AQA GCSE Maths Higher Paper 1: Every Question Type Worked Through

A Higher tier non-calculator paper can look intimidating, but most of it is built from a small set of recurring question types. Once you can spot them, a lot of questions take seconds instead of minutes. Below we work through a complete AQA Higher Paper 1, grouped by topic, and point out the quick method and the trap in each.

Tip: Attempt each question before you read the solution. Recognising a method isn’t the same as being able to produce it yourself in the exam hall.

The paper at a glance

Topic area	Question types on this paper	Typical marks
Number	Fractions, reciprocals, decimals, indices, surds, ordering	1–3 each
Algebra	Sequences, simultaneous equations, quadratics, iteration, proportion	2–4 each
Geometry	Exact trig, arc lengths, cyclic quadrilaterals, similar shapes, vectors	1–4 each
Statistics	Scatter graphs, Venn diagrams, histograms	1–4 each
Graphs	Straight-line models, trig graphs, transformations	1–3 each

Quick-fire opening questions

The first few questions are 1-mark multiple choice. Bank them fast and don’t overthink them.

• 4.75 as a fraction: 5 = 20/4, and you need a quarter less, so the answer is 19/4.
• Column vector: 3 right and 2 down gives a top number of 3 and a bottom number of −2.
• Square and cube number: 1,000,000, because 1000² = 100³ = 1,000,000.
• Reciprocal of 5/6: flip it to get 6/5.
• 3x = ½y, ratio x : y: multiply by 2 to get 6x = y. So y is six times x, and x : y = 1 : 6. Ratios written the wrong way round lose easy marks, so always check which quantity comes first.

Exact trig values

You’re asked to find angle x in a right-angled triangle with adjacent 9 and hypotenuse 18. Adjacent and hypotenuse means cosine:

cos x = 9/18 = ½

There’s no calculator, so you need your exact values. Sketch a 30-60-90 triangle with sides 1, 2 and √3. The angle with adjacent 1 and hypotenuse 2 is 60°. Our trigonometry for beginners guide covers SOH CAH TOA from the start if you need it.

Venn diagrams and ratio sharing

Venn diagram (80 people): 12 are singers but not guitar players, and 30% are neither (10% of 80 is 8, so 30% is 24). That accounts for 36 people, which leaves 44 guitar players. A quarter of those also sing, so 11 go in the intersection and 33 go in guitar only.

Holiday cost: £2400 is split in the ratio 3 : 5 between deposit and monthly payments. 2400 ÷ 8 = 300, so the deposit is £900 and the payments total £1500. Over six months that’s £250 a month.

Parallelogram: the short side of 6.5 cm is 1/9 of the perimeter, so the perimeter is 6.5 × 9 = 58.5. Then 13 + 2x = 58.5, which gives 2x = 45.5 and x = 22.75 cm.

Sequences

Type	Rule	Example on this paper
Geometric	Multiply by the same number each time	2nd term 4 and 4th term 16, so ×2 each step: 1st term 2, 3rd term 8
Arithmetic	Add the same amount each time	p, 5p, … goes up by 4p, so the 3rd term is 9p. If the three terms sum to 90, then 15p = 90 and p = 6

For more on arithmetic sequences, see finding the nth term.

Decimals, indices and surds

33/400 from 11/40 = 0.275: dividing the denominator by 10 more gives 11/400 = 0.0275. Multiply by 3: 0.0825.

Index laws: if c = 2¹⁰ × 3 × 5⁶, then 18c = 2 × 3² × 2¹⁰ × 3 × 5⁶ = 2¹¹ × 3³ × 5⁶.

Cube root of 2⁶ × 11³: divide each power by 3 to get 2² × 11 = 44.

32^(−3/5): the negative power means take the reciprocal, the denominator 5 means fifth root (which is 2), and the 3 means cube. 1/2³ = 1/8.

Rationalising 14/√7: multiply top and bottom by √7 to get 14√7/7, then simplify to 2√7. Stopping at 14√7/7 can cost the final mark, because the question asks for the form a√b.

2√10 × √80 × √18: √80 = 4√5 and √18 = 3√2. Multiply the whole numbers (2 × 4 × 3 = 24) and the roots (√(10 × 5 × 2) = √100 = 10), giving 240.

Remember: when an expression has the form a² − b², use the difference of two squares, (a + b)(a − b). It’s one of the most commonly missed marks on Higher papers because students don’t recognise it straight away.

