Mathematics

Simplify fully (2x³y²)³ ÷ (4x²y⁵)

Solve the simultaneous equations: 3x + 2y = 1 x² + y = 5

After a 20% reduction, a laptop costs £680. (a) Work out the original price. (b) The shop then increases the sale price by 20%. Show that this does not give the original price and explain why.

The first four terms of a quadratic sequence are: 5, 12, 23, 38, ... Find the nth term.

A frustum is formed by removing a small cone of height 6 cm from a large cone of height 18 cm. The radius of the base of the large cone is 12 cm. (a) Show that the radius of the top of the frustum is 4 cm. (b) Calculate the volume of the frustum in terms of π. [Volume of cone = ⅓πr²h]

Prove algebraically that (n + 3)² − (n − 2)² is always an odd multiple of 5 for all positive integers n.

A bag contains n counters. 4 are red and the rest are blue. Tom takes two counters at random without replacement. The probability that both are red is 2/15. (a) Show that n² − n − 90 = 0. (b) Find the number of blue counters.

The line L passes through (−1, 5) and is perpendicular to y = ²⁄₃x + 1. (a) Find the equation of L. (b) L intersects the x-axis at point P. Find the coordinates of P. (c) Calculate the area of the triangle formed by L, the line y = ²⁄₃x + 1, and the x-axis.

The graph of y = f(x) passes through the point (4, 7). Write down the coordinates of the corresponding point on the graph of: (a) y = f(x) + 3 (b) y = f(x − 2) (c) y = 2f(x) (d) y = f(−x)