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-PROOF-0012 marksL1 FoundationCalculatorCounterexample

Find the smallest positive integer

n

for which

2^n+1

is not prime.

Salam Academy Original Practice · Proposed mark scheme

SA-PROOF-0021 markL1 FoundationCalculatorCounterexample

Which value of

x

is a counterexample to "if

x^2>4

then

x>2

Salam Academy Original Practice · Proposed mark scheme

SA-PROOF1-0021 markL1 FoundationCalculatorCounterexample

Which pair is a counterexample to "the sum of two prime numbers is always even"?

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SA-PROOF1-0031 markL1 FoundationNon-calculatorDeduction & exhaustion

To prove by exhaustion that no square number ends in the digit

7

, you check the last digit of

n^2

for each possible last digit of

n

. How many cases are there?

Salam Academy Original Practice · Proposed mark scheme

SA-PROOF-G012 marksL1 FoundationCalculatorCounterexample

The statement "

n^2-n+1

is prime for every integer

n\ge2

" is false. Find the smallest integer

n\ge2

that is a counterexample.

Salam Academy Original Practice · Proposed mark scheme

SA-PROOF-Q011 markL1 FoundationCalculatorCounterexample

How many counterexamples do you need to disprove a statement?

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SA-PROOF-Q021 markL1 FoundationCalculatorDeduction & exhaustion

Checking that a statement is true for

n=1

n=100

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SA-PROOF-0031 markL2 StandardCalculatorDeduction & exhaustion

Which line correctly starts a proof that the sum of any three consecutive integers is divisible by

3

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SA-PROOF-0041 markL2 StandardCalculatorDeduction & exhaustion

To prove that the sum of two odd numbers is even, the two odd numbers should be written as

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SA-PROOF-0051 markL2 StandardCalculatorDeduction & exhaustion

To prove by exhaustion that every square number leaves remainder

0

1

when divided by

3

, how many cases for

n

do you need?

Salam Academy Original Practice · Proposed mark scheme

SA-PROOF1-0012 marksL2 StandardNon-calculatorDeduction & exhaustion

For every integer

n

(n+1)^2-(n-1)^2=kn

. Find

k

, which proves the difference is always a multiple of

k

Salam Academy Original Practice · Proposed mark scheme

SA-PROOF-0061 markL3 ExamCalculatorDeduction & exhaustion

In the proof above, the case

n=3k+2

gives

n^2=9k^2+12k+4=3(3k^2+4k+1)+r

. Find

r

Salam Academy Original Practice · Proposed mark scheme