If b is the mean proportion between a and c, show that

$\frac{{\mathrm{a}}^4 + {\mathrm{a}}^2 {\mathrm{b}}^2 + {\mathrm{b}}^4}{{\mathrm{b}}^4 + {\mathrm{b}}^2 {\mathrm{c}}^2 + {\mathrm{c}}^4} = \frac{{\mathrm{a}}^2}{{\mathrm{c}}^2}$

Solve the equation 4x² – 5x – 3 = 0 and give your answer correct to two decimal places.

AB and CD are two parallel chords of a circle such that AB = 24 cm and CD = 10 cm. If the radius of the circle is 13 cm. find the distance between the two chords.

Evaluate without using trigonometric tables,

sin² 28⁰  +  sin² 62⁰  +  tan² 38⁰  -  cot² 52⁰  +  $\frac{1}{4}$ sec² 30⁰.

If $\mathrm{A} = \left[ \begin{array}{cc}1& 3\\ 3& 4\end{array} \right] \mathrm{and} \mathrm{B} = \left[ \begin{array}{cc}- 2& 1\\ - 3& 2\end{array} \right]$  and  A² - 5 B² = 5 C. Find matrix  C, where C is a 2 by 2 matrix.

Jaya borrowed Rs. 50,000 for 2 years. The rates of interest for two successive years are 12% and 15% respectively. She repays 33,000 at the end of the first year. Find the amount she must pay at the end of the second year to clear her debt.

The catalogue price of a computer set is Rs. 42,000. The shopkeeper gives a discount of 10% on the listed price. He further gives an off-season discount of 5% on the discounted price. However, sales tax at 8% is charged on the remaining price after the two successive discounts. Find 

(i) the amount of sales tax a customer has to pay

(ii) the total price to be paid by the customer for the computer set.

P(1, -2) is a point on the line segment A(3, -6) and B(x, y) such that AP : PB is equal to 2 : 3. Find the coordinates of B

The marks of 10 students of a class in an examination arranged in ascending order is as follows: 

13, 35, 43, x, x + 4, 55, 61, 71, 80

If the median marks is 48, find the value of x. Hence find the mode of the given data.

What must be subtracted from 16x³ – 8x² + 4x + 7 so that the resulting expression has 2x + 1 as a factor?