Pure Mathematics Practice Pack: Past Paper Style Questions
11th Grade Pure Mathematics Practice Pack
Past paper style questions with full mark scheme, written for 11th grade students aged 16
to 17. Total: 205 marks. Suggested time: 3 hours 25 minutes in total, best sat as four term
papers.
Instructions
These questions are original, written in the style of Grade 11 pure mathematics exams and
AS-level past papers (Edexcel, AQA, OCR). They are not reproductions of real exam
papers. Attempt every question in exam conditions first, then mark yourself using the
mark scheme at the end.
Show all working. Method marks (M) are awarded for a correct approach even if the
final answer is wrong.
Give exact answers unless asked for decimals. Otherwise give answers to 3 significant
figures.
Calculators are allowed, but write down any formula you use.
The mark for each part is shown in square brackets, e.g. [3].
How to Use This Pack at Your Level
Every question is tagged Foundation, Core or Stretch, so you can start where you are and
work up.
Level Who it suits What to attempt Marks Support to use
Foundation Finding Grade 11 Q1, Q2, Q7, Q10, Q12, 53 Hints, glossary,
maths hard, or Q19, Q22, Q25 and as much time
catching up from as you need
Grade 10
Core Following the All Foundation and 181 Hints only after a
standard Grade 11 Core questions first attempt
course and wanting
exam practice
Stretch Aiming for top marks The whole pack 205 Mark scheme only
or preparing for timed, then the plus 16 once you have
Grade 12 and extension problems finished
university maths
Page 1 of 20
,Pure Mathematics Practice Pack: Past Paper Style Questions
Move up a level when you score about 75% at your current one. At Foundation that is
about 40 out of 53. Core questions are Q3, Q4, Q5, Q6, Q8, Q9, Q11, Q13, Q16, Q20, Q21,
Q23, Q24, Q26, Q27 and Q28. Stretch questions are Q14, Q15, Q17 and Q18. Sections F, G
and L (differentiation, integration and kinematics, Q13 to Q17, Q27 and Q28) are
introductory calculus, which some Grade 11 courses leave until Grade 12. Skip them if your
class has not covered them yet.
Term-by-Term Map
Terms 1 to 4 run from algebra, through functions and geometry, to trigonometry and
exponentials, and finish with calculus and proof. This follows a standard four-term order. If
your school sequences topics differently, use the Topics column to find the right
questions.
Term Topics Questions Marks
Term 1 Surds and indices; quadratics and Q1, Q2, Q3, Q19, Q20, Q21 46
completing the square; inequalities;
polynomials and the factor theorem;
simultaneous equations and the
discriminant; modulus and graph sketching
Term Functions and transformations; coordinate Q4, Q5, Q6, Q7, Q8, Q12, 57
2 geometry and circles; sequences and Q25, Q26
series; binomial expansion; vectors
Term Trigonometry (equations, identities, sine Q9, Q10, Q11, Q22, Q23, Q24 42
3 and cosine rules, radians, graphs);
exponentials and logarithms
Term Differentiation, integration and Q13 to Q18, Q27, Q28 60
4 kinematics; proof; full-paper revision
Sit each term's questions as a short paper at the end of that term. Then sit the whole pack
at the end of term 4 as a mock exam.
Contents
1. Section A: Algebra and Functions (Q1 to Q4)
2. Section B: Coordinate Geometry (Q5 to Q6)
3. Section C: Sequences, Series and the Binomial Expansion (Q7 to Q8)
4. Section D: Trigonometry (Q9 to Q10)
Page 2 of 20
, Pure Mathematics Practice Pack: Past Paper Style Questions
5. Section E: Exponentials, Logarithms and Graphs (Q11 to Q12)
6. Section F: Differentiation (Q13 to Q15)
7. Section G: Integration (Q16 to Q17)
8. Section H: Proof (Q18)
9. Section I: Surds, Indices and Simultaneous Equations (Q19 to Q21)
10. Section J: Radians and Trigonometric Graphs (Q22 to Q24)
11. Section K: Vectors (Q25 to Q26)
12. Section L: More Differentiation, Integration and Kinematics (Q27 to Q28)
13. Extension problems, hints, mark scheme and worked solutions
14. Formula reference, plain-English glossary and self-assessment
Section A: Algebra and Functions
Question 1 [8 marks] · Foundation
f(x) = 3x² − 12x + 7
(a) Write f(x) in the form a(x + b)² + c, where a, b and c are constants. [3]
(b) State the minimum value of f(x) and the value of x at which it occurs. [2]
(c) Hence solve f(x) = 0, giving your answers as exact values. [3]
Question 2 [7 marks] · Foundation
(a) Solve the inequality 3x − 7 < 5x + 1. [2]
(b) Solve the inequality x² − x − 6 > 0. [3]
(c) Hence find the set of values of x for which both inequalities are true. [2]
Question 3 [6 marks] · Core
f(x) = 2x³ − 3x² − 11x + 6
(a) Use the factor theorem to show that (x − 3) is a factor of f(x). [2]
(b) Hence factorise f(x) completely. [3]
(c) Write down all the solutions of f(x) = 0. [1]
Question 4 [9 marks] · Core
The functions f and g are defined for all real x by
f(x) = 2x + 3 and g(x) = x² − 1
Page 3 of 20
11th Grade Pure Mathematics Practice Pack
Past paper style questions with full mark scheme, written for 11th grade students aged 16
to 17. Total: 205 marks. Suggested time: 3 hours 25 minutes in total, best sat as four term
papers.
