A Level Math Revision

Edexcel IAL P1 Maths Revision: Pure Mathematics 1

Edexcel International A Level P1 (Pure Mathematics 1): every topic with an exam-style question and its mark scheme, and 348 practice questions with mark schemes.

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Differentiation · P1 paper · 6 marks · hard

The tangents to the curve y=x2−4x at the points where x=1 and x=3 intersect at a point P. Find the coordinates of P.

Worked answer
  1. M1 for finding the y-coordinates: at x=1,y=−3 and at x=3,y=−3.
  2. M1 for differentiating dydx=2x−4 to find the gradients: m1=−2 (at x=1) and m3=2 (at x=3).
  3. M1 for finding the equation of the first tangent: y−(−3)=−2(x−1)⟹y=−2x−1.
  4. M1 for finding the equation of the second tangent: y−(−3)=2(x−3)⟹y=2x−9.
  5. M1 for equating the two tangent equations: −2x−1=2x−9⟹4x=8⟹x=2.
  6. A1 for finding the corresponding y-coordinate y=−5, giving P(2,−5).

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P1 topics

"Tested in" counts come from our question-by-question mapping of the 4 Pearson P1 papers from June 2024 to June 2025.

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P1 past papers

Official Pearson P1 papers and mark schemes, with a timer: all IAL P1 past papers. Most recent: June 2025 · January 2025 · October 2024 · June 2024.

Examiner Insights for IAL P1: every examiner report since June 2024, the top mistakes and what to focus on next →

Knowledge organiser: Pure Mathematics 1 (P1) on one page (key facts, formulas and mistakes to avoid; printable PDF)

P1 Algebra

Indices, surds, quadratics (discriminant, completing the square), simultaneous equations, inequalities.

119 practice questions with mark schemes · Worked examples → · Notes → · Calculator steps → · Practise algebra →

Example question · 7 marks · hard

Let f(x)=5−4x−2x2.

(a) Express f(x) in the form A−B(x+C)2, where A, B, and C are integers.
(b) State the maximum value of f(x) and the value of x at which it occurs.
(c) Solve f(x)=0, giving your answers in exact form.

Show the mark scheme

(a) M1 Factors out −2 from the x terms: 5−2(x2+2x).

  1. M1 Completes the square inside the bracket: 5−2((x+1)2−1).
  2. A1 Expands and simplifies to obtain 7−2(x+1)2.

(b) B1 States the maximum value is 7.

  1. B1 States this occurs at x=−1.

(c) M1 Sets 7−2(x+1)2=0 and rearranges to (x+1)2=72.

  1. A1 Solves for x by square rooting, yielding x=−1±72 (or equivalent exact surd).

P1 Coordinate Geometry

Straight lines, gradient, midpoint, perpendicular/parallel, equation of a line, distance formula.

68 practice questions with mark schemes · Worked examples → · Practise coordinate geometry →

Example question · 8 marks · hard


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.

Show the mark scheme

  1. Gradient mAB=7−(−1)4−(−2)=86=43. A1
    Gradient mBC=4−78−4=−34. A1
    Since mAB×mBC=43×−34=−1, [AB] and [BC] are perpendicular. M1
    Thus, triangle ABC has a right angle at B. B1

  2. Length AB=(4−(−2))2+(7−(−1))2=36+64=100=10. A1
    Length BC=(8−4)2+(4−7)2=16+9=25=5. A1

  3. Since the triangle is right-angled at B, AB and BC act as the base and height. M1
    Area =12×base×height=12×10×5=25 sq units. A1

P1 Trigonometry

Sine and cosine rules; area of a triangle = ½ab sin C; radian measure, arc length and sector area; graphs of sin, cos and tan and their symmetries; exact values.

67 practice questions with mark schemes · Worked examples → · Calculator steps → · Practise trigonometry →

Example question · 5 marks · hard

Triangle ABC has sides AB=4 cm and BC=5 cm, with angle ABC=60∘. Triangle ACD shares side AC and has a right angle at C. If CD=7 cm, find the exact length of AD.

Show the mark scheme
  1. M1 for using the cosine rule in triangle ABC to find AC2: AC2=42+52−2(4)(5)cos⁡60∘.
  2. M1 for evaluating to get AC2=16+25−40(0.5)=21.
  3. M1 for recognising that triangle ACD is right-angled at C, so Pythagoras' theorem applies: AD2=AC2+CD2.
  4. M1 for substituting the known values: AD2=21+72=21+49=70.
  5. A1 for finding AD=70 cm.

