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 calculator Paper 2, calculator

Difficulty

Easy Medium Hard

Question 1

Paper 1, no calculatorMedium7 marks

(a)

Show that $3^3-2^3=3(2)^2+3(2)+1$

[2]

(b)

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

[4]

(c)

Hence determine $21^3-20^3$

[1]

Question 2

Paper 1, no calculatorHard6 marks

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

(a)

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

[1]

(b)

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

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 $u^3+v^3+w^3$ for the numbers in the stem.

[1]

Question 3

Paper 1, no calculatorHard6 marks

(a)

Use mathematical induction to prove that $5$ divides $2^n+3^n$ for every positive odd integer $n$

[4]

(b)

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

[2]

Question 4

Paper 1, no calculatorMedium12 marks

Let

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

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

(a)

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

[2]

(b)

Write $f(x)$ in partial fractions.

[4]

(c)

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

(i)

Hence determine an antiderivative $F$ of $f$

[3]

(ii)

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

[3]

Question 5

Paper 1, no calculatorMedium7 marks

Consider the equation

z^3=8(\cos\pi+i\sin\pi).

(a)

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

[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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