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 calculatorEasy5 marks

The concentration CC in mg L−1\text{mg L}^{-1} of a dye in a tank tt hours after treatment begins is modelled by

C(t)=32(14)t.C(t)=32\left(\frac14\right)^t.
(a)

Find the concentration after 32\frac32 hours.

[2]
(b)

Determine the time at which the concentration reaches 3 mg L−13\text{ mg L}^{-1} Use logarithms and give your answer exactly in terms of ln⁡\ln including units.

[3]

Question 2

Paper 2, calculatorEasy6 marks

A laboratory measures the mass of two layers in a material sample as 6.40×10−8 kg6.40\times10^{-8}\,\text{kg} and 3.60×10−9 kg3.60\times10^{-9}\,\text{kg}

(a)

Determine the combined mass of the two layers. Give your answer in scientific notation.

[2]
(b)

A batch contains five identical samples, each with the combined mass found in part (a).

Determine the total mass of the batch. Give your answer in scientific notation.

[2]
(c)

The storage limit for the batch is 5.00×10−7 kg5.00\times10^{-7}\,\text{kg}

Using your answer to part (b), determine the mass by which the batch is below this limit. Give your answer in scientific notation.

[2]

Question 3

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 4

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]

Question 5

Paper 2, calculatorEasy6 marks
(a)

Expand (1+2x)3(1+2x)^3

[2]
(b)

For integer rr where 0≤r≤60\leq r\leq 6 use your GDC to generate a table of (6r)\binom{6}{r} Hence determine the value of rr for which (6r)=20\binom{6}{r}=20 Using the formula for (6r)\binom{6}{r} verify your answer.

[4]

Question 6

Paper 1, no calculatorEasy6 marks

Let

R(x)=3x−2(x−4)(x+1).R(x)=\frac{3x-2}{(x-4)(x+1)}.
(a)

Express R(x)R(x) in partial fractions.

[3]
(b)

Hence determine the exact value of

∫01R(x) dx.\int_0^1 R(x)\,dx.
[3]

Question 7

Paper 1, no calculatorEasy6 marks

The rational function

R(x)=3x−1(x−3)(x+1)R(x)=\frac{3x-1}{(x-3)(x+1)}

is used as an integrand.

(a)

State the real values of xx for which R(x)R(x) is undefined.

[1]
(b)

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

[3]
(c)

Hence determine an antiderivative of R(x)R(x)

[2]

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