i) a positive integer.
(ii) an integer greater than - 3.
(iii) the smallest integer.

Find x, y if $\left[\begin{array}{cc}-2& 0\\ 3& 1\end{array}\right]\left[\begin{array}{c}- 1\\ 2\mathrm{x}\end{array}\right] + 3\left[\begin{array}{c}- 2\\ 1\end{array}\right] = 2\left[\begin{array}{c}\mathrm{y}\\ 3\end{array}\right]$.

Shahrukh opened a ‘Recurring Deposit’ account in a bank and deposited Rs. 800 per month for 1½ years. If he received Rs. 15,084 at the time of maturity, find the rate of interest per annum.

Without using trigonometric tables, evaluate

${\sin }^2\left({34}^{°}\right) + {\sin }^2\left({56}^{°}\right) + 2\tan \left({18}^{°}\right)\tan \left({72}^{°}\right) - {\cot }^2\left({30}^{°}\right)$

Using the Remainder and Factor Theorem, factorise the following polynomial:
x³ + 10x² - 37x + 26

The numbers 6, 8, 10, 12, 13 and x are arranged in an ascending order.
If the mean of the observations is equal to the median, find the value of x.

In the figure, m∠DBC = 58°. BD is the diameter of the circle. Calculate:

(i) m∠BDC

(ii) m∠BEC

(iii) m∠BAC

8.

Use graph paper to answer the following questions. (Take 2 cm = 1 unit on both axes)
(i) Plot the points A( - 4, 2) and B(2, 4)
(ii) A' is the image of A when reflected at the y-axis. Plot it on the graph paper and write
the co-ordinates of A'.
(iii) B' is the image of B when reflected on the line AA'. Write the co-ordinates of B'.
(iv) Write the geometric name of the figure ABA'B'.
(v) Name a line of symmetry of the figure formed.

A shopkeeper bought a washing machine at a discount of 20% from a wholesaler, the printed price of the washing machine being Rs. 18,000. The shopkeeper sells it to a consumer at a discount of 10% on the printed price. If the rate of sales tax is 8%, find:
(i) the VAT paid by the shopkeeper.
(ii) the total amount that the consumer pays for the washing machine.

If $\frac{{\mathrm{x}}^2 + {\mathrm{y}}^2}{{\mathrm{x}}^2 - {\mathrm{y}}^2} = \frac{17}{8}$ then find the value of :

(i) x:y

(ii) $\frac{{\mathrm{x}}^3 + {\mathrm{y}}^3}{{\mathrm{x}}^3 - {\mathrm{y}}^3}$