AQA GCSE Mathematics Paper 1 (Higher), June 2025: Question 18

Mark scheme

Show the mark scheme

Q Answer Mark Comments

1 oe

sin 30 =

21 1 1 2

× ×

and 2 2 2 2 1

eg or or

33 2 6

cos 45 = or

22 M2 may be seen in a table

and implied by position in a calculation

11 1

tan 60 = 3 eg × × is correct for M2

22 3

M1 1 or 2 correct values

6 a 24

with all three correct values oe in the form eg

12 b 24

A1

seen

condone a = 6 and b = 12 with all

18 three correct values seen

Additional Guidance

Allow 1 for 1 throughout

Allow, eg 2 for tan 60

×

Correct answer from scores only M1 for tan 60 unless the correct

values are attributed to sin 30 and cos 45 elsewhere in the working

Do not allow further work eg = (with all three correct values seen) M2A0

12 2

How to answer it

Exact Trigonometric Values and Rationalising Surds

📋 What This Question Tests

This non-calculator question assesses your ability to combine exact trigonometric values with algebraic manipulation of surds:

Recalling exact values: sin 30°, cos 45°, and tan 60°
Multiplying and dividing fractional expressions containing surds
Rationalising denominators to present an answer in the targeted form √a / b where a and b are integers

Question 18 • 3 Marks

Show that (sin 30° × cos 45°) / tan 60° can be written in the form √a / b

AQA GCSE Mathematics (Higher Tier)

💡 Key Knowledge: Exact Trig Values

You must memorise these standard values for non-calculator papers:

sin 30° = 1/2
cos 45° = 1/√2 (or √2 / 2)
tan 60° = √3 (or (√3 / 2) ÷ (1/2))

Surd Rule: √x × √y = √(x × y), and (√x)² = x.

📐 Step-by-Step Solution

Step 1: Substitute the exact values

sin 30° × cos 45° / tan 60° = ( (1/2) × (√2 / 2) ) / √3

Step 2: Simplify the numerator

(1/2) × (√2 / 2) = √2 / 4

Step 3: Divide by tan 60° (√3)

(√2 / 4) ÷ √3 = √2 / (4√3)

Step 4: Rationalise the denominator

Multiply top and bottom by √3:

(√2 × √3) / (4√3 × √3) = √6 / (4 × 3) = √6 / 12

Here, a = 6 and b = 12, both of which are integers.

📐 Alternative Working (Using 1/√2)

Step 1: Substitute values:

(1/2 × 1/√2) ÷ √3 = 1 / (2√6)

Step 2: Rationalise directly:

Multiply numerator and denominator by √6:

(1 × √6) / (2√6 × √6) = √6 / (2 × 6) = √6 / 12

Equally valid: √24 / 24 (obtained by multiplying top and bottom of 1/(2√6) by √24).

✅ Final Answer & Mark Breakdown

√6 / 12 (or a = 6, b = 12)

Mark Scheme Breakdown:
• M1: 1 or 2 correct values seen (e.g. sin 30° = 1/2 or tan 60° = √3).
• M2: All 3 correct values seen or implied in calculation: (1/2 × √2/2) / √3 or 1/(2√6) .
• A1: Correct final form √6 / 12 (or unsimplified equivalent like √24 / 24 ) with all 3 trig values correct.

🧠 Exam Technique: "Show That" Proofs

Write down each trig value separately first: State sin 30° = 1/2 , cos 45° = √2/2 , and tan 60° = √3 explicitly. Even if your surd arithmetic goes wrong later, you instantly secure M2.
Check the target format: The question asks for √a / b. Leaving an answer as 1 / (2√6) loses the final accuracy mark because the root is still in the denominator.
Do not write decimal approximations: This is a non-calculator surd question; decimals receive 0 marks.

❌ Common Examiner Traps

Swapping trig values: Mixing up sin 30° (1/2) with sin 60° (√3/2), or confusing tan 30° (1/√3) with tan 60° (√3).
Attribution error: If you write (1/√2 × 1/2) / √3 without labeling which is which, examiners must penalise value mix-ups (caps at M1).
Invalid "cancelling": Continuing beyond the final answer by incorrectly doing √6 / 12 = √1 / 2 . The mark scheme explicitly states: "Do not allow further work: M2 A0"!

Topics

Geometry and measures · Number · 3.1.1 Structure and calculation · 3.4.2 Mensuration and calculation