IB Maths Applications & Interpretation — Interactive Practice Paper 1
Paper 1 for SL AI is short-response with a GDC. Real-world data, financial calculations, geometry and statistics. Two modes: instant per-part feedback, or a full 90-minute timed exam.
Priya invests €4000 in an account paying 3.2% interest per year, compounded monthly.
(a) Find the value of her investment after 5 years. Give your answer to the nearest cent (2 d.p.). [3]
(b) How many complete years must Priya wait for her investment to first exceed €6000? [3]
(c) Comment on whether "3.2% per year, compounded monthly" is a better deal than "3.25% per year, compounded annually". Type a for monthly, b for annually. [1]
(a) $V = 4000\left(1 + \tfrac{0.032}{12}\right)^{12 \times 5} = 4000(1.00267)^{60} \approx \mathbf{€4692.87}$ (GDC TVM: $N=60,\;I=3.2,\;P/Y=C/Y=12$). (b) Solve $4000(1.00267)^{12n} > 6000 \Rightarrow 12n > \log_{1.00267}(1.5) \approx 152.2$, so $n > 12.68$; the first whole year is $\mathbf{13}$. (c) Effective annual rate of monthly plan $= (1.00267)^{12}-1 \approx 3.247\%$, which is less than $3.25\%$. So (b) annually wins by a whisker — a great discrimination question because the nominal 3.2% "sounds better with more compounding" but the maths says otherwise.
Question 2
8 marks
Bivariate data — linear regression
A café records the outside temperature $T$ (°C) and the number of iced coffees sold $N$ on 7 mornings.
$T$ (°C)
18
21
24
26
29
32
34
$N$
22
31
35
44
52
63
71
(a) Find the Pearson product-moment correlation coefficient $r$. Give your answer to 3 s.f. [2]
(b) Write the regression line of $N$ on $T$ in the form $N = aT + b$. Give $a$ and $b$ to 3 s.f. [3]
(c) Use your regression line to estimate $N$ when $T = 27$°C. Round to the nearest whole number. [2]
(d) Explain in one word whether extrapolating this model to $T = 45$°C is reliable. Type yes or no. [1]
(a) GDC linear regression: $r \approx \mathbf{0.996}$ — strong positive. (b) $a \approx \mathbf{2.93},\;b \approx \mathbf{-30.5}$, so $N = 2.93T - 30.5$. (c) $N \approx 2.93(27) - 30.5 \approx 48.6 \Rightarrow \mathbf{49}$ iced coffees. (d) $T=45$°C is outside the data range (18–34 °C) — no, extrapolation is unreliable.
Question 3
8 marks
Geometry — 3-D right triangle & volume
A grain silo consists of a right cylinder of radius $r = 3$ m and height $h = 8$ m, topped by a hemisphere of the same radius.
(a) Find the total volume of the silo. Give your answer to 3 s.f. [3]
m³
(b) Find the total outer surface area of the silo, including the flat base but excluding the top of the cylinder (which is covered by the hemisphere). Give your answer to 3 s.f. [3]
m²
(c) The silo is currently 30% full. How many m³ of grain does it contain? [2]
A logistics company has three warehouses at coordinates $A(2, 6)$, $B(8, 10)$ and $C(6, 2)$ (units in km).
(a) Find the distance $AB$. Give your answer to 3 s.f. [2]
km
(b) Find the equation of the perpendicular bisector of $AB$, giving your answer in the form $y = mx + c$ with $m$ and $c$ to 3 s.f. [3]
(c) A new warehouse $D$ is to be built at the point equidistant from all three existing warehouses (the "toxic-waste" / Voronoi vertex). Find its coordinates. Give each to 3 s.f. [3]
(a) $AB = \sqrt{6^2 + 4^2} = \sqrt{52} \approx \mathbf{7.21}$ km. (b) Midpoint of $AB = (5, 8)$; gradient of $AB = \tfrac{10-6}{8-2} = \tfrac{2}{3}$; perpendicular gradient $= -\tfrac{3}{2}$. Line: $y - 8 = -\tfrac{3}{2}(x - 5) \Rightarrow y = -1.5x + 15.5$. So $m = \mathbf{-1.5},\;c = \mathbf{15.5}$. (c) Repeat for perpendicular bisector of $BC$: midpoint $(7, 6)$, gradient of $BC = \tfrac{2-10}{6-8} = 4$, perpendicular gradient $= -\tfrac{1}{4}$; line $y = -0.25x + 7.75$. Solve simultaneously with $y = -1.5x + 15.5$: $-1.5x + 15.5 = -0.25x + 7.75 \Rightarrow -1.25x = -7.75 \Rightarrow x = \mathbf{6.2}$, and $y = -1.5(6.2) + 15.5 = \mathbf{6.2}$. Verify: distance from $(6.2, 6.2)$ to each of $A,B,C$ equals $\sqrt{17.68} \approx 4.20$ km. ✓