2018 ACJC P1 Q11

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. Find a cartesian equation of the plane \(p\) which passes through the points \(Q\), \(R\) and \(S\).[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]