IB Maths AA HL AA 1.3 Proof questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper
Paper 1, no calculatorPaper 2, calculator
Difficulty
EasyMediumHard

Question 1

Paper 1, no calculatorMedium7 marks
(a)

Show that 33−23=3(2)2+3(2)+13^3-2^3=3(2)^2+3(2)+1

[2]
(b)

For x∈Rx\in\mathbb R prove by starting with the left-hand side and transforming it that (x+1)3−x3(x+1)^3-x^3 simplifies to 3x2+3x+13x^2+3x+1 In your final line, use the appropriate symbol to indicate that this is an identity.

[4]
(c)

Hence determine 213−20321^3-20^3

[1]

Question 2

Paper 1, no calculatorHard6 marks

For real numbers u,v,wu,v,w it is given that u+v+w=0u+v+w=0 and uvw=5uvw=5

(a)

Write down the value of u+v+wu+v+w

[1]
(b)

For arbitrary real numbers u,v,wu,v,w starting with the left-hand side, prove that

u3+v3+w3−3uvw≡(u+v+w)(u2+v2+w2−uv−vw−wu).u^3+v^3+w^3-3uvw \equiv (u+v+w)(u^2+v^2+w^2-uv-vw-wu).

In your final line, use ≡\equiv to indicate an identity.

[4]
(c)

Hence determine u3+v3+w3u^3+v^3+w^3 for the numbers in the stem.

[1]

Question 3

Paper 1, no calculatorHard6 marks
(a)

Use mathematical induction to prove that 55 divides 2n+3n2^n+3^n for every positive odd integer nn

[4]
(b)

A student conjectures that 2n+3n2^n+3^n is prime for every positive integer nn Use your result from part (a) to disprove this conjecture, explaining why your example is a counterexample.

[2]

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