Challenging +1.8 This is a substantial Further Maths question requiring multiple advanced techniques: finding perpendicular vectors via cross products, calculating plane intersections with parametric lines, computing angles between planes, and working with polar coordinate arc length formulas including the derivative identity for ds/dθ. While the individual steps follow standard procedures, the multi-part nature, extended reasoning across 14+ marks, and Further Maths content place this well above average A-level difficulty.
Answer only one of the following two alternatives.
EITHER
The points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{i}\), \(2\mathbf{j}\) and \(4\mathbf{k}\) respectively, relative to an origin \(O\). The point \(N\) is the foot of the perpendicular from \(O\) to the plane \(ABC\). The point \(P\) on the line-segment \(ON\) is such that \(OP = \frac{3}{4}ON\). The line \(AP\) meets the plane \(OBC\) at \(Q\). Find a vector perpendicular to the plane \(ABC\) and show that the length of \(ON\) is \(\frac{1}{\sqrt{(21)}}\). [4]
Find the position vector of the point \(Q\). [5]
Show that the acute angle between the planes \(ABC\) and \(ABQ\) is \(\cos^{-1}\left(\frac{4}{5}\right)\). [5]
OR
The curve \(C\) has polar equation \(r = a(1 - \cos\theta)\) for \(0 \leqslant \theta < 2\pi\). Sketch \(C\). [2]
Find the area of the region enclosed by the arc of \(C\) for which \(\frac{1}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi\), the half-line \(\theta = \frac{1}{3}\pi\) and the half-line \(\theta = \frac{2}{3}\pi\). [5]
Show that
$$\left(\frac{\mathrm{d}s}{\mathrm{d}\theta}\right)^2 = 4a^2\sin^2\left(\frac{1}{2}\theta\right),$$
where \(s\) denotes arc length, and find the length of the arc of \(C\) for which \(\frac{1}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi\). [7]
Answer only one of the following two alternatives.
\textbf{EITHER}
The points $A$, $B$ and $C$ have position vectors $\mathbf{i}$, $2\mathbf{j}$ and $4\mathbf{k}$ respectively, relative to an origin $O$. The point $N$ is the foot of the perpendicular from $O$ to the plane $ABC$. The point $P$ on the line-segment $ON$ is such that $OP = \frac{3}{4}ON$. The line $AP$ meets the plane $OBC$ at $Q$. Find a vector perpendicular to the plane $ABC$ and show that the length of $ON$ is $\frac{1}{\sqrt{(21)}}$. [4]
Find the position vector of the point $Q$. [5]
Show that the acute angle between the planes $ABC$ and $ABQ$ is $\cos^{-1}\left(\frac{4}{5}\right)$. [5]
\textbf{OR}
The curve $C$ has polar equation $r = a(1 - \cos\theta)$ for $0 \leqslant \theta < 2\pi$. Sketch $C$. [2]
Find the area of the region enclosed by the arc of $C$ for which $\frac{1}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi$, the half-line $\theta = \frac{1}{3}\pi$ and the half-line $\theta = \frac{2}{3}\pi$. [5]
Show that
$$\left(\frac{\mathrm{d}s}{\mathrm{d}\theta}\right)^2 = 4a^2\sin^2\left(\frac{1}{2}\theta\right),$$
where $s$ denotes arc length, and find the length of the arc of $C$ for which $\frac{1}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi$. [7]
\hfill \mbox{\textit{CAIE FP1 2015 Q11 [28]}}