Year 13 Pure Maths Booklet
Contents
Page 2: Proof
Page 4: Algebraic and partial fractions
Page 6: Functions and Modelling
Page 9: Sequences and Series
Page 11: The Binomial Theorem
Page 13: Trigonometry
Page 17: Parametric Equations
Page 21: Differentiation
Page 24: Numerical Methods
Page 28: Integration (part 1)
Page 32: Integration (Part 2)
Page 37: Vectors
1
,Proof
1 Prove by exhaustion that for positive integers from 1 to 6 inclusive. (3 marks)
2 Use proof by contradiction to prove the statement: ‘The product of two odd numbers is
odd.’
3 Prove by contradiction that if n is odd, n3 + 1 is even. (5 )
4 Use proof by contradiction to show that there exist no integers a and b for which
2
,25a + 15b = 1.
5 Use proof by contradiction to show that there is no greatest positive rational number.
6 Use proof by contradiction to show that, given a rational number a and an irrational
number b, a − b is irrational.
7 Use proof by contradiction to show that there are no positive integer solutions to the
statement
8 a Use proof by contradiction to show that if n2 is an even integer then n is also an even
integer.
3
, b Prove that is irrational.
9 Prove by contradiction that there are infinitely many prime numbers.
Algebraic and partial fractions
1 Given that
find the values of the constants A, B and C, where A, B and C are integers.
2 Show that can be written in the form
Find the values of the constants A and B.
4
Contents
Page 2: Proof
Page 4: Algebraic and partial fractions
Page 6: Functions and Modelling
Page 9: Sequences and Series
Page 11: The Binomial Theorem
Page 13: Trigonometry
Page 17: Parametric Equations
Page 21: Differentiation
Page 24: Numerical Methods
Page 28: Integration (part 1)
Page 32: Integration (Part 2)
Page 37: Vectors
1
,Proof
1 Prove by exhaustion that for positive integers from 1 to 6 inclusive. (3 marks)
2 Use proof by contradiction to prove the statement: ‘The product of two odd numbers is
odd.’
3 Prove by contradiction that if n is odd, n3 + 1 is even. (5 )
4 Use proof by contradiction to show that there exist no integers a and b for which
2
,25a + 15b = 1.
5 Use proof by contradiction to show that there is no greatest positive rational number.
6 Use proof by contradiction to show that, given a rational number a and an irrational
number b, a − b is irrational.
7 Use proof by contradiction to show that there are no positive integer solutions to the
statement
8 a Use proof by contradiction to show that if n2 is an even integer then n is also an even
integer.
3
, b Prove that is irrational.
9 Prove by contradiction that there are infinitely many prime numbers.
Algebraic and partial fractions
1 Given that
find the values of the constants A, B and C, where A, B and C are integers.
2 Show that can be written in the form
Find the values of the constants A and B.
4
