2025 YIJC P1 Q10
2025 YIJC P1 Q10
Junior College 2
10 marks
Planes \(p\) and \(q\) are perpendicular. Plane \(p\) has equation \(\mathbf{r} \cdot (\mathbf{i} + 2\mathbf{j} - \mathbf{k}) = 6\). Plane \(q\) contains the line \(l\) with equation \(\mathbf{r} = \mathbf{j} - 12\mathbf{k} + \lambda(4\mathbf{i} + \mathbf{j} + 10\mathbf{k})\) where \(\lambda\) is a parameter. The point \(A\) has coordinates \((0, 1, -12)\).
- Find a cartesian equation of \(q\).[2]
- Find a vector equation of the line \(m\), where \(p\) and \(q\) meet.[2]
\(l\) passes through \(p\) at point \(B\). Point \(C\) is the foot of the perpendicular of \(A\) to \(p\).
- Find the position vector of \(B\).[2]
- Find the position vector of \(C\).[2]
- Find the exact area of \(\Delta ABC\).[2]
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Answer:(a) \(q:3x-2y-z=10\) (b) \(\mathbf r=\begin{pmatrix}4\\1\\0\end{pmatrix}+\mu\begin{pmatrix}2\\1\\4\end{pmatrix}\) (c) \(\overrightarrow{OB}=\begin{pmatrix}8\\3\\8\end{pmatrix}\) (d) \(\overrightarrow{OC}=-\frac13\begin{pmatrix}4\\5\\32\end{pmatrix}\) (e) \(\frac{28\sqrt{14}}3\) units\({}^2\)
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