Indices, surds and quadratic functions – Year 12 A level maths test to download as PDF

Do not use a calculator.

1. Write \(\dfrac{(3x^2)^3}{9x^{-1}}\) in the form \(ax^n\). (3 marks)
2. Show that \(\dfrac{10}{\sqrt5-1}\) can be written as \(\dfrac{5(\sqrt5+1)}2\). (2 marks)

\(f(x)=x^2+10x+7\).

1. Write \(f(x)\) in the form \((x+a)^2+b\). (2 marks)
2. Write down the coordinates of the minimum point and solve \(f(x)=0\) exactly. (2 marks)

The equation \(x^2+(k-1)x+4=0\) has equal roots.

1. Find the possible values of \(k\). (3 marks)
2. For one of these values, solve the equation. (1 mark)

Solve \(x^4-10x^2+9=0\).

A rectangular poster has perimeter 100 cm. Its width is \(w\) cm.

1. Show that its area is \(A=50w-w^2\). (1 mark)
2. By completing the square find the maximum area and the width that gives it. (2 marks)
3. The printer needs the area to be at least 600 cm\(^2\). Find the range of possible widths. (1 mark)