Practise

Question bank

Original exam-style questions, mapped to your syllabus. Answers are checked for mathematical equivalence and every attempt updates your mastery.

TopicSkillDifficulty

12 questions

SA-PROOF2-0011 markL1 FoundationCalculatorThe assumption

To prove by contradiction that there is no largest even number, you should start by assuming

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SA-PROOF2-0021 markL1 FoundationCalculatorThe assumption

Assume

N

is the largest even number. Which fact gives the contradiction?

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SA-PROOF2-0071 markL1 FoundationCalculatorClassic proofs

A rational number is one that can be written as

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SA-PROOF-0071 markL2 StandardCalculatorClassic proofs

A proof by contradiction that

\sqrt2

is irrational begins with which assumption?

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SA-PROOF2-0031 markL2 StandardNon-calculatorClassic proofs

In proving

\sqrt3

irrational, assume

\sqrt3=\dfrac ab

in lowest terms, so

a^2=3b^2

and

a=3c

. Substituting gives

9c^2=3b^2

, so

b^2=kc^2

. Find

k

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SA-PROOF2-0041 markL2 StandardCalculatorThe assumption

To prove "if

n^2

is odd then

n

is odd" by contradiction, assume

n^2

is odd and

n

is even,

n=2k

. Which line gives the contradiction?

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SA-PROOF2-0051 markL2 StandardNon-calculatorClassic proofs

To prove there are no integers

a,b

with

6a+9b=1

, write

6a+9b=m(2a+3b)

. Find

m

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SA-PROOF2-0082 marksL2 StandardNon-calculatorThe assumption

To prove

x^2-6x+10>0

for all real

x

, assume

x^2-6x+10\le0

and write it as

(x-3)^2+q\le0

. Find

q

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SA-PROOF-0081 markL3 ExamCalculatorClassic proofs

In the proof that there are infinitely many primes, the list

p_1,p_2,\dots,p_n

is assumed to contain every prime. Which number is then considered?

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SA-PROOF2-0061 markL3 ExamCalculatorClassic proofs

To show

x^2-y^2=2

has no integer solutions, write

(x-y)(x+y)=2

. Why is this impossible?

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SA-PROOF2-0092 marksL3 ExamNon-calculatorClassic proofs

Suppose the only primes were

2,3,5,7

. Find Euclid's number

N=2\times3\times5\times7+1

, and use it to see why the list must be incomplete.

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SA-PROOF-0091 markL4 ChallengingCalculatorThe assumption

To prove "if

n^2

is even, then

n

is even" by contradiction, assume

n

is odd,

n=2k+1

. Which step gives the contradiction?

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