AQA A Level Maths June 2023 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. Coordinate geometry · 8 marks

   Three points define the vertices of a triangle: $A(-2,-1)$, $B(4,7)$, and $C(8,4)$.

   1. By calculating the gradients of $[AB]$ and $[BC]$, prove that triangle $ABC$ is a right-angled triangle.

   2. Calculate the exact lengths of $[AB]$ and $[BC]$.

   3. Hence, find the exact area of the triangle $ABC$.

   Worked solution

   1. Gradient $m_{AB}=\frac{7-(-1)}{4-(-2)}=\frac{8}{6}=\frac{4}{3}$. A1
      Gradient $m_{BC}=\frac{4-7}{8-4}=\frac{-3}{4}$. A1
      Since $m_{AB}\times m_{BC}=\frac{4}{3}\times -\frac{3}{4}=-1$, $[AB]$ and $[BC]$ are perpendicular. M1
      Thus, triangle $ABC$ has a right angle at $B$. B1

   2. Length $AB=\sqrt{(4-(-2){)}^2+(7-(-1){)}^2}=\sqrt{36+64}=\sqrt{100}=10$. A1
      Length $BC=\sqrt{(8-4{)}^2+(4-7{)}^2}=\sqrt{16+9}=\sqrt{25}=5$. A1

   3. Since the triangle is right-angled at $B$, $AB$ and $BC$ act as the base and height. M1
      Area $=\frac{1}{2}\times \text{base}\times \text{height}=\frac{1}{2}\times 10\times 5=25$ sq units. A1

2. Exponentials and logarithms · 5 marks

   Solve the equation ${\log }_{2}x+{\log }_{x}2=2.5$ for $x>0,x\ne 1$.

   Worked solution

   1. M1 for applying the reciprocal rule: ${\log }_{x}2=\frac{1}{{\log }_{2}x}$.
   2. M1 for making the substitution $u={\log }_{2}x$ to form $u+\frac{1}{u}=2.5$.
   3. M1 for multiplying by $2u$ to form the quadratic $2u^2-5u+2=0$ and solving it.
   4. M1 for finding the roots $u=2$ and $u=\frac{1}{2}$.
   5. A1 for converting back to $x$ and finding both solutions: $x=4$ and $x=\sqrt{2}$.

3. Proof · 4 marks

   Prove by contradiction that if $a$ and $b$ are integers, then $a^2-4b\ne 2$.

   Worked solution

   1. M1 for assuming that there exist integers $a$ and $b$ such that $a^2-4b=2$, which rearranges to $a^2=4b+2=2(2b+1)$.
   2. M1 for deducing that since $a^2$ is an even number, $a$ must also be an even number.
   3. M1 for letting $a=2k$, substituting to get $(2k{)}^2=4b+2⟹4k^2=4b+2$, and dividing by $2$ to get $2k^2=2b+1⟹2(k^2-b)=1$.
   4. A1 for observing that the left-hand side $2(k^2-b)$ is even while the right-hand side $1$ is odd, which is a contradiction, so $a^2-4b\ne 2$.

More on these topics: Coordinate geometry · Exponentials and logarithms · Proof

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 14 marks.
• Biggest question: Q16, 14 marks: partial fractions and filling a container.
• Marks by main topic: Algebra and functions 27, Trigonometry 17, Integration 17, Sequences and series 10, Differentiation 10, Coordinate geometry 9, Exponentials and logarithms 6, Numerical methods 4.
• More Algebra and functions in other sittings: June 2022 Q11 (10 marks), November 2021 Q13 (8 marks), November 2020 Q13 (15 marks).
• Against the other 5 sittings we mapped: more marks on Algebra and functions (27 marks here, 14 on average) and Integration (17 marks here, 8 on average); fewer on Differentiation (10 marks here, 23 on average) and Sequences and series (10 marks here, 16 on average).
• Not on this paper: Proof, 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. Q1Coefficient in a binomial expansion (multiple choice) (1 marks) · Sequences and series
2. Q2Differentiating a power of x (multiple choice) (1 marks) · Differentiation
3. Q3Stretch of y = ln x (multiple choice) (1 marks) · Exponentials and logarithms
4. Q4Small-angle approximation for cos 2x (multiple choice) (1 marks) · Trigonometry
5. Q5Trapezium rule with 6 ordinates (4 marks) · Numerical methods
6. Q6Log equation with exactly one solution (5 marks) · Exponentials and logarithms
7. Q7Rationalising a surd expression (4 marks) · Algebra and functions
8. Q8Integration by parts with sin 4x (6 marks) · Integration, Trigonometry
9. Q9Midpoint and perpendicular bisector (9 marks) · Coordinate geometry
10. Q10Points on y = sin x and a transformation (8 marks) · Trigonometry
11. Q11Recurrence relation with a constant p (9 marks) · Sequences and series
12. Q12Theme park ride modelled with cosine (8 marks) · Trigonometry, Differentiation
13. Q13Arccos function and its domain (9 marks) · Algebra and functions, Trigonometry
14. Q14Integrating 2^x and an area (11 marks) · Integration, Exponentials and logarithms
15. Q15Implicit differentiation, stationary point (9 marks) · Differentiation
16. Q16Partial fractions and filling a container (14 marks) · Algebra and functions, Integration