Teach Me Maths

Surds

This section is in development. More questions and features coming shortly.

AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen and paper handy, to work out your answers.

Question 11 mark

Do not use a calculator.

Which of these is equal to (3+\sqrt7)(3-\sqrt7)?

Hint

Expand the brackets fully: the two middle terms cancel each other out.

Worked solution
  1. (3+\sqrt7)(3-\sqrt7)=9-3\sqrt7+3\sqrt7-\sqrt7\times\sqrt7
  2. The \sqrt7 terms cancel
  3. \sqrt7\times\sqrt7=7, so this is 9-7
  4. Answer: 2

Question 22 marks

Do not use a calculator.

Rationalise the denominator and simplify fully

\frac{10\sqrt3}{\sqrt{15}}

Give your answer in the form a\sqrt b, where a and b are integers.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Multiply the top and bottom by \sqrt{15}, then simplify the surd on top before cancelling.

Worked solution
  1. Multiply top and bottom by \sqrt{15}: \dfrac{10\sqrt3\times\sqrt{15}}{15}
  2. \sqrt3\times\sqrt{15}=\sqrt{45}=3\sqrt5
  3. So the fraction is \dfrac{30\sqrt5}{15}
  4. Cancel: 30\div15=2
  5. Answer: 2\sqrt5

Question 3Challenge5 marks

Do not use a calculator.

A cuboid has a rectangular base.

The base measures (2+\sqrt2) cm by (1+\sqrt2) cm.

(a)

Work out the area of the base.

Give your answer in the form a+b\sqrt2, where a and b are integers.

2 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

The volume of the cuboid is (18+13\sqrt2) cm^3.

Work out the height of the cuboid.

Give your answer in the form a+b\sqrt2, where a and b are integers.

3 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Height = volume ÷ base area; to divide by 4+3\sqrt2, multiply top and bottom by 4-3\sqrt2, and watch the sign of the denominator.

Worked solution

Part (a)

  1. Area =(2+\sqrt2)(1+\sqrt2)
  2. =2+2\sqrt2+\sqrt2+\sqrt2\times\sqrt2
  3. \sqrt2\times\sqrt2=2, so =2+3\sqrt2+2
  4. Answer: 4+3\sqrt2 cm^2

Part (b)

  1. Height = volume \div base area =\dfrac{18+13\sqrt2}{4+3\sqrt2}
  2. Multiply top and bottom by the conjugate 4-3\sqrt2 (change the sign between the terms)
  3. Bottom: (4+3\sqrt2)(4-3\sqrt2)=16-18=-2
  4. Top: (18+13\sqrt2)(4-3\sqrt2)=72-54\sqrt2+52\sqrt2-78=-6-2\sqrt2
  5. \dfrac{-6-2\sqrt2}{-2}=3+\sqrt2
  6. Answer: 3+\sqrt2 cm

More on this topic: Surds worksheet with full solutions

All AQA Level 2 Further Maths practice questions

Unofficial practice questions written by Teach Me Maths. Not produced or endorsed by AQA.