AQA A Level Maths November 2020 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 · 7 marks

   Express ${\displaystyle \frac{x^3+1}{x^2-1}}$ in the form $Ax+B+{\displaystyle \frac{C}{x-1}}+{\displaystyle \frac{D}{x+1}}$ and find $A,B,C,D$.

   Worked solution

   Polynomial division: $\frac{x^3+1}{x^2-1}=x+\frac{x+1}{x^2-1}=x+\frac{1}{x-1}$ (since $\frac{x+1}{(x+1)(x-1)}=\frac{1}{x-1}$)   M1 M1 A1. So $A=1$, $B=0$, $C=1$, $D=0$   A1 A1 A1 A1.

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. Numerical methods · 6 marks

   The equation $\cos x=x$ has a unique root $\alpha$ in $[0,1]$. Using the iteration $x_{n+1}=\cos x_n$ with $x_0=0.5$, find $x_1$, $x_2$, $x_3$, and $x_4$ each to $4$ d.p. (Angle in radians.)

   Worked solution

   $x_1=\cos 0.5=0.8776$   M1 A1. $x_2=\cos 0.8776=0.6390$   A1. $x_3=\cos 0.6390=0.8027$   A1. $x_4=\cos 0.8027=0.6948$   A1 A1.

More on these topics: Algebra and functions · Trigonometry · Numerical methods

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: Q13, 15 marks: inverse of a rational function and its range.
• Marks by main topic: Algebra and functions 27, Numerical methods 19, Sequences and series 17, Integration 14, Differentiation 13, Trigonometry 8, Proof 2.
• More Algebra and functions in other sittings: November 2021 Q13 (8 marks), June 2019 Q6 (8 marks), June 2022 Q11 (10 marks).
• Against the other 5 sittings we mapped: more marks on Algebra and functions (27 marks here, 14 on average) and Numerical methods (19 marks here, 6 on average); fewer on Trigonometry (8 marks here, 19 on average) and Differentiation (13 marks here, 22 on average).
• Not on this paper: Coordinate geometry, Exponentials and logarithms, 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. Q1Binomial expansion with a fractional power (2 marks) · Sequences and series
2. Q2Change of sign and a misleading interval (1 marks) · Numerical methods
3. Q3Sector perimeter (multiple choice) (1 marks) · Trigonometry
4. Q4Modulus graph and an inequality (5 marks) · Algebra and functions
5. Q5Proof by exhaustion with powers (2 marks) · Proof
6. Q6Spotting errors in four attempts at an integral (4 marks) · Integration
7. Q7Recurrence relation terms (4 marks) · Sequences and series
8. Q8Trig model for hours of darkness (7 marks) · Trigonometry
9. Q9Correcting a partial fractions attempt (7 marks) · Algebra and functions
10. Q10Arithmetic series in sigma notation (11 marks) · Sequences and series
11. Q11Trapezium rule for the area under ln(8 - x) (9 marks) · Numerical methods, Integration
12. Q12Implicit differentiation with sin y and cos y (13 marks) · Differentiation, Trigonometry
13. Q13Inverse of a rational function and its range (15 marks) · Algebra and functions
14. Q14Newton-Raphson on a function with a square root (9 marks) · Numerical methods, Differentiation
15. Q15Area between two exponential curves (10 marks) · Integration, Exponentials and logarithms