| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Session | Specimen |
| Marks | 11 |
| Topic | Vectors 3D & Lines |
| Type | Line intersection verification |
| Difficulty | Standard +0.3 This is a standard 3D vectors question covering routine techniques: checking line intersection by equating components (straightforward algebra), finding angle between direction vectors using dot product formula, using perpendicularity conditions with cross products, and applying the standard volume formula. All parts follow textbook methods with no novel insight required, though it does require careful algebraic manipulation across multiple parts. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10g Problem solving with vectors: in geometry4.04g Vector product: a x b perpendicular vector |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) Use of \(\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{ | \mathbf{a} |
(i) The lines intersect [B1] **1 mark**
(ii) Use of $\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$ [M1]
$= \frac{2\times(-1) + (-3)\times(-1) + 6\times2}{\sqrt{2^2+(-3)^2+6^2} \times \sqrt{(-1)^2+(-1)^2+2^2}}$ [A1]
$\theta = 0.710$ or $40.7°$ [A1] **3 marks**
(iii) Use of $\mathbf{c} \cdot \mathbf{v} = 0$ for two vectors [M1]
Obtain two equations: $2p - 3 + 6r = 0$ [A1]
$-p - 1 + 2r = 0$ [A1]
Solve and obtain $p = 0$, $r = \frac{1}{2}$ [A1] **4 marks**
(iv) Use of volume $= \frac{1}{3}$(area of triangle $OAB$)(height $OC$) [M1]
Use of area of triangle $OAB = \frac{1}{2}OA \times OB \sin\theta$ [M1]
Obtain 2.08 [A1] **3 marks**7 With respect to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf { a }$ and $\mathbf { b }$ respectively, where $\mathbf { a } = 2 \mathbf { i } - 3 \mathbf { j } + 6 \mathbf { k }$ and $\mathbf { b } = - \mathbf { i } - \mathbf { j } + 2 \mathbf { k }$.
The lines $L _ { 1 }$ and $L _ { 2 }$ have the vector equations
$$\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } , \quad \mathbf { r } = 2 \mathbf { b } + \mu \mathbf { a }$$
respectively.\\
(i) Determine whether or not $L _ { 1 }$ and $L _ { 2 }$ intersect.\\
(ii) Find the acute angle between the directions of $L _ { 1 }$ and $L _ { 2 }$.
The point $C$ has position vector $\mathbf { c } = p \mathbf { i } + \mathbf { j } + r \mathbf { k }$.\\
(iii) Given that $O C$ is perpendicular to the triangle $O A B$, determine $p$ and $r$.\\
(iv) Determine the volume of the tetrahedron $O A B C$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 Q7 [11]}}