Anderson Junior College H2 Math papers, with video solutions

A farmer claims that the mean weight of the melons grown in his farm is at least \(1.2\) kg. A random sample of \(10\) melons is chosen and the weight, \(x\) kg, of each melon is recorded. The results are as follows:

\[ \begin{matrix} 1.037 \quad & 0.914 \quad & 1.019 \quad & 1.234 \quad & 1.110 \\ 1.417 \quad & 1.008 \quad & 0.846 \quad & 1.105 \quad & 1.331 \end{matrix} \]

1. By combining the \(2\) sets of data, find unbiased estimates of the population mean and population variance of the weight of a melon grown by this farmer.[2]
2. The farmer decides to change the claim to “the mean weight of the melons grown in his farm is at least \(m\) kg”. Based on the combined sample of size \(50\), find the set of values of \(m\) for which the farmer’s claim is not rejected at the \(5\%\) significance level.[3]

The graph of \(y=\sqrt{x+40}\) is shown in the diagram below.

1. By considering the shaded rectangle, show that
   \(\sqrt{n+40}<\displaystyle\displaystyle\int\limits_{n}^{n+1}{\sqrt{x+40}\mathrm{ \; d}x}\).[1]
2. Deduce that
   \(\sqrt{1}+\sqrt{2}+\sqrt{3}+...+\sqrt{80}<\displaystyle\displaystyle\int\limits_{-40}^{41}{\sqrt{x+40}\mathrm{\; d}x}\).[2]
3. Show also that
   \(\sqrt{n+41}>\displaystyle\displaystyle\int\limits_{n}^{n+1}{\sqrt{x+40}\mathrm{\; d}x}\).[2]
4. Deduce that
   \(\sqrt{1}+\sqrt{2}+\sqrt{3}+...+\sqrt{81}>\displaystyle\displaystyle\int\limits_{-40}^{41}{\sqrt{x+40}\mathrm{\; d}x}\).[1]
5. Hence, deduce that the value \(a\), where \(a\in \mathbb{Z}\), that satisfies the following inequality,
   \(9a<\sqrt{1}+\sqrt{2}+\sqrt{3}+...+\sqrt{81}<9\left( a+1 \right)\).[2]

The diagram shows a shaded region $R$ bounded by the curve ${{\left( y-2 \right)}^{2}}=x+1$ and the line $y+2x=6$.

Find the volume generated when $R$ is rotated through $2\pi $ radians about the $x$-axis, leaving your answer correct to $3$ significant figures.

[4]

2. A curve is defined by the parametric equations
   $x=a{{t}^{2}},\,y=2at.$
   Show that the equation of the normal to the curve at the point $(a{{t}^{2}},\,2at)$ is at $y+tx=a{{t}^{3}}+2at.$
   If the normal to the curve at the point $(a{{t}^{2}},\,2at)$ meets the curve again at the point $(a{{q}^{2}},\,2aq),$ show that ${{t}^{2}}+qt+2=0.$
   Deduce that ${{q}^{2}}$ cannot be less than $8$.