HL AI · Practice Paper 1 · Interactive

IB Maths Applications & Interpretation HL — Interactive Practice Paper 1

HL AI Paper 1 is GDC-allowed short-response — matrices, chi-squared testing, graph theory, coupled & differential equations, financial maths. Two modes: per-part feedback, or a full timed exam.

30 marks · 5 questions · GDC required · 3 s.f. answers
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Question 1

6 marks
Matrices — determinant, inverse, linear systems

Let $A = \begin{pmatrix} 3 & 1 \\ 2 & -1 \end{pmatrix}$.

(a) Find $\det(A)$. [1]

(b) Write down the entries of $A^{-1}$ as decimals. Reading left-to-right, top-to-bottom, type each entry. [2]

(c) Use $A^{-1}$ to solve the system $3x + y = 5,\;2x - y = 0$. [3]

(a) $\det(A) = 3(-1) - (1)(2) = \mathbf{-5}$.
(b) $A^{-1} = \dfrac{1}{-5}\begin{pmatrix} -1 & -1 \\ -2 & 3 \end{pmatrix} = \begin{pmatrix} 0.2 & 0.2 \\ 0.4 & -0.6 \end{pmatrix}$.
(c) $\mathbf{x} = A^{-1}\mathbf{b} = \begin{pmatrix} 0.2 & 0.2 \\ 0.4 & -0.6 \end{pmatrix} \begin{pmatrix} 5 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$. Check: $3(1)+2 = 5$ ✓, $2(1)-2 = 0$ ✓.

Question 2

7 marks
Chi-squared goodness-of-fit test

A researcher surveys 200 IB students about their favourite Maths paper. She hypothesises that preferences are equally split between Paper 1, Paper 2 and Paper 3. The observed counts are:

P1P2P3
Observed557867

(a) State the expected count for each paper under the null hypothesis. Give your answer to 3 s.f. [1]

(b) Calculate the chi-squared test statistic $\chi^{2}_{\text{calc}}$. Give to 3 s.f. [3]

(c) State the number of degrees of freedom. [1]

(d) Given the critical value at the 5% level with 2 d.o.f. is $\chi^{2}_{\text{crit}} = 5.991$, state whether the null hypothesis should be rejected. Type reject or accept. [2]

(a) Under $H_0$ (uniform), expected $= 200/3 \approx \mathbf{66.7}$.
(b) $\chi^{2}_{\text{calc}} = \dfrac{(55-66.67)^2}{66.67} + \dfrac{(78-66.67)^2}{66.67} + \dfrac{(67-66.67)^2}{66.67} \approx 2.04 + 1.93 + 0.00 = \mathbf{3.97}$.
(c) $\mathrm{d.o.f.} = 3 - 1 = \mathbf{2}$.
(d) $3.97 < 5.991$, so accept (fail to reject) $H_0$ — no significant evidence against uniform preference.

Question 3

6 marks
Graph theory — minimum spanning tree & shortest path

Four cities $A,B,C,D$ have the following direct road lengths (km):

edgeABACBCBDCD
km12851015

(a) Using Kruskal's algorithm, find the total length of the minimum spanning tree. [3]

km

(b) Find the length of the shortest path from $A$ to $D$. [2]

km

(c) Type the vertex sequence of that shortest path (e.g. ABD). [1]

(a) Sorted edges: $BC(5),\,AC(8),\,BD(10),\,AB(12),\,CD(15)$. Add $BC$ (5), $AC$ (8), $BD$ (10) — all four vertices connected. Total $= 5+8+10 = \mathbf{23}$ km.
(b) Possible $A\to D$ routes: $A\to B\to D = 22$; $A\to C\to D = 23$; $A\to B\to C\to D = 32$; $A\to C\to B\to D = 23$. Shortest $= \mathbf{22}$ km.
(c) Path: $\mathbf{A \to B \to D}$.

Question 4

5 marks
Differential equations — Euler's method

The differential equation $\dfrac{dy}{dx} = x + y$ has initial condition $y(0) = 1$. Use Euler's method with step $h = 0.2$ to estimate the following:

(a) $y(0.2)$. Give your answer to 3 d.p. [2]

(b) $y(0.4)$. Give your answer to 3 d.p. [3]

Euler recursion: $y_{n+1} = y_n + h \cdot f(x_n, y_n)$ with $f(x, y) = x + y$.
(a) $y(0.2) = 1 + 0.2(0 + 1) = \mathbf{1.200}$.
(b) $y(0.4) = 1.2 + 0.2(0.2 + 1.2) = 1.2 + 0.28 = \mathbf{1.480}$.

Question 5

6 marks
Financial mathematics — compound & continuous

Alex invests €5000 in a savings account paying 4% per year, compounded quarterly.

(a) Find the value of the investment after 3 years, to the nearest cent (2 d.p.). [3]

(b) Find the interest earned. [1]

(c) Suppose instead the €5000 earns 4% p.a. compounded continuously. Find the value after 3 years, to the nearest cent. [2]

(a) $V = 5000\bigl(1 + \tfrac{0.04}{4}\bigr)^{4 \cdot 3} = 5000(1.01)^{12} \approx \mathbf{€5634.13}$ (GDC TVM: $N=12,\;I=4,\;P/Y=C/Y=4$).
(b) Interest $= 5634.13 - 5000 = \mathbf{€634.13}$.
(c) $V = 5000 e^{0.04 \cdot 3} = 5000 e^{0.12} \approx \mathbf{€5637.48}$.

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