IB Maths AA HL Topic 1: Number and algebra 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 calculatorMedium12 marks

Let

f(x)=x+8x2+x−6,x∈R,f(x)=\frac{x+8}{x^2+x-6}, \qquad x\in\mathbb R,

where f(x)f(x) is defined only when its denominator is non-zero.

(a)

Write x2+x−6x^2+x-6 as a product of linear factors, and state the values of xx for which f(x)f(x) is undefined.

[2]
(b)

Write f(x)f(x) in partial fractions.

[4]
(c)

For parts (i) and (ii), consider −3<x<2-3<x<2

(i)

Hence determine an antiderivative FF of ff

[3]
(ii)

Hence evaluate ∫01f(x) dx\displaystyle\int_0^1 f(x)\,dx giving your answer exactly. Explain why the sign of your answer is consistent with the sign of f(x)f(x) on [0,1][0,1]

[3]

Question 3

Paper 1, no calculatorMedium7 marks

Consider the equation

z3=8(cos⁡π+isin⁡π).z^3=8(\cos\pi+i\sin\pi).
(a)

Determine all distinct solutions, giving each in the form a+bia+bi

[4]
(b)

Hence determine the monic quadratic polynomial with real coefficients whose roots are the two non-real solutions of the equation.

[3]

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