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2026 O-Level E-Maths Paper 2 Answers (4052/02) – Worked Solutions
Suggested answers to all 9 questions from Acer Academy's Maths team, with every final answer below and the first 3 worked in full. Answers to 3 s.f., angles to 1 d.p., as the paper instructs.
9 questions · 90 marksUpdated
Answers at a glance
Final answers to every question. Questions 1–3 are worked in full below; the full PDF has the working, diagrams and tips for all of them.
| 1 | (a)(i) y = 2.4 (ii) see graph (iii) (1, 6) (b) ∩-shaped through (−4, 0), (2, 0), (0, 8) | ||
| 2 | (a) a = 3, b = 196 (b) 92 (c)(i) 148 (ii) overestimate (d)(i) 6n − 8 (ii) 6n − 8 = 3(2n − 3) + 1 (iii)(a) 17 408 (b) 17 410 | ||
| 3 | (a) x < 2.5 (b) 4 + 48x−2 (c) x = −0.19 or 1.79 (d)(i) t = 4y + 3 (ii) y = 7 − 4x²4x² − 8 | ||
| 4 | (a)(i) 105 m³ (ii) 31.5 minutes (b)(i) d = 22.56 (ii)(a) 0.7744 : 1 (b) 0.88 : 1 | ||
| 5 | (a) construction (RS ≈ 6.1 cm) (b)(i) congruent by AAS (ii) ∠ACD = (x + y2)° | ||
| 6 | (a)(i) 276 (ii) 12.8 (b)(i) T =
| ||
| 7 | (b) x = 32 or −78 (c) x = −78 gives a negative length (d) 103.1° | ||
| 8 | (a)(i) 35 (ii) 2.50 (b)(i) −2a + b (iii) MN : NC = 5 : 4 | ||
| 9 | (a) $32 000 (b) 10 900 (c) the claim is correct: interest ≈ $162 045 |
Questions 1–3, fully worked
Working is in red; the final answer sits on the answer line, with the marks in brackets.
1
(a)(i)
Complete the table of values for y = 4x − 2x² + 4, to 1 decimal place.
| x | −3 | −2 | −1 | −0.5 | 0.25 | 0.5 | 1 | 2 | 3 |
| y | 2.4 | 1.5 | −2 | −12 | −12 | 4 | 6 | 5.5 | 5.1 |
At x = −3: y = 4−3 − 2(−3)² + 4 = −1.333 − 0.222 + 4 = 2.444
To 1 decimal place, y = 2.4
Answer y =2.4[1]
Watch out: The table asks for 1 decimal place, so write 2.4, not 229.
(a)(ii)
Draw the graph of y = 4x − 2x² + 4 for −3 ≤ x ≤ 3.
Plot the points from the table and join them with a smooth curve.
The curve is not defined at x = 0, so it is drawn as two separate branches.
Answersee graph[3]
Watch out: Don't join the two branches across x = 0: y is not defined there.
(a)(iii)
Write down the coordinates of the turning point on the graph.
The right-hand branch has its highest point at x = 1.
y = 41 − 21² + 4 = 6
Answer (1,6)[1]
(b)
Sketch y = −(x − 2)(x + 4), showing where the curve crosses the axes.
y-intercept: x = 0 gives y = −(0 − 2)(0 + 4) = 8
x-intercepts: y = 0 gives x = 2 or x = −4
Line of symmetry: x = 2 + (−4)2 = −1
Turning point: y = −(−1 − 2)(−1 + 4) = 9, so (−1, 9)
The x² term is negative, so the curve is ∩-shaped.
Answer∩-shaped through (−4, 0), (2, 0) and (0, 8)[2]
2
(a)
a is an integer, b is a square number and 3a + b = 223. Find a and b.
Try values of a:
a = 0: b = 223 − 1 = 222, not a square number
a = 1: b = 223 − 3 = 220, not a square number
a = 2: b = 223 − 9 = 214, not a square number
a = 3: b = 223 − 27 = 196 = 14² ✓
a = 4: b = 223 − 81 = 142, not a square number; a ≥ 5 makes b negative
Answer a =3 b =196[1]
Watch out: 196 = 14². Check by squaring: 14 × 14 = 196.
(b)
n is an integer and ∛n is 5, correct to the nearest whole number. Find the smallest possible n.
4.5 ≤ ∛n < 5.5
n ≥ 4.5³ = 91.125
n is an integer, so the smallest possible value is 92.
Check: ∛92 = 4.51, which rounds to 5; ∛91 = 4.498, which rounds to 4.
Answer92[1]
Watch out: n must be an integer, so 91.125 is not allowed. Round up to 92.