Instructions
These questions are original, written in the style of Grade 11 pure mathematics exams and
AS-level past papers (Edexcel, AQA, OCR). They are not reproductions of real exam
papers. Attempt every question in exam conditions first, then mark yourself using the
mark scheme at the end.
Show all working. Method marks (M) are awarded for a correct approach even if the
final answer is wrong.
Give exact answers unless asked for decimals. Otherwise give answers to 3 significant
figures.
Calculators are allowed, but write down any formula you use.
The mark for each part is shown in square brackets, e.g. [3].
How to Use This Pack at Your Level
Every question is tagged Foundation, Core or Stretch, so you can start where you are and
work up.
Level Who it suits What to attempt Marks Support to use
Foundation Finding Grade 11 Q1, Q2, Q7, Q10, Q12, 53 Hints, glossary,
maths hard, or Q19, Q22, Q25 and as much time
catching up from as you need
Grade 10
Core Following the All Foundation and 181 Hints only after a
standard Grade 11 Core questions first attempt
course and wanting
exam practice
Stretch Aiming for top marks The whole pack 205 Mark scheme only
or preparing for timed, then the plus 16 once you have
Grade 12 and extension problems finished
university maths
Page 1 of 20
,Pure Mathematics Practice Pack: Past Paper Style Questions
Move up a level when you score about 75% at your current one. At Foundation that is
about 40 out of 53. Core questions are Q3, Q4, Q5, Q6, Q8, Q9, Q11, Q13, Q16, Q20, Q21,
Q23, Q24, Q26, Q27 and Q28. Stretch questions are Q14, Q15, Q17 and Q18. Sections F, G
and L (differentiation, integration and kinematics, Q13 to Q17, Q27 and Q28) are
introductory calculus, which some Grade 11 courses leave until Grade 12. Skip them if your
class has not covered them yet.
Term-by-Term Map
Terms 1 to 4 run from algebra, through functions and geometry, to trigonometry and
exponentials, and finish with calculus and proof. This follows a standard four-term order. If
your school sequences topics differently, use the Topics column to find the right
questions.
Term Topics Questions Marks
Term 1 Surds and indices; quadratics and Q1, Q2, Q3, Q19, Q20, Q21 46
completing the square; inequalities;
polynomials and the factor theorem;
simultaneous equations and the
discriminant; modulus and graph sketching
Term Functions and transformations; coordinate Q4, Q5, Q6, Q7, Q8, Q12, 57
2 geometry and circles; sequences and Q25, Q26
series; binomial expansion; vectors
Term Trigonometry (equations, identities, sine Q9, Q10, Q11, Q22, Q23, Q24 42
3 and cosine rules, radians, graphs);
exponentials and logarithms
Term Differentiation, integration and Q13 to Q18, Q27, Q28 60
4 kinematics; proof; full-paper revision
Sit each term's questions as a short paper at the end of that term. Then sit the whole pack
at the end of term 4 as a mock exam.
Contents
1. Section A: Algebra and Functions (Q1 to Q4)
2. Section B: Coordinate Geometry (Q5 to Q6)
3. Section C: Sequences, Series and the Binomial Expansion (Q7 to Q8)
4. Section D: Trigonometry (Q9 to Q10)
Page 2 of 20
, Pure Mathematics Practice Pack: Past Paper Style Questions
5. Section E: Exponentials, Logarithms and Graphs (Q11 to Q12)
6. Section F: Differentiation (Q13 to Q15)
7. Section G: Integration (Q16 to Q17)
8. Section H: Proof (Q18)
9. Section I: Surds, Indices and Simultaneous Equations (Q19 to Q21)
10. Section J: Radians and Trigonometric Graphs (Q22 to Q24)
11. Section K: Vectors (Q25 to Q26)
12. Section L: More Differentiation, Integration and Kinematics (Q27 to Q28)
13. Extension problems, hints, mark scheme and worked solutions
14. Formula reference, plain-English glossary and self-assessment
Section A: Algebra and Functions
Question 1 [8 marks] · Foundation
f(x) = 3x² − 12x + 7
(a) Write f(x) in the form a(x + b)² + c, where a, b and c are constants. [3]
(b) State the minimum value of f(x) and the value of x at which it occurs. [2]
(c) Hence solve f(x) = 0, giving your answers as exact values. [3]
Question 2 [7 marks] · Foundation
(a) Solve the inequality 3x − 7 < 5x + 1. [2]
(b) Solve the inequality x² − x − 6 > 0. [3]
(c) Hence find the set of values of x for which both inequalities are true. [2]
Question 3 [6 marks] · Core
f(x) = 2x³ − 3x² − 11x + 6
(a) Use the factor theorem to show that (x − 3) is a factor of f(x). [2]
(b) Hence factorise f(x) completely. [3]
(c) Write down all the solutions of f(x) = 0. [1]
Question 4 [9 marks] · Core
The functions f and g are defined for all real x by
f(x) = 2x + 3 and g(x) = x² − 1
Page 3 of 20
