AQA A Level Maths June 2018 Paper 1: question paper and mark scheme (7357/1)

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 · 4 marks

   Show that $\frac{4x}{x^2-9}-\frac{2}{x-3}$ can be simplified to the form $\frac{a}{x+b}$, where $a$ and $b$ are integers.

   Worked solution

   1. M1 for factorising the denominator $x^2-9=(x-3)(x+3)$.
   2. M1 for finding the common denominator and forming the numerator: $4x-2(x+3)$.
   3. M1 for expanding the numerator to $4x-2x-6=2x-6$ and factorising it to $2(x-3)$.
   4. A1 for cancelling $(x-3)$ to achieve $\frac{2}{x+3}$, meaning $a=2$ and $b=3$.

2. Integration · 4 marks

   Find $\int (\sin x+\cos x{)}^{2}dx$.

   Worked solution

   1. M1 for expanding the bracket to ${\sin }^{2}x+2\sin x\cos x+{\cos }^{2}x$.
   2. M1 for simplifying using ${\sin }^{2}x+{\cos }^{2}x\equiv 1$ and $2\sin x\cos x\equiv \sin 2x$.
   3. M1 for integrating $1+\sin 2x$.
   4. A1 for $x-\frac{1}{2}\cos 2x+c$.

3. Vectors · 8 marks

   Consider the following two lines in 3D space: $$L_1:r=\left(\begin{array}{c}1\\ -1\\ 2\end{array}\right)+\lambda \left(\begin{array}{c}2\\ 1\\ -3\end{array}\right)\text{and}{L}_2:r=\left(\begin{array}{c}4\\ 2\\ -1\end{array}\right)+\mu \left(\begin{array}{c}-1\\ 2\\ 1\end{array}\right)$$ Show algebraically that these two lines are skew (they are neither parallel nor do they intersect).

   Worked solution

   Set the components equal:
   $1+2\lambda =4-\mu ⟹2\lambda +\mu =3$ (Eq 1). M1
   $-1+\lambda =2+2\mu ⟹\lambda -2\mu =3$ (Eq 2). M1
   $2-3\lambda =-1+\mu ⟹-3\lambda -\mu =-3$ (Eq 3). M1
   Solve Eq 1 and Eq 2: $2(3+2\mu )+\mu =3⟹6+5\mu =3⟹\mu =-0.6$. M1
   Substitute $\mu =-0.6⟹\lambda =1.8$. A1
   Check in Eq 3: $-3(1.8)-(-0.6)=-5.4+0.6=-4.8$. M1
   However, the RHS of Eq 3 is $-3$. Since $-4.8\ne -3$, there is no consistent solution. B1
   Therefore, the lines do not intersect. Since their direction vectors are not scalar multiples, they are skew. A1

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

What this paper covers

What this paper is like

• Shape: 15 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 1 to 12 marks.
• Biggest question: Q6, 12 marks: binomial expansion used for an estimate.
• Marks by main topic: Trigonometry 25, Sequences and series 22, Algebra and functions 14, Differentiation 13, Numerical methods 10, Coordinate geometry 8, Exponentials and logarithms 8.
• More Sequences and series in other sittings: June 2019 Q5 (7 marks), November 2020 Q10 (11 marks), November 2021 Q6 (7 marks).
• Against the other 5 sittings we mapped: more marks on Trigonometry (25 marks here, 16 on average) and Sequences and series (22 marks here, 14 on average); fewer on Integration (0 marks here, 11 on average) and Differentiation (13 marks here, 22 on average).
• Not on this paper: Proof, Integration, Vectors.
• Calculator: allowed.

7357/1: question by question

Labels and topics are ours, read from the 7357/1 paper (checked 2026-11-01); marks as printed. Tap a number to practise that topic.

1. Q1Differentiating a power of x (multiple choice) (1 marks) · Differentiation
2. Q2Stretch of an exponential graph (1 marks) · Algebra and functions
3. Q3Period of a sequence defined by sine (1 marks) · Sequences and series, Trigonometry
4. Q4Inverse of an exponential function and its domain (3 marks) · Algebra and functions, Exponentials and logarithms
5. Q5Parametric curve: gradient and a tangent (6 marks) · Differentiation, Coordinate geometry
6. Q6Binomial expansion used for an estimate (12 marks) · Sequences and series
7. Q7Right angle in a triangle and a circle through three points (8 marks) · Coordinate geometry
8. Q8Sector with a midpoint: area and angle (8 marks) · Trigonometry
9. Q9Arithmetic series with two conditions (9 marks) · Sequences and series
10. Q10Exponential decay model for caffeine (8 marks) · Exponentials and logarithms
11. Q11Oil production model: locating when it reaches zero (10 marks) · Numerical methods
12. Q12Factor theorem, factorising and a proof (10 marks) · Algebra and functions, Proof
13. Q13Logo design: maximising a rectangle in a circle (10 marks) · Trigonometry, Differentiation
14. Q14Geometric proof of the sin(A + B) identity (7 marks) · Trigonometry, Proof
15. Q15Cubic curve: gradients at two points (6 marks) · Differentiation, Coordinate geometry