OCR A Level Mathematics A
Pure Mathematics – Complete Notes
By Rujul Nayak (University of Cambridge O er-Holder)
Section 1: Proof 2
Section 2: Algebra and Functions 5
Section 3: Coordinate Geometry 16
Section 4: Sequences and Series 17
Section 5: Trigonometry 21
Section 6: Exponentials and Logarithms 27
Section 7: Di erentiation 28
Section 8: Integration 35
Section 9: Numerical Methods 43
Section 10: Vectors 45
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, OCR A Level Mathematics A Notes by Rujul Nayak
Section 1: Proof
Notation for Proofs
A ⟹ B is the statement “A implies B”
B ⟹ A is the converse of “A implies B”
A′ ⟹ B′ is the inverse of “A implies B”
B′ ⟹ A′ is the contrapositive of “A implies B”
A statement is logically equivalent to its contrapositive, but not linked to its converse or inverse.
Also, A ⟺ B means “A is true if and only if B is true”
More symbols:
≡ is the symbol for equivalency
∃ means “there exists”
∀ means “for all values of”
Types of Numbers
The integers are the set of all positive whole numbers, zero and all negative whole numbers.
The real numbers are any number on the number line.
The rational numbers are any number that can be expressed as a fraction a /b, where a and b are
both integers (and b ≠ 0 ) and do not share any factors other than 1 (if this is true, we say that
they are coprime).
The irrational numbers are real numbers which are not rational.
Proof by Deduction
Proof by deduction (also called direct proof) involves making simple logical statements to come
to a conclusion.
Example:
Prove that the di erence between any two consecutive odd square numbers is a multiple of 8.
Solution:
Let the two consecutive odd square numbers be (2m − 1)2 and (2m + 1)2.
Expanded, these numbers are (4m 2 − 4m + 1) and (4m 2 + 4m + 1).
Their di erence is 8m, which is clearly a multiple of 8.
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, OCR A Level Mathematics A Notes by Rujul Nayak
Proof by Exhaustion
Proof by exhaustion involves breaking the proof down into di erent categories and proving each
one separately.
Example:
Prove that all square numbers leave a remainder of 0, 1 or 4 when divided by 5.
Solution:
The last digit of n 2 depends solely on the last digit of n.
Last digit of n Last digit of n 2
0 0
1 1
2 4
3 9
4 6
5 5
6 6
7 9
8 4
9 1
All last digits are either 0 , 1 , 4 , 5 , 6 or 9 . These last digits leave remainders of only 0 , 1 and 4
when divided by 5.
Proof by Contradiction
Proof by contradiction involves assuming that the statement is false, then showing that the
assumption leads to a logical contradiction, which means that the statement must be true.
You will need to be familiar with the following two proofs for the exam.
Example:
Prove that there are an in nite number of prime numbers.
Solution:
Assume that there is a nite set of primes, {2,3,5,…, P}.
Then, we can construct a number Q = (2 × 3 × 5 × … × P) + 1.
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, OCR A Level Mathematics A Notes by Rujul Nayak
Q is not a multiple of any of the primes in our nite set, so it must either be prime itself, or have a
prime factor which is not in the set.
Therefore there are other primes which are not in our set, so it is impossible for the set to be nite.
This means that there is an in nite number of prime numbers.
Example:
Prove that 2 is irrational.
Solution:
a
Assume that 2 is rational, i.e. that it can be written as the ratio for some coprime integers
b
a and b.
a2
Therefore, = 2 ⟹ a 2 = 2b 2
b 2
This means that a 2 must be even, which means that a is even.
This means that a 2 is a multiple of 4, which means that b 2 is even, meaning that b is also even.
Therefore, a and b share a common factor of 2, and are therefore not coprime. This means that
2 cannot be written as the ratio of two coprime integers, so it is irrational.
This argument works for the square root of any prime number, not just 2.
