Anglo-Chinese Junior College H2 Math papers, with video solutions

For events $A$ and $B$, it is given that $P(A\cup B)=0.8$ and $P(\left. A \right|B)=0.5$. It is also given that events $A$ and $B$ are independent.

1. Find $\mathrm{P(}B\mathrm{)}$.

   [2]

A third event $C$ is such that events $A$ and $C$ are independent and events $B$ and $C$ are independent.

2. Given also that $\mathrm{P}(C)=0.5$, find exactly the maximum and minimum possible values of $\mathrm{P}(A\cap B\cap C)$.

   [3]

Use differentiation to find the area of the largest rectangle with sides parallel to the coordinate axes, lying above the \(x\)- axis and below the curve with equation \(y=44+4x-{{x}^{2}}\).[5]

Find the general solution of the differential equation $\frac{\mathrm{d}m}{\mathrm{d}v}=\frac{{{m}^{2}}+m+1}{m+1}$.

[5]

The random variable $X$ is the number of successes in $n$ independent trials of an experiment in which the probability of a success at any trial is $p$.

Denoting $\mathrm{P}(X=k)$ by ${{p}_{k}}$, it is given that $\frac{{{p}_{k}}}{{{p}_{k-1}}}=\frac{(n-k+1)p}{k(1-p)}$, $k=1$, $2$, …, $n$.

A distribution is said to be bimodal if it has two modes.

Find the least value of $n$, and the corresponding modes of $X$, given that $X$ is bimodal and that $p=\frac{18}{25}$.

Figure 1 shows a methane molecule consisting of a carbon atom with four hydrogen atoms symmetrically placed around it. Figure 2 shows the tetrahedron structure of the methane molecule with the centre of the hydrogen atoms represented by points \(Q\), \(R\), \(S\) and \(T\) and the centre of the carbon atom represented by point \(V\).

The points \(Q\), \(R\) and \(S\) has coordinates \(\left( 8,1,8 \right)\), \(\left( 8,7,2 \right)\) and \(\left( 2,1,2 \right)\) respectively and form an equilateral triangle.

1. [4]
2. Find a cartesian equation of the plane \({{p}_{1}}\) which passes through the midpoint of \(QR\) and is perpendicular to \(QR\).[2]

Plane \({{p}_{2}}\) which passes through the midpoint of \(RS\)and is perpendicular to \(RS\) has equation \(x+y=9\).

3. Find a vector equation of the line \(l\) where \({{p}_{1}}\) and \({{p}_{2}}\) meet.[1]
4. The point \(T\) is on the line \(l\) such that \(QRST\) is a regular tetrahedron with \(QR=QT\).
   Show that the possible coordinates for the point \(T\) is \(\left( 2,7,8 \right)\). Hence, or otherwise, find the coordinates of a point on plane \(p\) that is the closest to point \(T\).[5]
5. Given that \(TV\) is \(\frac{3}{4}\) of the vertical height of tetrahedron \(QRST\). Find the coordinates of point \(V\) and hence show that the bonding angle \(TVQ\) of the methane molecule is \(109.5{}^\circ \) (correct to 1 decimal place).[4]

1. A student defines a function \(\mathrm{f}\) and its inverse \({{\mathrm{f}}^{-1}}\) by the following:
   \(\mathrm{f}:x\mapsto \sin x,\) \(\qquad\,\,\,\) \(x\in \mathbb{R},\) \(\quad\) \(-\pi \le x\le \pi \)
   \({{\mathrm{f}}^{-1}}:x\mapsto {{\sin }^{-1}}x,\) \(\;\;\) \(x\in \mathbb{R},\) \(\;\;\) \(-1\le x\le 1\).
   With the aid of a sketch, explain what is wrong with the definition of \(\mathrm{f}\), and suggest a suitable restriction to its domain such that the definition of \({{\mathrm{f}}^{-1}}\) is correct.[2]
2. The function \(\mathrm{g}\) is defined by \(\mathrm{g}:x\mapsto {{e}^{-{{\left( x+1 \right)}^{2}}}}\), \(x\in \mathbb{R}\), \(-1\le x\le 0\). Find the inverse function \({{\mathrm{g}}^{-1}}\) in similar form.[3]
3. Explain if the composite function \({{\mathrm{g}}^{-1}}{{\mathrm{f}}^{-1}}\) exists.[1]
4. Write down \({{\mathrm{f}}^{-1}}{{\mathrm{g}}^{-1}}\left( x \right)\)and its domain, and find its range. Hence, or otherwise, find the value of \(x\) such that \(\mathrm{gf}\left( x \right)=\frac{1}{e}\).[4]