| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2016 |
| Session | June |
| Marks | 7 |
| Topic | Vectors 3D & Lines |
| Type | Angle between two vectors/lines (direct) |
| Difficulty | Moderate -0.8 This question requires straightforward application of the dot product formula to find an angle, and simple distance calculations to verify the triangle is isosceles. Both parts are routine 3D vector techniques with no problem-solving insight needed—easier than a typical A-level question which would involve more steps or conceptual challenge. |
| Spec | 1.10c Magnitude and direction: of vectors4.04c Scalar product: calculate and use for angles |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Angle \(AOB = \cos^{-1}\frac{16+6+20}{\sqrt{38} \times 84} = 42.0°\) | M1, M1, M1, A1 [4] | Attempt \(a.b\) for \(\vec{OA}\) and \(\vec{OB}\) (at least 2 elements correct). Use correct formula for their vectors. Attempt evaluation, with correct two vectors. Obtain \(42.0°\) (allow \(42°\)) or \(0.733\) rad. |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) \(\vec{BA} = -6i + j + k\); \( | \vec{BA} | = \sqrt{38}\) |
| \( | \vec{BA} | = |
**(i)** Angle $AOB = \cos^{-1}\frac{16+6+20}{\sqrt{38} \times 84} = 42.0°$ | M1, M1, M1, A1 [4] | Attempt $a.b$ for $\vec{OA}$ and $\vec{OB}$ (at least 2 elements correct). Use correct formula for their vectors. Attempt evaluation, with correct two vectors. Obtain $42.0°$ (allow $42°$) or $0.733$ rad.
If using cosine rule, then: M1 – attempt sides (at least 2 correct); M1 – attempt cosine rule; M1 – rearrange to attempt angle; A1 – obtain $42.0°$.
**(ii)** $\vec{BA} = -6i + j + k$; $|\vec{BA}| = \sqrt{38}$ | B1, M1 | State correct $\vec{BA}$ or $\vec{AB}$. Find one side length or one angle other than those found in part (i). If $\vec{BA}$ or $\vec{AB}$ has been stated then sufficient to just state $\sqrt{38}$. If $\vec{BA}$ or $\vec{AB}$ has not been stated then a minimum of $\sqrt{(36+1+1)}$ must be seen.
$|\vec{BA}| = |\vec{OA}|$ hence isosceles ($\neq |\vec{OB}|$ not nec) | A1 [3] | Conclude convincingly. NB Angles and sides must be given in exact form to demonstrate equality.
B0M1A1 if $\vec{BA}$ or $\vec{AB}$ not explicit. B0M1A1 if $\vec{BA}$ or $\vec{AB}$ incorrect, as long as of form $= 6i \pm j \pm k$.
If using cosine rule, then: B1 – state correct cosine rule; M1 – attempt evaluation; A1 – conclude convincingly, including use of surd value for $\cos 42°$.The points $A$ and $B$ are at $(2, 3, 5)$ and $(8, 2, 4)$ with respect to the origin $O$.
\begin{enumerate}[label=(\roman*)]
\item Find the size of angle $AOB$. [4]
\item Show that triangle $AOB$ is isosceles. [3]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2016 Q6 [7]}}