Page 4 of 46
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Pure Mathematics – Complete Notes
By Rujul Nayak (University of Cambridge O er-Holder)
Section 1: Proof 2
Section 2: Algebra and Functions 5
Section 3: Coordinate Geometry 16
Section 4: Sequences and Series 17
Section 5: Trigonometry 21
Section 6: Exponentials and Logarithms 27
Section 7: Di erentiation 28
Section 8: Integration 35
Section 9: Numerical Methods 43
Section 10: Vectors 45
ff ff
, OCR A Level Mathematics A Notes by Rujul Nayak
Section 1: Proof
Notation for Proofs
A ⟹ B is the statement “A implies B”
B ⟹ A is the converse of “A implies B”
A′ ⟹ B′ is the inverse of “A implies B”
B′ ⟹ A′ is the contrapositive of “A implies B”
A statement is logically equivalent to its contrapositive, but not linked to its converse or inverse.
Also, A ⟺ B means “A is true if and only if B is true”
More symbols:
≡ is the symbol for equivalency
∃ means “there exists”
∀ means “for all values of”
Types of Numbers
The integers are the set of all positive whole numbers, zero and all negative whole numbers.
The real numbers are any number on the number line.
The rational numbers are any number that can be expressed as a fraction a /b, where a and b are
both integers (and b ≠ 0 ) and do not share any factors other than 1 (if this is true, we say that
they are coprime).
The irrational numbers are real numbers which are not rational.
Proof by Deduction
Proof by deduction (also called direct proof) involves making simple logical statements to come
to a conclusion.
Example:
Prove that the di erence between any two consecutive odd square numbers is a multiple of 8.
Solution:
Let the two consecutive odd square numbers be (2m − 1)2 and (2m + 1)2.
Expanded, these numbers are (4m 2 − 4m + 1) and (4m 2 + 4m + 1).
Their di erence is 8m, which is clearly a multiple of 8.
Page 2 of 46
 
ff ff
, OCR A Level Mathematics A Notes by Rujul Nayak
Proof by Exhaustion
Proof by exhaustion involves breaking the proof down into di erent categories and proving each
one separately.
Example:
Prove that all square numbers leave a remainder of 0, 1 or 4 when divided by 5.
Solution:
The last digit of n 2 depends solely on the last digit of n.
Last digit of n Last digit of n 2
0 0
1 1
2 4
3 9
4 6
5 5
6 6
7 9
8 4
9 1
All last digits are either 0 , 1 , 4 , 5 , 6 or 9 . These last digits leave remainders of only 0 , 1 and 4
when divided by 5.
Proof by Contradiction
Proof by contradiction involves assuming that the statement is false, then showing that the
assumption leads to a logical contradiction, which means that the statement must be true.
You will need to be familiar with the following two proofs for the exam.
Example:
Prove that there are an in nite number of prime numbers.
Solution:
Assume that there is a nite set of primes, {2,3,5,…, P}.
Then, we can construct a number Q = (2 × 3 × 5 × … × P) + 1.
Page 3 of 46
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, OCR A Level Mathematics A Notes by Rujul Nayak
Q is not a multiple of any of the primes in our nite set, so it must either be prime itself, or have a
prime factor which is not in the set.
Therefore there are other primes which are not in our set, so it is impossible for the set to be nite.
This means that there is an in nite number of prime numbers.
Example:
Prove that 2 is irrational.
Solution:
a
Assume that 2 is rational, i.e. that it can be written as the ratio for some coprime integers
b
a and b.
a2
Therefore, = 2 ⟹ a 2 = 2b 2
b 2
This means that a 2 must be even, which means that a is even.
This means that a 2 is a multiple of 4, which means that b 2 is even, meaning that b is also even.
Therefore, a and b share a common factor of 2, and are therefore not coprime. This means that
2 cannot be written as the ratio of two coprime integers, so it is irrational.
This argument works for the square root of any prime number, not just 2.
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