AQA A Level Maths June 2019 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. Trigonometry · 5 marks

   Solve the equation $2{\cos }^{2}x-\sin x-1=0$ for $0^{\circ }\le x\le {360}^{\circ }$.

   Worked solution

   1. M1 for using the identity ${\cos }^{2}x=1-{\sin }^{2}x$ to rewrite the equation: $2(1-{\sin }^{2}x)-\sin x-1=0$.
   2. M1 for simplifying to a quadratic in $\sin x$: $2{\sin }^{2}x+\sin x-1=0$.
   3. M1 for factorising the quadratic to $(2\sin x-1)(\sin x+1)=0$.
   4. A1 for finding the solutions from $\sin x=0.5$, which are $x={30}^{\circ },{150}^{\circ }$.
   5. A1 for finding the solution from $\sin x=-1$, which is $x={270}^{\circ }$.

3. Integration · 5 marks

   The gradient of a curve is given by $\frac{dy}{dx}=4\sqrt{x}-1$. The curve passes through the point $(9,20)$. Find the equation of the curve.

   Worked solution

   1. M1 for rewriting as $4x^{\frac{1}{2}}-1$ and integrating: $y=\frac{4x^{\frac{3}{2}}}{\frac{3}{2}}-x+c$.
   2. A1 for simplifying the integral to $y=\frac{8}{3}{x}^{\frac{3}{2}}-x+c$.
   3. M1 for substituting the point $(9,20)$: $20=\frac{8}{3}(9{)}^{\frac{3}{2}}-9+c$.
   4. M1 for evaluating $9^{\frac{3}{2}}=27$ to solve for $c$: $20=72-9+c⟹c=-43$.
   5. A1 for the final equation $y=\frac{8}{3}{x}^{\frac{3}{2}}-x-43$.

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

What this paper covers

What this paper is like

• Shape: 16 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 1 to 16 marks.
• Biggest question: Q16, 16 marks: product rule, then integrating e^x sin x.
• Marks by main topic: Differentiation 42, Trigonometry 19, Sequences and series 11, Numerical methods 10, Algebra and functions 9, Proof 5, Coordinate geometry 4.
• More Differentiation in other sittings: June 2018 Q5 (6 marks), November 2020 Q12 (13 marks), November 2021 Q14 (10 marks).
• Against the other 5 sittings we mapped: more marks on Differentiation (42 marks here, 16 on average) and Proof (5 marks here, 1 on average); fewer on Integration (0 marks here, 11 on average) and Algebra and functions (9 marks here, 18 on average).
• Not on this paper: Exponentials and logarithms, 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. Q1Equivalent index expressions (multiple choice) (1 marks) · Algebra and functions
2. Q2Differentiating e^(kx) (multiple choice) (1 marks) · Differentiation, Exponentials and logarithms
3. Q3Arc length of a sector (multiple choice) (1 marks) · Trigonometry
4. Q4Line through two points with unknowns (4 marks) · Coordinate geometry
5. Q5Arithmetic sequence: first term and difference (7 marks) · Sequences and series
6. Q6Range and inverse of a function (8 marks) · Algebra and functions
7. Q7Graph of sec 2x and a root by iteration (11 marks) · Trigonometry, Numerical methods
8. Q8Sigma notation for a sum of cubes (4 marks) · Sequences and series
9. Q9Proof that rational plus irrational is irrational (5 marks) · Proof
10. Q10Rate of change of a bubble's radius (4 marks) · Differentiation
11. Q11Checking a first-principles proof for sin x (4 marks) · Differentiation, Trigonometry
12. Q12Cot and cosec equation solved as a quadratic (7 marks) · Trigonometry
13. Q13Showing a curve with e^(3x) has one stationary point (7 marks) · Differentiation, Exponentials and logarithms
14. Q14Trapezium rule and whether it over- or underestimates (10 marks) · Numerical methods, Integration
15. Q15Parametric model of tide height and flow (10 marks) · Differentiation, Coordinate geometry
16. Q16Product rule, then integrating e^x sin x (16 marks) · Differentiation, Integration