| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2017 |
| Session | June |
| Marks | 10 |
| Topic | Vectors Introduction & 2D |
| Type | Angle between two vectors |
| Difficulty | Standard +0.3 This is a straightforward 3D vectors question requiring standard techniques: finding position vectors of P and Q using scalar multiplication, writing parametric equations for two lines, solving simultaneously for intersection, then using the scalar product formula to find the angle between direction vectors. All steps are routine A-level procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.10g Problem solving with vectors: in geometry4.04c Scalar product: calculate and use for angles |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain either \( | \overrightarrow{AQ} | = \sqrt{42}\) or \( |
**Question 11(i)**
State $\overrightarrow{OQ} = 6\mathbf{i} - 4\mathbf{j} - 2\mathbf{k}$ and $\overrightarrow{OP} = 6\mathbf{i} + 3\mathbf{j} - 9\mathbf{k}$
or $AQ = 4\mathbf{i} - 5\mathbf{j} + \mathbf{k}$ and $BP = 3\mathbf{i} + 5\mathbf{j} - 8\mathbf{k}$ **B1**
Form equation of line $AQ$ and $BP$ in form $\mathbf{a} + \lambda\mathbf{b}$ **M1**
Obtain $\mathbf{r}_{AQ} = (2\mathbf{i} + \mathbf{j} - 3\mathbf{k}) + \lambda(4\mathbf{i} - 5\mathbf{j} + \mathbf{k})$ **A1**
Obtain $\mathbf{r}_{BP} = (3\mathbf{i} - 2\mathbf{j} - \mathbf{k}) + \mu(3\mathbf{i} + 5\mathbf{j} - 8\mathbf{k})$ **A1**
Equate line equations and solve two equations simultaneously to find some value of $\lambda$ or $\mu$ **M1**
Obtain either $\mu = \dfrac{1}{5}$ or $\lambda = \dfrac{2}{5}$ **A1**
State $\left(\dfrac{18}{5}, -1, \dfrac{-13}{5}\right)$ Must be in coordinate form **B1**
**Question 11(ii)**
Use dot product correctly to find an angle **M1**
Obtain either $|\overrightarrow{AQ}| = \sqrt{42}$ or $|\overrightarrow{BP}| = \sqrt{98}$ **B1**
Obtain $70.9°$ **A1**
**Total: 10 marks**11 The points $A$ and $B$ have position vectors $2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }$ and $3 \mathbf { i } - 2 \mathbf { j } - \mathbf { k }$ respectively, relative to the origin $O$. The point $P$ lies on $O A$ extended so that $\overrightarrow { O P } = 3 \overrightarrow { O A }$ and the point $Q$ lies on $O B$ extended so that $\overrightarrow { O Q } = 2 \overrightarrow { O B }$.\\
(i) Find the coordinates of the point of intersection of the lines $A Q$ and $B P$.\\
(ii) Find the acute angle between the lines $A Q$ and $B P$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2017 Q11 [10]}}