AQA A Level Maths November 2021 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

   Given that $f(x)=\frac{x+1}{x-2}$, $x\ne 2$, find an expression for $ff(x)$ in its simplest form.

   Worked solution

   1. M1 for substituting $f(x)$ into itself: $ff(x)=\frac{\frac{x+1}{x-2}+1}{\frac{x+1}{x-2}-2}$.
   2. M1 for multiplying the numerator and the denominator by $(x-2)$ to clear the nested fractions.
   3. M1 for obtaining the expanded fraction $\frac{(x+1)+(x-2)}{(x+1)-2(x-2)}$.
   4. A1 for fully simplifying to $\frac{2x-1}{5-x}$ (or $\frac{1-2x}{x-5}$).

2. Trigonometry · 5 marks

   Solve $2{\sin }^{2}x+3\cos x-3=0$ for $0^{\circ }\le x\le {360}^{\circ }$.

   Worked solution

   1. M1 for substituting ${\sin }^{2}x$ with $1-{\cos }^{2}x$ to get $2(1-{\cos }^{2}x)+3\cos x-3=0$.
   2. M1 for expanding and rearranging to form the quadratic $2{\cos }^{2}x-3\cos x+1=0$.
   3. M1 for factorising the quadratic as $(2\cos x-1)(\cos x-1)=0$.
   4. A1 for finding the solutions from $\cos x=0.5$ giving $x={60}^{\circ }$ and ${300}^{\circ }$.
   5. A1 for finding the solutions from $\cos x=1$ giving $x=0^{\circ }$ and ${360}^{\circ }$.

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: 15 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 1 to 15 marks.
• Biggest question: Q9, 15 marks: plastics production fitted to an exponential model.
• Marks by main topic: Differentiation 27, Trigonometry 19, Exponentials and logarithms 15, Algebra and functions 9, Sequences and series 8, Integration 8, Numerical methods 7, Coordinate geometry 6, Proof 1.
• More Exponentials and logarithms in other sittings: June 2022 Q3 (1 marks), June 2023 Q6 (5 marks), June 2018 Q10 (8 marks).
• Against the other 5 sittings we mapped: more marks on Exponentials and logarithms (15 marks here, 3 on average) and Differentiation (27 marks here, 19 on average); fewer on Sequences and series (8 marks here, 17 on average) and Algebra and functions (9 marks here, 18 on average).
• Not on this paper: 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. Q1Solving a quadratic inequality (multiple choice) (1 marks) · Algebra and functions
2. Q2Derivative of ln(5x) (multiple choice) (1 marks) · Differentiation, Exponentials and logarithms
3. Q3Sum to infinity of a scaled series (1 marks) · Sequences and series
4. Q4Proof by contradiction setup (multiple choice) (1 marks) · Proof
5. Q5Perpendicular line through a point (6 marks) · Coordinate geometry
6. Q6Arithmetic series with S21 given (7 marks) · Sequences and series
7. Q7Change of sign for a cubic equation (7 marks) · Numerical methods
8. Q8Equation in sin^2 and sin 2x via cot (9 marks) · Trigonometry
9. Q9Plastics production fitted to an exponential model (15 marks) · Exponentials and logarithms
10. Q10Quotient rule for tan x, then an area (8 marks) · Differentiation, Integration
11. Q11Separable differential equation through a point (8 marks) · Integration
12. Q12Implicit differentiation and a tangent (8 marks) · Differentiation
13. Q13Factor theorem with a cubic (8 marks) · Algebra and functions
14. Q14Parametric curve: tangent and turning point (10 marks) · Differentiation, Coordinate geometry
15. Q15Small-angle approximation in an area (10 marks) · Trigonometry, Integration