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:

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 $8$.