Robinson Tuition
Grade 9 · Higher · free GCSE maths worksheet

Exact value problems with surds and trigonometry

Answers kept as roots or fractions.

16 original questions, 57 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Work out the exact value of sin 60° × cos 30°.

Give your answer as a fraction.

2 marks
Show answer
34
2

In triangle ABC, AB = 8 cm, AC = 6 cm and angle BAC = 60°.
Work out the exact area of triangle ABC. Give your answer in the form a√3 cm2.

3 marks
Show answer
12√3 cm2
3

Work out the exact value of (sin 30° + cos 60°) × tan 60°.

2 marks
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√3
4

In triangle ABC, AB = 3 cm, AC = 5 cm and BC = √19 cm.

Find the size of angle BAC.

3 cm√19 cm5 cmABCDiagram NOT accurately drawn
4 marks
Show answer
60°
5

In triangle ABC, angle ABC = 90° and angle BAC = 30°.

AC = 14 cm.

Without using a calculator, work out the length of BC.

You must show your working.

14 cm30°ABCDiagram NOT accurately drawn
3 marks
Show answer
7 cm
6

Work out the exact value of (sin 45° + cos 30°)2
Give your answer in the form a + b√64 where a and b are integers.

3 marks
Show answer
(5 + 2√6)/4
7

In triangle PQR, angle PQR = 90° and angle QPR = 45°.

PQ = 8 cm.

Without using a calculator, work out the exact length of PR.

Give your answer in the form k√2.

8 cm45°PQRDiagram NOT accurately drawn
3 marks
Show answer
8√2 cm
8

In triangle ABC, angle BAC = 30°, BC = 5 cm and AB = 8 cm.

Work out the exact value of sin C.

8 cm5 cm30°ABCDiagram NOT accurately drawn
4 marks
Show answer
45
9

In triangle DEF, angle DEF = 90° and angle EDF = 60°.

DE = 5 cm.

Without using a calculator, work out the exact length of EF.

5 cm60°DEFDiagram NOT accurately drawn
3 marks
Show answer
5√3 cm
10

In triangle JKL, angle JKL = 90°.

JK = 6 cm and JL = 12 cm.

Without using a calculator, find the size of angle JLK.

You must show your working.

6 cm12 cmJKLDiagram NOT accurately drawn
3 marks
Show answer
30°
11

In triangle ABC, angle ABC = 90°, AB = √5 cm and BC = 2 cm. Angle BAC = x.

  1. Work out the exact value of sin x and of cos x.

  2. Hence find the exact value of tan x, giving your answer with a rational denominator.

4 marks
Show answer
a) sin x = 2/3, cos x = √5/3 b) 2√5/5
12

Work out the exact value of

tan 30° × tan 60° + cos 45° × sin 45°

Give your answer as a fraction.

3 marks
Show answer
32
13

In triangle ABC, AB = 6 cm, BC = 4√2 cm and angle ABC = 45°.
Without using a calculator, work out the exact perimeter of triangle ABC. Give your answer in the form a + b√2 + c√5.

5 marks
Show answer
6 + 4√2 + 2√5 cm
14

An equilateral triangle has side length 2√3 cm.
Without using a calculator, work out the exact height and the exact area of the triangle.

4 marks
Show answer
height) 3 cm area) 3√3 cm2
15

ABCD is a trapezium in which AB is parallel to DC.

Angle ADC = 90°, angle BCD = 60°, AB = 6 cm and DC = 10 cm.

Without using a calculator, work out the exact area of the trapezium.

Give your answer in the form k√3 cm2.

6 cm10 cm60°ABCDDiagram NOT accurately drawn
5 marks
Show answer
32√3 cm²
16

A trapezium has parallel sides of length (3 + √2) cm and (5 + 3√2) cm.

The perpendicular distance between the parallel sides is (4 − √2) cm.

Work out the exact area of the trapezium.

Give your answer in the form (a + b√2) cm², where a and b are integers.

(5 + 3√2) cm(3 + √2) cm(4 − √2) cmDiagram NOT accurately drawn
6 marks
Show answer
12 + 4√2

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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