(c)(i)
Round each number to 1 significant figure to estimate 3.99³ + √95.20.517, showing your working.
3.99 → 4, 95.2 → 100, 0.517 → 0.5
4³ + √1000.5 = 64 + 100.5 = 740.5 = 148
Answer148[1]
(c)(ii)
Without working out the exact value, is your estimate an underestimate or an overestimate? Give a reason.
overestimatebecause
both numbers in the numerator were rounded up (3.99 → 4 and 95.2 → 100)
and the denominator was rounded down (0.517 → 0.5), so the estimate is too big.
[1]
(d)(i)
Sequence S has a constant difference. Its 5th term is 22 and its 6th term is 28. Find the nth term of S.
Common difference d = 28 − 22 = 6
First term = 22 − 4 × 6 = −2
nth term = −2 + (n − 1) × 6 = 6n − 8
Answer6n − 8[2]
(d)(ii)
Explain why each term of S is one greater than a multiple of 3.
6n − 8 = 6n − 9 + 1 = 3(2n − 3) + 1
3(2n − 3) is a multiple of 3, so every term is one greater than a multiple of 3.
Answer6n − 8 = 3(2n − 3) + 1[1]
(d)(iii)(a)
Sequence T has a constant positive ratio. Its 1st term is 17 and its 3rd term is 272. Find the 6th term of T.
Third term = first term × r²: 272 = 17r²
r² = 16, so r = 4 (the ratio is positive)
Sixth term = 17 × 4⁵ = 17 × 1024 = 17 408
Answer17 408[2]
(d)(iii)(b)
Find the smallest term of S that is greater than the 6th term of T.
6n − 8 > 17 408
6n > 17 416
n > 2902.67, so n = 2903
Term = 6(2903) − 8 = 17 410
Answer17 410[2]
Watch out: The question asks for the term, not its position. n = 2903 gives the term 17 410.
3
(a)
Solve 4x + 2x − 152 < 5.
Multiply every term by 2: 8x + 2x − 15 < 10
10x < 25
x < 2.5
Answerx < 2.5[2]
(b)
Write 12x−3(13x³ + 4x) in the form a + bxn.
12x−3 × 13x³ = 4x⁰ = 4
12x−3 × 4x = 48x−2
Answer4 + 48x−2[2]
(c)
Solve 12 − x + 14x + 1 = 5, giving your solutions to 2 decimal places.
(4x + 1) + (2 − x)(2 − x)(4x + 1) = 5
3x + 3 = 5(2 − x)(4x + 1)
3x + 3 = 5(−4x² + 7x + 2) = −20x² + 35x + 10
20x² − 32x − 7 = 0
x = 32 ± √(32² − 4(20)(−7))2(20) = 32 ± √158440
x = 1.7949… or x = −0.1949…
Answer x =−0.19or x =1.79[5]
Watch out: 1.7949… rounds to 1.79, not 1.80. Keep the full calculator value until the last step.
(d)(i)
Find t in terms of y.
4y = t − 3
t = 4y + 3
Answer t =4y + 3[1]
(d)(ii)
Hence find y in terms of x, as a single fraction in its simplest form.
x² = 2t + 1t + 1 = 2(4y + 3) + 1(4y + 3) + 1 = 8y + 74y + 4
x²(4y + 4) = 8y + 7
4x²y + 4x² = 8y + 7
4x²y − 8y = 7 − 4x²
y(4x² − 8) = 7 − 4x²
Answer y =7 − 4x²4x² − 8[4]
Also correct: y = 4x² − 78 − 4x² is the same fraction, with top and bottom multiplied by −1.
Questions 4–9
Topic and marks for each. The final answers are in the table above; the full working is in the PDF.
Question 4 – Mensuration [11 marks]
Question 5 – Construction and geometry [7 marks]
Question 6 – Statistics, matrices and probability [11 marks]
Question 7 – Trigonometry [11 marks]
Question 8 – Vectors [8 marks]
Question 9 – Problem solving [10 marks]
Unlock Q4–9 with full working
Get the full answer key (PDF)
All 9 questions with full working, emailed to you as a PDF.
Questions about this answer key
Are these the official SEAB answers?
No. These are Acer Academy's own worked solutions; SEAB does not publish answer keys.
How were the answers rounded?
To 3 significant figures, with angles to 1 decimal place, as the paper instructs.
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Where is Paper 1?
It has its own post, with all 28 questions worked and the full PDF free. See the Paper 1 worked solutions
Questions from the 2026 GCE O-Level Mathematics (4052) Paper 2 © SEAB / Cambridge. Worked solutions and diagrams: Acer Academy.
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