AQA AS Level Mathematics Paper 1, June 2025: Question 4

Mark scheme

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Q Marking instructions AO Marks Typical solution

4 Obtains one correct value for 3θ 1.1a M1 3

θ tan 3θ =

or 2

Condone missing or incorrect 3θ = 56 3.°,236 3.°,416 3.°

labelling

Obtains AWRT 56, 236 and 416 1.1a M1 θ = 19°,79°,139°

and no others within the range

Condone missing or incorrect

labelling

Divides their solutions for 3θ by 1.1b A1F

3 to obtain solutions for θ

FT their values for 3θ

Must be at least 2 solutions for

3θ

Question 4 Total 3

AQA AS Level Mathematics • Paper 1 & 2 Pure

Rearranging linear trigonometric equations into the standard form tan(kθ) = c .
Adjusting the working domain to find all valid intermediate values for the compound argument ( 3θ ).
Exploiting the 180° periodicity of the tangent function to capture all roots within the interval.
Converting intermediate values back to the original variable θ and following exact rounding instructions (nearest integer degree).

Rearrange to isolate tan 3θ:
2 tan 3θ − 3 = 0
2 tan 3θ = 3
tan 3θ = 3/2 = 1.5
Adjust the domain for 3θ:
Given interval: 0° ≤ θ ≤ 180°
Multiply by 3: 0° ≤ 3θ ≤ 540°
Find the principal angle using arctan:
3θ = tan⁻¹(1.5) = 56.3099...°
M1 awarded: For obtaining at least one correct value of 3θ (e.g. AWRT 56.3°) or an equivalent correct θ value.
Find all solutions within the expanded range (0° to 540°):
Since tan repeats every 180°, keep adding 180°:
• 3θ₁ = 56.31°
• 3θ₂ = 56.31° + 180° = 236.31°
• 3θ₃ = 236.31° + 180° = 416.31°
(Note: 416.31° + 180° = 596.31°, which is greater than 540°, so stop here.)
M1 awarded: For obtaining AWRT 56°, 236°, and 416° with no extra solutions in the range for 3θ.
Divide by 3 to find θ and round to the nearest whole degree:
• θ₁ = 56.31° / 3 = 18.77° → 19°
• θ₂ = 236.31° / 3 = 78.77° → 79°
• θ₃ = 416.31° / 3 = 138.77° → 139°
A1F awarded: Follow-through mark for dividing solutions for 3θ by 3 to reach final values of θ (must have at least 2 solutions for 3θ).

θ = 19°, 79°, 139°

Symmetry of Tangent: The general rule for tangent is simply tan(α + 180°k) = tan α . You only need to add/subtract multiples of 180°.
Multiple Angles rule: When solving tan(kθ) = c , always adjust the interval to k × interval first, find all values of kθ , and only then divide by k .
Degree Mode: Ensure your calculator is in DEGREES ( D ), not Radians ( R ).

Mark 1 (M1): Isolating the trig function and finding the base angle: 3θ ≈ 56.3° .
Mark 2 (M1): Finding all 3 values in the range: 56.3°, 236.3°, 416.3° with none missing and no invalid values outside 540°.
Mark 3 (A1F): Dividing by 3 accurately and observing the rounding requirement: "to the nearest degree". Leaving unrounded decimals loses this mark!

Premature Division (The Most Common Pitfall): Finding 3θ = 56.3° , immediately dividing by 3 to get θ = 18.8° , and then adding 180° to get θ = 198.8° (which is outside the range). This completely misses the other two valid solutions!
Missing the Third Solution: Forgetting to multiply the upper limit 180° × 3 = 540° , leading students to stop after 236.3° and only finding two values instead of three.
Ignoring Rounding Instructions: Writing 18.8°, 78.8°, 138.8° instead of rounding to the nearest degree ( 19°, 79°, 139° ).
Calculator in Radian Mode: Yielding tan⁻¹(1.5) = 0.983 radians rather than 56.3°, resulting in immediate loss of accuracy marks.

Pure Mathematics · E: Trigonometry