AQA A Level Maths June 2022 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 · 5 marks

   Solve the inequality $|2x+1|\le |x-5|$.

   Worked solution

   1. M1 for squaring both sides to remove the modulus signs: $(2x+1{)}^2\le (x-5{)}^2$.
   2. M1 for expanding the brackets: $4x^2+4x+1\le x^2-10x+25$.
   3. M1 for rearranging to form a quadratic inequality: $3x^2+14x-24\le 0$.
   4. M1 for factorising to find the critical values: $(3x-4)(x+6)\le 0$, yielding critical values $-6$ and $\frac{4}{3}$.
   5. A1 for stating the correct closed interval $-6\le x\le \frac{4}{3}$.

2. Integration · 6 marks

   Find the exact particular solution to the differential equation: $$\frac{dy}{dx}=y\cos x$$ given the boundary condition that $y(0)=e^2$.

   Worked solution

   Separate variables: $\frac{1}{y}dy=\cos xdx$. M1
   Integrate both sides: $\int \frac{1}{y}dy=\int \cos xdx$. M1
   $\ln |y|=\sin x+C$. A1A1
   Convert to exponential form: $y=e^{\sin x+C}=A{e}^{\sin x}$ (where $A=e^C$). M1
   Substitute initial condition $y(0)=e^2$:
   $e^2=A{e}^{\sin (0)}⟹e^2=A{e}^0⟹A=e^2$.
   Exact particular solution: $y=e^2{e}^{\sin x}$ or $y=e^{\sin x+2}$. A1

3. Proof · 5 marks

   (a) Prove by exhaustion that the square of any integer leaves a remainder of 0, 1 or 4 when it is divided by 5. [4]

   (b) Hence show that the last digit of a square number is never 2, 3, 7 or 8. [1]

   Worked solution

   (a) Considers every case: $n=5k,5k\pm 1,5k\pm 2$ (or $5k,5k+1,5k+2,5k+3,5k+4$), $k$ an integer M1

   Squares at least two cases correctly, e.g. $(5k\pm 1{)}^2=25k^2\pm 10k+1$ M1

   All cases correct: $(5k{)}^2=5(5k^2)$; $(5k\pm 1{)}^2=5(5k^2\pm 2k)+1$; $(5k\pm 2{)}^2=5(5k^2\pm 4k)+4$ A1

   Conclusion: every integer is in one of the cases, so the remainder is always 0, 1 or 4 A1 (cso)

   (b) A whole number ending in 2 or 7 leaves remainder 2, and one ending in 3 or 8 leaves remainder 3, on division by 5; by (a) a square leaves remainder 0, 1 or 4, so it cannot end in 2, 3, 7 or 8 B1

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

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 16 marks.
• Biggest question: Q15, 16 marks: derivative of cosec and a stationary point.
• Marks by main topic: Sequences and series 24, Differentiation 19, Integration 18, Trigonometry 15, Coordinate geometry 12, Algebra and functions 11, Exponentials and logarithms 1.
• More Differentiation in other sittings: November 2021 Q14 (10 marks), June 2023 Q15 (9 marks), November 2020 Q12 (13 marks).
• Against the other 5 sittings we mapped: more marks on Sequences and series (24 marks here, 14 on average) and Integration (18 marks here, 8 on average); fewer on Numerical methods (0 marks here, 10 on average) and Algebra and functions (11 marks here, 17 on average).
• Not on this paper: Proof, Numerical methods, 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. Q1Parametric equations of a circle (multiple choice) (1 marks) · Coordinate geometry
2. Q2Period of a sequence (multiple choice) (1 marks) · Sequences and series
3. Q3Stretch of a log graph (multiple choice) (1 marks) · Exponentials and logarithms
4. Q4Transformed quadratic graph (multiple choice) (1 marks) · Algebra and functions
5. Q5Tangent to a quartic at x = 0 (3 marks) · Differentiation
6. Q6Binomial expansion and an approximation (6 marks) · Sequences and series
7. Q7Sketching y = cot x (3 marks) · Trigonometry
8. Q8Two parallel lines and a triangle (11 marks) · Coordinate geometry
9. Q9Arithmetic sequence with terms in x (9 marks) · Sequences and series
10. Q10Sector with a perpendicular: area and angle (12 marks) · Trigonometry
11. Q11Factor theorem with a constant b (10 marks) · Algebra and functions
12. Q12Geometric sequence and sum to infinity (8 marks) · Sequences and series
13. Q13Model of a cave entrance and its area (9 marks) · Integration, Algebra and functions
14. Q14Area under (2x - 8) ln x by parts (9 marks) · Integration, Exponentials and logarithms
15. Q15Derivative of cosec and a stationary point (16 marks) · Differentiation, Trigonometry