P1 Differentiation

Gradient of a tangent as a limit; differentiating xⁿ and sums of such terms; gradients, tangents and normals; the second derivative.

32 practice questions with mark schemes · Worked examples → · Notes → · Calculator steps → · Practise differentiation →

Example question · 6 marks · hard

The tangents to the curve y=x2−4x at the points where x=1 and x=3 intersect at a point P. Find the coordinates of P.

Show the mark scheme
  1. M1 for finding the y-coordinates: at x=1,y=−3 and at x=3,y=−3.
  2. M1 for differentiating dydx=2x−4 to find the gradients: m1=−2 (at x=1) and m3=2 (at x=3).
  3. M1 for finding the equation of the first tangent: y−(−3)=−2(x−1)⟹y=−2x−1.
  4. M1 for finding the equation of the second tangent: y−(−3)=2(x−3)⟹y=2x−9.
  5. M1 for equating the two tangent equations: −2x−1=2x−9⟹4x=8⟹x=2.
  6. A1 for finding the corresponding y-coordinate y=−5, giving P(2,−5).

P1 Integration

Indefinite integration of xⁿ (n ≠ −1) and sums of such terms; finding the constant of integration from a point on the curve.

32 practice questions with mark schemes · Worked examples → · Calculator steps → · Practise integration →

Example question · 6 marks · hard

Given that f″(x)=12x−4, f′(1)=2 and f(−1)=5, find f(x).

Show the mark scheme
  1. M1 for integrating f″(x) to find f′(x)=6x2−4x+c1.
  2. M1 for substituting f′(1)=2 to find c1: 2=6(1)2−4(1)+c1⟹c1=0.
  3. M1 for substituting c1=0 and integrating again to find f(x)=2x3−2x2+c2.
  4. M1 for substituting f(−1)=5: 5=2(−1)3−2(−1)2+c2.
  5. M1 for solving for c2: 5=−2−2+c2⟹c2=9.
  6. A1 for the final correct function f(x)=2x3−2x2+9.

P1 Graphs and Transformations

Sketching cubics, quartics and reciprocals; transformations of graphs (translations, stretches, reflections); function notation.

30 practice questions with mark schemes · Worked examples → · Calculator steps → · Practise graphs and transformations →

Example question · 4 marks · hard

A cubic curve crosses the x-axis at x=−2, x=1, and x=4. It crosses the y-axis at the point (0,16). Find the equation of the curve in the form y=ax3+bx2+cx+d.

Show the mark scheme
  1. M1 for writing the equation in the factored form y=k(x+2)(x−1)(x−4).
  2. M1 for substituting (0,16) to find k: 16=k(2)(−1)(−4)⟹16=8k.
  3. A1 for finding k=2.
  4. A1 for expanding to get y=2(x3−3x2−6x+8)=2x3−6x2−12x+16.

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FAQ

What is on the Edexcel IAL P1 exam?

P1 Pure Mathematics 1 is assessed by a 1h 30min written paper (75 marks), calculator permitted. It covers Algebra, Coordinate Geometry, Trigonometry, Differentiation, Integration, Graphs and Transformations.

Can I use a calculator in P1?

Yes, a calculator is permitted in P1, as in every IAL maths unit. You still need to show full method to earn the M marks.

How many P1 practice questions are there?

348 exam-style P1 questions across 6 topics, each with an Edexcel-style mark scheme (M, A and B marks).

Which P1 topics come up most?

In our mapping of the 4 Pearson P1 papers from June 2024 to June 2025, Algebra (12 questions), Differentiation (7 questions), Integration (7 questions) were tested most. Every topic can be examined in any series.

Where can I find P1 past papers?

Pearson publishes them. Our IAL P1 past papers page links to each official paper and mark scheme by series, with an exam timer.

How should I revise for P1?

Read the note for a topic, study the worked example and its mark scheme, practise that topic's questions, then sit a timed P1 past paper and check the common mistakes examiners report.

What is free on this page?

Everything here, with no sign-in: one example question with its mark scheme for every P1 topic, the worked examples on the topic pages and a quick quiz.

What does a free account add?

A free account (no card) adds 5 more practice questions on every topic with instant feedback, answers and mark schemes on free questions, 3 AI-marked answers a week, free, and your streak and progress on any device. The rest of the P1 bank with its mark schemes is part of the paid plans.

Other IAL units: IAL P2 Pure Mathematics 2 · IAL P3 Pure Mathematics 3 · IAL P4 Pure Mathematics 4 · IAL S1 Statistics 1 · IAL M1 Mechanics 1

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