AQA A Level Maths June 2019 Paper 3: question paper and mark scheme (7357/3)

3 A Level questions like these, with worked solutions

Written and checked by us, not taken from the AQA paper. Try each one, then tap "Worked solution" to see every step and its marks.

1. Algebra and functions · 5 marks

   Let $f(x)=x^3+c{x}^2+dx-4$. Given $f(x)$ has a factor $(x+2)$ and leaves a remainder of $-10$ when divided by $(x-3)$, find the values of $c$ and $d$.

   Worked solution

   1. M1 for applying the Factor Theorem for $(x+2)$: $f(-2)=0⟹-8+4c-2d-4=0⟹4c-2d=12$.
   2. M1 for applying the Remainder Theorem for $(x-3)$: $f(3)=-10⟹27+9c+3d-4=-10⟹9c+3d=-33$.
   3. M1 for simplifying both equations (e.g., $2c-d=6$ and $3c+d=-11$).
   4. M1 for solving the simultaneous equations (e.g., adding them gives $5c=-5$).
   5. A1 for finding the correct values $c=-1$ and $d=-8$.

2. Numerical methods · 7 marks

   $f(x)=x^3-3x-4$.
   (a) Show that $f(x)=0$ has a root in $(2,3)$. (2)
   (b) Rewrite the equation in the form $x=\sqrt[3]{3x+4}$ and use this iteration with $x_0=2$ to find $x_1,x_2,x_3,x_4$ each to 4 d.p. State the value of the root to 3 d.p. (5)

   Worked solution

   (a) $f(2)=8-6-4=-2<0$; $f(3)=27-9-4=14>0$   M1 A1.
   (b) $x_1=\sqrt[3]{10}=2.1544$; $x_2=\sqrt[3]{10.4633}=2.1872$; $x_3=\sqrt[3]{10.5616}=2.1940$; $x_4=\sqrt[3]{10.5821}=2.1955$   M1 A1 A1 A1. Root $\approx 2.196$   A1.

3. Sampling and data · 5 marks

   A histogram is drawn. The class $0\le x<20$ has frequency $10$, and the class $20\le x<30$ has frequency $30$. If the height of the bar for the first class is $2\text{ cm}$, find the height of the bar for the second class, assuming the width of the bars is proportional to the class width.

   Worked solution

   1. M1 for stating the class widths: $20$ and $10$.
   2. M1 for establishing the width ratio: if class width $20$ has a certain drawn width, class width $10$ has half that drawn width. Let drawn width of first bar be $2w$, then second bar is $w$.
   3. M1 for using Area $\propto$ Frequency for the first bar: Area $=2w\times 2=4w$. So $4w$ represents frequency $10$, meaning $w$ represents frequency $2.5$.
   4. M1 for applying to the second bar: Area $=w\times h$. Frequency $=30$. So $w\times h$ represents $30$. Since $w$ represents $2.5$, Area must be $12w$.
   5. A1 for solving for height: $w\times h=12w⟹h=12\text{ cm}$.

More on these topics: Algebra and functions · Numerical methods · Sampling and data

What this paper covers

What this paper is like

• Shape: 17 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 1 to 15 marks.
• Biggest question: Q9, 15 marks: implicit curve that misses both axes.
• Marks by main topic: Statistical distributions 22, Sampling and data 21, Differentiation 15, Algebra and functions 12, Exponentials and logarithms 12, Probability 7, Coordinate geometry 5, Proof 4, Sequences and series 2.
• More Differentiation in other sittings: June 2018 Q2 (1 marks), November 2021 Q9 (9 marks), June 2022 Q8 (7 marks).
• Against the other 5 sittings we mapped: more marks on Sampling and data (21 marks here, 10 on average) and Differentiation (15 marks here, 7 on average); fewer on Trigonometry (0 marks here, 10 on average) and Integration (0 marks here, 7 on average).
• Not on this paper: Trigonometry, Integration, Numerical methods, Vectors, Hypothesis testing.
• Calculator: allowed.

7357/3: question by question

Labels and topics are ours, read from the 7357/3 paper (checked 2026-11-01); marks as printed. Tap a number to practise that topic.

1. Q1Domain of arcsin (tick one) (1 marks) · Algebra and functions, Trigonometry
2. Q2Factorials (multiple choice) (1 marks) · Sequences and series
3. Q3Increasing sequences (multiple choice) (1 marks) · Sequences and series
4. Q4Region defined by two inequalities (3 marks) · Algebra and functions
5. Q5Circle with a chord and an angle (5 marks) · Coordinate geometry, Trigonometry
6. Q6Proof about Pythagorean triples (4 marks) · Proof
7. Q7Partial fractions and an exact integral (8 marks) · Algebra and functions, Integration
8. Q8Cooling liquid modelled exponentially (12 marks) · Exponentials and logarithms
9. Q9Implicit curve that misses both axes (15 marks) · Differentiation
10. Q10Describing correlation (tick one) (1 marks) · Sampling and data
11. Q11Quota sampling (multiple choice) (1 marks) · Sampling and data
12. Q12Heights in a rowing club: mean and spread (6 marks) · Sampling and data
13. Q13Skateboard tricks: binomial and a test (10 marks) · Statistical distributions, Hypothesis testing
14. Q14Survey of teachers' health: conditional probability (7 marks) · Probability
15. Q15Rainfall and wheat yield: correlation (3 marks) · Sampling and data
16. Q16Large data set: salt purchases over time (10 marks) · Sampling and data
17. Q17Normal model for brownie masses (12 marks) · Statistical distributions

Some Paper 3 statistics questions use AQA's large data set. We don't write large data set questions, so use your class materials for those.