Given $\mathrm{A} = \left[\begin{array}{cc}2& - 6\\ 2& 0\end{array}\right], \mathrm{B} = \left[\begin{array}{cc}- 3& 2\\ 4& 0\end{array}\right], \mathrm{C} = \left[\begin{array}{cc}4& 0\\ 0& 2\end{array}\right]$

Find the matrix X such that A + 2X = 2B + C.

At what rate % p.a. will a sum of Rs. 4000 yield Rs. 1324 as compound interest in 3 years?

The median of the following observation 11, 12, 14, (x - 2), (x + 4), (x + 9), 32, 38, 47 arranged in ascending order is 24.

Find the value of x and hence find the mean.

What number must be added to each of the numbers 6, 15, 20 and 43 to make them proportional?

If (x – 2) is a factor of the expression 2x³ + ax² + bx – 14 and when the expression is divided by (x – 3), it leaves a remainder 52, find the values of a and b.

Draw a histogram from the following frequency distribution and find the mode from the graph:

Without using tables evaluate:
$3\cos \left(80°\right).\csc \left(10°\right) + 2\sin \left(59°\right).\sec \left(31°\right)$

In the given figure, $∆$BAD = 65$°$, $∆$ABD = 70$°$, $∆$BDC = 45$°$

(i) Prove that AC is a diameter of the circle.

(ii) Find $\angle$ACB

AB is a diameter of a circle with centre C = (– 2, 5). If A = (3, – 7). Find

(i) the length of radius AC

(ii) the coordinates of B.

Solve the following equation and calculate the answer correct to two decimal places:

x² – 5x – 10 = 0.