| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Topic | Vectors 3D & Lines |
| Type | Show lines intersect and find intersection point |
| Difficulty | Standard +0.3 This is a standard two-part question on 3D vector lines requiring routine techniques: (i) equating components to find intersection point (solving simultaneous equations in two parameters), and (ii) using the scalar product formula to find the angle between direction vectors. Both parts are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles |
| Answer | Marks | Guidance |
|---|---|---|
| Use \(\cos\theta = \dfrac{\mathbf{a} \cdot \mathbf{b}}{ | \mathbf{a} |
**(i)**
Find $\lambda$, $\mu$ or both from at least one correct equation e.g. $3 + \lambda = 1$ or $2 - 6\lambda = 5 + 3\mu$, $1 - 2\lambda = 2 + \mu$ [M1]
Obtain $\lambda = -2$ or $\mu = 3$ [A1]
Obtain $(1, 14, 5)$ [A1]
Show substitution of values of $\lambda$ and $\mu$ into third equation or both lines [M1]
Demonstrate consistency or obtain $(1, 14, 5)$ from both lines [A1]
**Subtotal: [5]**
**(ii)**
Use $\mathbf{i} - 6\mathbf{j} - 2\mathbf{k}$ and $3\mathbf{j} + \mathbf{k}$ **(N.B.** using $(3, 2, 1)$ and $(1, 5, 2)$ = 15 B0**)** [B1]
Use $\cos\theta = \dfrac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$ for their vectors $\mathbf{a}$ and $\mathbf{b}$ [M1]
Attempt evaluation of correct $\mathbf{a}$ and $\mathbf{b}$ $\left(= \dfrac{\pm 20}{\sqrt{41}\,\sqrt{10}}\right)$ [A1]
Obtain $9°$ or better ($8.984876°$) or $0.156853^c$ [A1]
**Subtotal: [4]**
**Total: [9]**7 The lines $l _ { 1 }$ and $l _ { 2 }$ have the following vector equations.
$$\begin{aligned}
& l _ { 1 } : \mathbf { r } = 3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } - 6 \mathbf { j } - 2 \mathbf { k } ) \\
& l _ { 2 } : \mathbf { r } = \mathbf { i } + 5 \mathbf { j } + 2 \mathbf { k } + \mu ( 3 \mathbf { j } + \mathbf { k } )
\end{aligned}$$
(i) Show that the lines $l _ { 1 }$ and $l _ { 2 }$ intersect and find the coordinates of their point of intersection.\\
(ii) Find the acute angle between the lines $l _ { 1 }$ and $l _ { 2 }$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2015 Q7 [9]}}