AQA A Level Maths Specimen 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 · 6 marks

   The circle $C$ has centre $(a,b)$ and passes through the points $P(0,0)$, $Q(6,0)$, and $R(0,8)$. Find the values of $a$, $b$, and the radius of $C$.

   Worked solution

   $P$ and $Q$ both on circle ⇒ centre lies on the perpendicular bisector of $PQ$, so $a=3$   M1 A1. Similarly $P$ and $R$ give $b=4$   M1 A1. Radius $=|CP|=\sqrt{9+16}=5$   M1 A1.

2. Exponentials and logarithms · 4 marks

   Given that ${\log }_{2}x=p$ and ${\log }_{2}y=q$, express ${\log }_2(8x^2{y}^3)$ in terms of $p$ and $q$.

   Worked solution

   1. M1 for splitting the logarithm into a sum: ${\log }_{2}8+{\log }_2{x}^2+{\log }_2{y}^3$.
   2. B1 for evaluating ${\log }_{2}8=3$.
   3. M1 for applying the power law: $2{\log }_{2}x+3{\log }_{2}y$.
   4. A1 for the final expression $3+2p+3q$.

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: Coordinate geometry · Exponentials and logarithms · Vectors

What this paper covers

What this paper is like

• Shape: 17 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 1 to 10 marks.
• Biggest question: Q10, 10 marks: range and inverse of a function.
• Marks by main topic: Differentiation 25, Trigonometry 22, Algebra and functions 20, Integration 19, Coordinate geometry 9, Proof 5.
• More Algebra and functions in other sittings: June 2018 Q12 (10 marks), June 2019 Q6 (8 marks), November 2020 Q13 (15 marks).
• Against the other 6 sittings we mapped: more marks on Integration (19 marks here, 10 on average) and Trigonometry (22 marks here, 17 on average); fewer on Sequences and series (0 marks here, 15 on average) and Numerical methods (0 marks here, 8 on average).
• Not on this paper: Sequences and series, Exponentials and logarithms, 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. Q1Gradient of a line (multiple choice) (1 marks) · Coordinate geometry
2. Q2Differentiating a root (multiple choice) (1 marks) · Differentiation
3. Q3Small-angle approximation (3 marks) · Trigonometry
4. Q4Factor theorem and a cubic (6 marks) · Algebra and functions
5. Q5Sector area and perimeter (8 marks) · Trigonometry
6. Q6Differential equation for weight loss (4 marks) · Integration
7. Q7Quadratic with equal roots and a substitution (4 marks) · Algebra and functions
8. Q8Exact value of a definite integral (7 marks) · Integration
9. Q9Increasing function (8 marks) · Differentiation
10. Q10Range and inverse of a function (10 marks) · Algebra and functions
11. Q11Circle centre and radius, then a tangent (8 marks) · Coordinate geometry
12. Q12Prism sculpture with an angle (8 marks) · Trigonometry
13. Q13Proving an identity with cot and cos (3 marks) · Trigonometry
14. Q14Minimising the material for a fish tank (10 marks) · Differentiation
15. Q15Differential equation for a fountain's height (8 marks) · Integration, Trigonometry
16. Q16Disproving a claim about rational and irrational numbers (5 marks) · Proof
17. Q17Differentiation from first principles for sin x (6 marks) · Differentiation, Trigonometry