| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Topic | Vectors 3D & Lines |
| Type | Parallel and perpendicular lines |
| Difficulty | Standard +0.3 This is a straightforward 3D vectors question requiring standard techniques: finding direction vectors, using the scalar product formula for angles, and applying the parallel lines condition. Both parts are routine applications of core methods with no conceptual challenges, making it slightly easier than average. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10c Magnitude and direction: of vectors4.04c Scalar product: calculate and use for angles |
(i) $\overrightarrow{AB} = \begin{pmatrix}1\\3\\5\end{pmatrix}$, $\overrightarrow{CB} = \begin{pmatrix}-1\\0\\2\end{pmatrix}$
B1 Any two relevant vectors. Allow vectors of inconsistent directions e.g. $AB$ and $BC$
$(\pm)9 = \sqrt{35}\sqrt{5}\cos A\hat{B}C$
M1 Attempt scalar product – allow inconsistent directions
$\cos A\hat{B}C = \dfrac{9}{\sqrt{35}\sqrt{5}}$
A1 Correct expression involving $\cos ABC$ – not necessarily with $\cos ABC$ as the subject
so $A\hat{B}C = 47.13...= 47.1°$ to 1 dp
A1 [4] CWO
**Using cosine rule:**
B1 Three correct vectors soi
M1 Attempt correct cosine rule
A1 Correct expression involving $\cos ABC$. CWO so A0 if correct surd from incorrect vector
A1 Obtain $47.1°$
(ii) $k\overrightarrow{AB} = k\begin{pmatrix}1\\3\\5\end{pmatrix} = \overrightarrow{CD} = \begin{pmatrix}-5\\a-2\\b-3\end{pmatrix}$
M1 Attempt to find at least one of $a$ and $b$, by considering at least two components of parallel vectors, including attempt at $k$
$3 \times -5 = a - 2$ so $a = -13$
$5 \times -5 = b - 3$ so $b = -22$
A1
A1 [3]4 The points $A , B , C$ and $D$ have coordinates $( 2 , - 1,0 ) , ( 3,2,5 ) , ( 4,2,3 )$ and $( - 1 , a , b )$ respectively, where $a$ and $b$ are constants.\\
(i) Find the angle $A B C$.\\
(ii) Given that the lines $A B$ and $C D$ are parallel, find the values of $a$ and $b$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2014 Q4 [7]}}