A Level Pure Maths Question Booklet

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