Edexcel IAL P3 January 2023 mark scheme and question paper (WMA13/01)

3 P3 questions like these, with worked solutions

Written and checked by us, not taken from the Pearson paper. Try each one, then tap "Worked solution" to see every step and its marks.

1. Further Differentiation · 4 marks

   Differentiate $y=\frac{e^{2x}}{x^2+1}$ with respect to $x$, giving your answer as a single algebraic fraction in its simplest form.

   Worked solution

   1. M1 for correctly identifying $u=e^{2x}$ and $v=x^2+1$, and their derivatives $2e^{2x}$ and $2x$.
   2. M1 for substituting into the quotient rule: $\frac{(x^2+1)2e^{2x}-e^{2x}(2x)}{(x^2+1{)}^2}$.
   3. M1 for factorising out $2e^{2x}$ in the numerator.
   4. A1 for the correct simplified expression: $\frac{2e^{2x}(x^2-x+1)}{(x^2+1{)}^2}$.

2. Further Integration · 4 marks

   Find $\int (\cos x-\sin x{)}^{2}dx$.

   Worked solution

   1. M1 for expanding the bracket to ${\cos }^{2}x-2\sin x\cos x+{\sin }^{2}x$.
   2. M1 for simplifying to $1-\sin 2x$.
   3. M1 for integrating the expression term by term.
   4. A1 for $x+\frac{1}{2}\cos 2x+c$.

3. Exponentials and Logarithms · 4 marks

   Solve the equation $e^{2x}+4e^x-12=0$, giving your answer(s) in exact form.

   Worked solution

   1. M1 for identifying the hidden quadratic and factorising: $(e^x+6)(e^x-2)=0$.
   2. M1 for extracting the roots $e^x=2$ and $e^x=-6$.
   3. M1 for rejecting $e^x=-6$ since an exponential function must be positive.
   4. A1 for the exact valid solution $x=\ln 2$.

More on these topics: Further Differentiation · Further Integration · Exponentials and Logarithms

What's on this paper

What this paper is like

• Shape: 10 questions, 75 marks in 1 h 30 min (about 1.2 minutes a mark); questions from 5 to 11 marks.
• Biggest question: Q9, 11 marks: tangent to a curve with e^x and an iteration.
• Marks by main topic: Further Differentiation 20, Trigonometric Addition Formulae 15, Functions 14, Exponentials and Logarithms 14, Algebraic Methods 7, Further Integration 5.
• More Further Differentiation in other sittings: October 2022 Q6 (6 marks), June 2023 Q8 (9 marks), June 2022 Q8 (12 marks).
• Against the other 14 sittings we mapped: more marks on Trigonometric Addition Formulae (15 marks here, 10 on average).
• Not on this paper: Numerical Methods, Trigonometric Functions.
• Calculator: allowed.

WMA13/01: question by question

Labels and topics are ours, read from the WMA13/01 paper (checked 2026-11-01); marks as printed. Tap a number to practise that topic.

1. Q1Range and composite functions (6 marks) · P3 Functions
2. Q2R sin(x + a) form (6 marks) · P3 Trigonometric Addition Formulae
3. Q3Linear graph of log y (5 marks) · P3 Exponentials and Logarithms
4. Q4Algebraic division and partial fractions (7 marks) · P3 Algebraic Methods
5. Q5Proving an identity, then solving (9 marks) · P3 Trigonometric Addition Formulae
6. Q6Modulus graph with a constant a (8 marks) · P3 Functions
7. Q7Quotient rule with integer constants (9 marks) · P3 Further Differentiation
8. Q8Integrating a product of sin and cos (5 marks) · P3 Further Integration
9. Q9Tangent to a curve with e^x and an iteration (11 marks) · P3 Further Differentiation, P3 Numerical Methods
10. Q10Fruit fly population model (9 marks) · P3 Exponentials and Logarithms

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