AQA A Level Maths June 2018 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 · 7 marks

   Express ${\displaystyle \frac{x^3+1}{x^2-1}}$ in the form $Ax+B+{\displaystyle \frac{C}{x-1}}+{\displaystyle \frac{D}{x+1}}$ and find $A,B,C,D$.

   Worked solution

   Polynomial division: $\frac{x^3+1}{x^2-1}=x+\frac{x+1}{x^2-1}=x+\frac{1}{x-1}$ (since $\frac{x+1}{(x+1)(x-1)}=\frac{1}{x-1}$)   M1 M1 A1. So $A=1$, $B=0$, $C=1$, $D=0$   A1 A1 A1 A1.

2. Vectors · 7 marks

   Line $L_1$ has vector equation ${\mathbf{r}}_{\mathbf{1}}=\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)+\lambda \left(\begin{array}{c}2\\ 1\\ 1\end{array}\right)$ and line $L_2$ has equation ${\mathbf{r}}_{\mathbf{2}}=\left(\begin{array}{c}5\\ -3\\ 0\end{array}\right)+\mu \left(\begin{array}{c}-1\\ 2\\ 1\end{array}\right)$.

   1. Show that $L_1$ and $L_2$ intersect.
   2. Find the exact coordinates of the point of intersection.
   Worked solution

   1. M1 for equating the parametric equations: $1+2\lambda =5-\mu$, $\lambda =-3+2\mu$, and $1+\lambda =\mu$.
      M1 for taking two equations to solve for the parameters, e.g., $\lambda =\mu -1$ into $2\lambda +\mu =4$.
      M1 for $2(\mu -1)+\mu =4⟹3\mu -2=4⟹3\mu =6$.
      A1 for finding $\mu =2$ and $\lambda =1$.
      B1 for verifying with the unused equation: $\lambda -2\mu =1-4=-3$, which matches the second component equation, showing the lines intersect.
   2. M1 for substituting $\lambda =1$ into $L_1$ (or $\mu =2$ into $L_2$).
      A1 for the intersection point $(3,1,2)$.

3. Probability · 3 marks

   Three independent events $A$, $B$, and $C$ have probabilities $P(A)=0.5$, $P(B)=0.4$, and $P(C)=0.3$. Find the probability that exactly one of these events occurs.

   Worked solution

   1. M1 for identifying the three scenarios: $(A\cap B^{\prime }\cap C^{\prime })$, $(A^{\prime }\cap B\cap C^{\prime })$, and $(A^{\prime }\cap B^{\prime }\cap C)$.
   2. M1 for calculating the probabilities of these scenarios: $(0.5\times 0.6\times 0.7)+(0.5\times 0.4\times 0.7)+(0.5\times 0.6\times 0.3)$.
   3. A1 for evaluating and summing: $0.21+0.14+0.09=0.44$.

More on these topics: Algebra and functions · Vectors · Probability

What this paper covers

What this paper is like

• Shape: 18 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 1 to 13 marks.
• Biggest question: Q6, 13 marks: domain, range and inverse of a root function.
• Marks by main topic: Statistical distributions 20, Probability 18, Algebra and functions 16, Trigonometry 12, Hypothesis testing 8, Sequences and series 7, Proof 7, Exponentials and logarithms 5, Sampling and data 4, Coordinate geometry 2, Differentiation 1.
• More Algebra and functions in other sittings: June 2019 Q7 (8 marks), November 2020 Q4 (7 marks), November 2021 Q6 (4 marks).
• Against the other 5 sittings we mapped: more marks on Probability (18 marks here, 9 on average) and Proof (7 marks here, 2 on average); fewer on Sampling and data (4 marks here, 14 on average) and Differentiation (1 marks here, 10 on average).
• Not on this paper: Integration, Numerical methods, Vectors.
• 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. Q1Area of a circle from its equation (multiple choice) (1 marks) · Coordinate geometry
2. Q2Gradient of a polynomial curve (multiple choice) (1 marks) · Differentiation
3. Q3Perpendicular lines (tick one) (1 marks) · Coordinate geometry
4. Q4Modulus graph with a constant a (3 marks) · Algebra and functions
5. Q5Small-angle approximation of a trig curve (3 marks) · Trigonometry
6. Q6Domain, range and inverse of a root function (13 marks) · Algebra and functions
7. Q7Log laws and solving an exponential equation (5 marks) · Exponentials and logarithms
8. Q8Trig identity, then an integral (9 marks) · Trigonometry, Integration
9. Q9Mosaic tiles as an arithmetic series (7 marks) · Sequences and series
10. Q10Proof by contradiction for an irrational cube root (7 marks) · Proof
11. Q11Discrete probability distribution (multiple choice) (1 marks) · Statistical distributions
12. Q12Histogram of heights (multiple choice) (1 marks) · Sampling and data
13. Q13Large data set: percentage change (3 marks) · Sampling and data
14. Q14Students' other subjects: Venn diagram (6 marks) · Probability
15. Q15Binomial model for faulty light bulbs (7 marks) · Statistical distributions
16. Q16Survey of drink volumes: normal model and a test (12 marks) · Statistical distributions, Hypothesis testing
17. Q17Winning matches: binomial and conditional probability (12 marks) · Probability, Statistical distributions
18. Q18Testing an advertising campaign (8 marks) · Hypothesis testing

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.