Moderate -0.5 This is a choice question where students select the easier option. The mechanics alternative involves standard moment of inertia calculations using parallel axis theorem and simple harmonic motion formulas—routine A-level Further Maths techniques. The statistics alternative requires standard two-sample t-tests with straightforward calculations. Both are multi-step but follow well-practiced procedures without requiring novel insight, making this slightly easier than average for Further Maths.
Answer only one of the following two alternatives.
EITHER
\includegraphics{figure_10a}
An object is formed by attaching a thin uniform rod \(PQ\) to a uniform rectangular lamina \(ABCD\). The lamina has mass \(m\), and \(AB = DC = 6a\), \(BC = AD = 3a\). The rod has mass \(M\) and length \(3a\). The end \(P\) of the rod is attached to the mid-point of \(AB\). The rod is perpendicular to \(AB\) and in the plane of the lamina (see diagram). Show that the moment of inertia of the object about a smooth horizontal axis \(l_1\), through \(Q\) and perpendicular to the plane of the lamina, is \(3(8m + M)a^2\). [4]
Show that the moment of inertia of the object about a smooth horizontal axis \(l_2\), through the mid-point of \(PQ\) and perpendicular to the plane of the lamina, is \(\frac{3}{4}(17m + M)a^2\). [2]
Find expressions for the periods of small oscillations of the object about the axes \(l_1\) and \(l_2\), and verify that these periods are equal when \(m = M\). [8]
OR
A farmer \(A\) grows two types of potato plants, Royal and Majestic. A random sample of 10 Royal plants is taken and the potatoes from each plant are weighed. The total mass of potatoes on a plant is \(x\) kg. The data are summarised as follows.
$$\Sigma x = 42.0 \qquad \Sigma x^2 = 180.0$$
A random sample of 12 Majestic plants is taken. The total mass of potatoes on a plant is \(y\) kg. The data are summarised as follows.
$$\Sigma y = 57.6 \qquad \Sigma y^2 = 281.5$$
Test, at the 5% significance level, whether the population mean mass of potatoes from Royal plants is the same as the population mean mass of potatoes from Majestic plants. You may assume that both distributions are normal and you should state any additional assumption that you make. [9]
A neighbouring farmer \(B\) grows Crown potato plants. His plants produce 3.8 kg of potatoes per plant, on average. Farmer \(A\) claims that her Royal plants produce a higher mean mass of potatoes than Farmer \(B\)'s Crown plants. Test, at the 5% significance level, whether Farmer \(A\)'s claim is justified. [5]
Answer only one of the following two alternatives.
\textbf{EITHER}
\includegraphics{figure_10a}
An object is formed by attaching a thin uniform rod $PQ$ to a uniform rectangular lamina $ABCD$. The lamina has mass $m$, and $AB = DC = 6a$, $BC = AD = 3a$. The rod has mass $M$ and length $3a$. The end $P$ of the rod is attached to the mid-point of $AB$. The rod is perpendicular to $AB$ and in the plane of the lamina (see diagram). Show that the moment of inertia of the object about a smooth horizontal axis $l_1$, through $Q$ and perpendicular to the plane of the lamina, is $3(8m + M)a^2$. [4]
Show that the moment of inertia of the object about a smooth horizontal axis $l_2$, through the mid-point of $PQ$ and perpendicular to the plane of the lamina, is $\frac{3}{4}(17m + M)a^2$. [2]
Find expressions for the periods of small oscillations of the object about the axes $l_1$ and $l_2$, and verify that these periods are equal when $m = M$. [8]
\textbf{OR}
A farmer $A$ grows two types of potato plants, Royal and Majestic. A random sample of 10 Royal plants is taken and the potatoes from each plant are weighed. The total mass of potatoes on a plant is $x$ kg. The data are summarised as follows.
$$\Sigma x = 42.0 \qquad \Sigma x^2 = 180.0$$
A random sample of 12 Majestic plants is taken. The total mass of potatoes on a plant is $y$ kg. The data are summarised as follows.
$$\Sigma y = 57.6 \qquad \Sigma y^2 = 281.5$$
Test, at the 5% significance level, whether the population mean mass of potatoes from Royal plants is the same as the population mean mass of potatoes from Majestic plants. You may assume that both distributions are normal and you should state any additional assumption that you make. [9]
A neighbouring farmer $B$ grows Crown potato plants. His plants produce 3.8 kg of potatoes per plant, on average. Farmer $A$ claims that her Royal plants produce a higher mean mass of potatoes than Farmer $B$'s Crown plants. Test, at the 5% significance level, whether Farmer $A$'s claim is justified. [5]
\hfill \mbox{\textit{CAIE FP2 2015 Q10 [28]}}