Ordering without a calculator

The values are 48/3 − 1/3 recurring, 2.1⁴ and 3√23. Estimate each one:

• 16 − 0.3̇ = 15.6̇
• 2⁴ = 16, so 2.1⁴ is a bit above 16
• √23 is between 4 and 5, roughly 4.8, so 3√23 is about 14.4

Smallest to largest: 3√23, 15.6̇, 2.1⁴. For more estimation tricks, read the non-calculator paper: 10 tactics that save marks.

Circles and arc lengths

The earring has a circle of radius 21 mm and a quarter circle, with radii in the ratio 7 : 2. Since 21 = 7 × 3, the small radius is 2 × 3 = 6 mm.

Total wire length:

1. Full circle: π × 42 = 42π
2. Quarter circle: the arc is 12π ÷ 4 = 3π, plus the two straight radii of 6 + 6 = 12

Total = 45π + 12. Leaving out the two radii is the classic mistake here.

Algebra: quadratics, simultaneous equations and iteration

Comparing coefficients: (x + s)(x − t) = x² + kx − 40, so st = 40 and k = s − t. The factor pairs of 40 with a positive difference are 8 and 5, 10 and 4, and so on. The smallest positive difference is k = 3.

Spot the error: if a factor is (x + 2), the solution is x = −2, not +2. Substituting +2 gives 4, not 0.

Simultaneous equations: 2x + 4y = −9 and −4x + 2y = −7. Doubling the second equation gives −8x + 4y = −14. Subtracting it from the first eliminates y: 10x = 5, so x = 0.5 and y = −2.5.

Iteration: with u₁ = 9 and uₙ₊₁ = 4/(uₙ − 1), we get u₂ = 4/8 = ½ and u₃ = 4/(−½) = −8.

Plant growth graph: use y = mx + c. The plant starts at 20 cm and grows 12 cm in 4 days, so the gradient is 3 and h = 3n + 20.

Our complete guide to GCSE Algebra covers each of these techniques in more depth.

Proportion

Relationship	Equation	Result
y ∝ x³, with y = 17 when x = 4	y = kx³, so k = 17/64	y = (17/64)x³
m ∝ 1/√r, with r multiplied by 4	m = k/√r	√r doubles, so m halves

Geometry proof and trig graphs

Not a cyclic quadrilateral: opposite angles in a cyclic quadrilateral add up to 180°. If 4x + 92 = 180, then x = 22. Check the other pair: 5(22) + 30 = 140 and 22 + 36 = 58, which add up to 198, not 180. So ABCD is not cyclic. Write the circle theorem down as part of your answer, because it’s often worth a mark on its own. The 8 circle theorems guide lists every one you need.

Obtuse angle y: between 90° and 180°, the sine graph is above the axis and the cosine graph is below it. So sin y > 0 and cos y < 0.

Similar cylinders: an area ratio of 9 : 25 gives a curved surface area ratio of 9 : 25 as well, and a height ratio of 3 : 5 (square root the area ratio).

Histograms

In a histogram, the area of each bar is the frequency. The three bars shown are 5 × 2 = 10, 10 × 3 = 30 and 5 × 4 = 20, which makes 60. That leaves 74 − 60 = 14 for the 170–180 class. Its width is 10, so the frequency density is 1.4.

Graph transformations

y = x³ + 6 translated 4 units right becomes y = (x − 4)³ + 6. Expand two brackets first: (x − 4)² = x² − 8x + 16. Then multiply by (x − 4) again:

y = x³ − 12x² + 48x − 64 + 6 = x³ − 12x² + 48x − 58

Did you know? Translating “right by 4” means subtracting 4 inside the function. Getting the sign the wrong way round is one of the most common errors on transformation questions.

Turn walkthroughs into marks

A walkthrough shows you the methods, but you only remember them by practising them. Work through the topics you dropped marks on here using our apps:

• GCSE Maths puts Number, Algebra, Geometry and Statistics in one bundle, which suits mixed papers like this one.
• GCSE Number covers fractions, indices, surds and ratio drills for the quick-fire questions.
• GCSE Algebra covers sequences, simultaneous equations, quadratics and graphs.
• GCSE Geometry covers exact trig, circle theorems, similar shapes and arcs.
• GCSE Statistics covers Venn diagrams, histograms and scatter graphs.

That’s 4900+ questions across the four subject apps. For the wider exam strategy, read how to get a Grade 9 in GCSE Maths.