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

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Question 1

Paper 2, calculatorMedium12 marks

A small hydroelectric station records its annual electricity generation to the nearest GWh. Figure 1 shows the recorded generation in years 1 to 6.

Let $v_n$ be the generation, in GWh, predicted for year $n$ by an arithmetic sequence model. The model is chosen so that its first and sixth terms match the recorded generation in years 1 and 6.

(a)

Calculate the increase in recorded generation from year 2 to year 3, and from year 5 to year 6.

[2]

(b)

Determine the first term $a$ and common difference $d$ of the model.

[3]

(c)

Hence, estimate the generation predicted by the model for year 9.

[2]

(d)

Write the total generation predicted by the model over the first nine years in sigma notation, and determine its value.

[3]

(e)

Comment on the suitability of this model for estimating generation in year 9, referring to the plotted data.

[2]

Question 2

Paper 2, calculatorMedium6 marks

A convergent geometric series has first term 8 and common ratio $r$ The sum of its first three terms is 14.

(a)

Determine $r$

[4]

(b)

Hence determine the sum to infinity of the series.

[2]

An employer models an employee’s salary at the start of each year using a geometric sequence. The salary at the start of year 1 was $40,000, and the salary at the start of year 5 was $48,620.25. Let $u_n$ denote the modelled salary, in dollars, at the start of year $n$

(a)

State the first term and determine the positive common ratio.

[2]

(b)

Determine the modelled salary at the start of year 9, to the nearest cent.

[2]

(c)

Hence write the total of the modelled salaries for the first nine years as a sum in sigma notation, and determine this total to the nearest cent.

[2]

(d)

State one reason why this model may become unreliable when used to predict salaries many years into the future.

[1]

Question 4

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 5

Paper 2, calculatorMedium12 marks

Let $S$ be a set of 9 distinct elements. Let $N$ be the number of subsets of $S$ that contain exactly 4 elements, and define

k=\left(1+\frac{1}{N}\right)^{1/2}.

(a)

Determine $N$

[2]

(b)

Find the first four non-zero terms, in descending powers of $N$ in the binomial expansion of $k$

[4]

(c)

Hence use the first three non-zero terms of this expansion to estimate $k$ Give your answer to 9 significant figures.

[2]

(d)

Use your GDC to determine $k$ and hence find the absolute difference between this value and your estimate in part (c), correct to 2 significant figures.

[3]

(e)

State whether your estimate in part (c) gives $k$ correctly to 3 decimal places.

[1]

Question 6

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 7

Paper 2, calculatorMedium7 marks

Let $u=-8+8\sqrt{3}\,i$

(a)

Express $u$ in the form $r(\cos\theta+i\sin\theta)$ where $r>0$ and $0\leq\theta<2\pi$

[2]

(b)

Hence, or otherwise, determine both square roots of $u$ giving your answers in the form $a+bi$

[3]

(c)

A polynomial with real coefficients has $2+2\sqrt{3}\,i$ as a root. State the other root in its conjugate pair and hence determine the monic quadratic factor associated with this pair.

[2]

Question 8

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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