IB Maths AI exam readiness check

Question 1

Layla invests USD 6 000 in an account paying 4.2% nominal annual interest, compounded monthly.

(a) Find the value of the investment after 5 years, to the nearest cent.

(b) Find the number of complete years before the investment first exceeds USD 9 000.

Accepted Answer

(a) $6000\left(1 + \dfrac{0.042}{12}ight)^{60} = 6000(1.0035)^{60} = 	extbf{USD } 7\,399.42$.

(b) Solve $6000(1.0035)^{12n} = 9000 \Rightarrow (1.0035)^{12n} = 1.5$.

$12n = \dfrac{\ln 1.5}{\ln 1.0035} = 116.1 \Rightarrow n = 9.67$ years.

So after $	extbf{10}$ complete years the investment first exceeds USD 9 000.

Question 2

A study records the hours of revision $x$ and the test score $y$ for eight students. The GDC gives $r = 0.874$ and the regression line $y = 4.62x + 31.5$.

(a) Describe the correlation.

(b) Use the model to estimate the score of a student who revises for 6 hours.

(c) Explain why the model should not be used to estimate the score of a student who revises for 25 hours.

Accepted Answer

(a) $r = 0.874$: strong positive linear correlation.

(b) $y = 4.62(6) + 31.5 = 59.22 \approx 59.2$.

(c) 25 hours is well outside the range of the recorded data, so using the line there is extrapolation. The model predicts $147$, which is not a possible score.

Question 3

A triangular plot has sides $AB = 48$ m, $BC = 65$ m and angle $ABC = 108°$.

(a) Find the length $AC$, to 3 significant figures.

(b) Find the area of the plot, to the nearest square metre.

Accepted Answer

(a) $AC^{2} = 48^{2} + 65^{2} - 2(48)(65)\cos 108° = 2304 + 4225 + 1928.6 = 8457.6$

$AC = 91.96... \approx 91.9$ m (3 s.f.)

(b) Area $= 	frac12(48)(65)\sin 108° = 1483.8 \approx 	extbf{1484 m²}$.

Question 4

A $3 	imes 4$ contingency table is tested for independence at the 5% significance level. The GDC gives $\chi^{2}_{calc} = 11.3$ and the critical value is $12.592$.

(a) State the null and alternative hypotheses.

(b) Find the number of degrees of freedom.

(c) State, with a reason, the conclusion of the test.

Accepted Answer

(a) $H_0$: the two variables are independent. $H_1$: the two variables are not independent.

(b) $
u = (3 - 1)(4 - 1) = 6$.

(c) $\chi^{2}_{calc} = 11.3 < 12.592$, so we do not reject $H_0$: there is insufficient evidence at the 5% level that the variables are dependent.

Question 5

An open-topped box with a square base of side $x$ cm has volume $500$ cm³.

(a) Show that the external surface area is $A = x^{2} + \dfrac{2000}{x}$.

(b) Find the value of $x$ that minimises $A$, and the minimum surface area, each to 3 significant figures.

Accepted Answer

(a) $x^{2}h = 500 \Rightarrow h = \dfrac{500}{x^{2}}$.

Open box: $A = x^{2} + 4xh = x^{2} + 4x\cdot\dfrac{500}{x^{2}} = x^{2} + \dfrac{2000}{x}$.

(b) $\dfrac{dA}{dx} = 2x - \dfrac{2000}{x^{2}} = 0 \Rightarrow x^{3} = 1000 \Rightarrow x = 10$ cm.

$A_{min} = 100 + 200 = 300$ cm² (and $\dfrac{d^{2}A}{dx^{2}} > 0$, so it is a minimum).

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