Meridian Junior College H2 Math papers, with video solutions

2. A manufacturer produces closed hollow cans. The top part is a hemisphere made of tin and the bottom part is a cylinder made of aluminium of cross-sectional radius \(r\) cm and \(h\) cm. There is no material between the cylinder and the hemisphere so that any fluid can move freely within the container. At the beginning of an experiment, a similar-shaped can of dimensions \(r=4\)and \(h=10\), is filled to its capacity with water. Due to a hole at its base, water is leaking at a constant rate of 2 \(\mathrm{c}{{\mathrm{m}}^{3}}{{\mathrm{s}}^{-1}}\) when the can is standing upright. Find the exact rate at which the height of the water is decreasing 80 seconds after the start of the experiment.[5]
   

A point $A$ with position vector $\overrightarrow{OA}=\alpha \,\mathbf{i}+\beta \,\mathbf{j}+\gamma \,\mathbf{k},$ where $\alpha ,\,\beta $ and $\gamma $ are real constants, has direction cosines $\cos \theta ,\,\,\cos \phi $ and $\cos \delta ,$ where $\theta ,\,\phi $ and $\delta $ are the angles $\overrightarrow{OA}$ makes with the positive $x,\,y$ and $z$-axes respectively.

1. Express the direction cosines $\cos \theta ,\,\,\cos \phi $ and $\cos \delta $ in terms of $\alpha ,\,\beta $ and $\gamma .$ Hence find the value of ${{\cos }^{2}}\theta +{{\cos }^{2}}\phi +{{\cos }^{2}}\delta .$
2. The vector $\mathbf{d}$ makes angle of $45{}^\circ $ with the $x$-axis, $60{}^\circ $ with the $y$-axis and $\delta $ with the $z$-axis, where $0{}^\circ \le \delta \le 90{}^\circ .$ Find the value of $\delta .$ If the magnitude of $\mathbf{d}$ is $12,$ express $\mathbf{d}$ in cartesian form.

The curve $C$ has equation $y=\sqrt{5{{x}^{2}}+4}$.

1. Sketch curve $C$, indicating clearly the axial intercepts, the equations of the asymptotes and the coordinates of the stationary points.
2. Hence, by inserting a suitable graph, determine the range of values of $h$, where $h$ is a positive constant, such that the equation $\sqrt{5{{x}^{2}}+4}=h\sqrt{(1-{{x}^{2}})}$ has no real roots.

The lines ${{l}_{1}}$ and ${{l}_{2}}$ have equations $\mathbf{r}=\overrightarrow{OA}+\lambda \left( \begin{matrix} 1 \\-1 \\5 \\\end{matrix} \right)$ and $\mathbf{r}=\overrightarrow{OB}+\mu \left( \begin{matrix} 2 \\1 \\ 1 \\\end{matrix} \right)$, $\lambda $, $\mu \in \mathbb{R}$ respectively, where $\overrightarrow{OA}=\left( \begin{matrix} 1 \\ 2 \\ -1 \\\end{matrix} \right)$, $\overrightarrow{OB}=\left( \begin{matrix} 2 \\ 4 \\-5 \\\end{matrix} \right)$. The lines ${{l}_{1}}$ and ${{l}_{2}}$ intersect at point $C$.

1. Find the position of vector of $C$.

   [3]

2. Given that $\overrightarrow{AB}$ is perpendicular to ${{l}_{2}}$, find the equation of the reflection of ${{l}_{1}}$ in ${{l}_{2}}$.

   [4]