| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Angle between two lines |
| Difficulty | Standard +0.3 This is a straightforward vectors question requiring standard techniques: finding a direction vector from two points, verifying skew lines by checking if they intersect, and calculating an angle using the scalar product formula. All methods are routine for Further Maths P3 students, though the multi-step nature and skew lines verification adds slight complexity beyond basic recall. |
| Spec | 4.04c Scalar product: calculate and use for angles4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | ||
| Obtain \(\begin{pmatrix}2\\-3\\-4\end{pmatrix}\) as direction vector of \(l_1\) | B1 | |
| State that two direction vectors are not parallel | B1 | |
| Express general point of \(l_1\) or \(l_2\) in component form, e.g. \((2\lambda, 1-3\lambda, 5-4\lambda)\) or \((7+\mu, 1+2\mu, 1+5\mu)\) | B1 | |
| Equate at least two pairs of components and solve for \(\lambda\) or for \(\mu\) | M1 | |
| Obtain correct answers for \(\lambda\) and \(\mu\) | A1 | |
| Verify that all three component equations are not satisfied (with no errors seen) | A1 | [6] |
| (ii) | ||
| Carry out correct process for evaluating scalar product of \(\begin{pmatrix}1\\2\\5\end{pmatrix}\) and \(\begin{pmatrix}1\\0\\0\end{pmatrix}\) | M1 | |
| Use correct process for finding modulus and evaluating inverse cosine | M1 | |
| Obtain \(79.5°\) or 1.39 radians | A1 | [3] |
**(i)** |
Obtain $\begin{pmatrix}2\\-3\\-4\end{pmatrix}$ as direction vector of $l_1$ | B1 |
State that two direction vectors are not parallel | B1 |
Express general point of $l_1$ or $l_2$ in component form, e.g. $(2\lambda, 1-3\lambda, 5-4\lambda)$ or $(7+\mu, 1+2\mu, 1+5\mu)$ | B1 |
Equate at least two pairs of components and solve for $\lambda$ or for $\mu$ | M1 |
Obtain correct answers for $\lambda$ and $\mu$ | A1 |
Verify that all three component equations are not satisfied (with no errors seen) | A1 | [6]
**(ii)** |
Carry out correct process for evaluating scalar product of $\begin{pmatrix}1\\2\\5\end{pmatrix}$ and $\begin{pmatrix}1\\0\\0\end{pmatrix}$ | M1 |
Use correct process for finding modulus and evaluating inverse cosine | M1 |
Obtain $79.5°$ or 1.39 radians | A1 | [3]6 The straight line $l _ { 1 }$ passes through the points $( 0,1,5 )$ and $( 2 , - 2,1 )$. The straight line $l _ { 2 }$ has equation $\mathbf { r } = 7 \mathbf { i } + \mathbf { j } + \mathbf { k } + \mu ( \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } )$.\\
(i) Show that the lines $l _ { 1 }$ and $l _ { 2 }$ are skew.\\
(ii) Find the acute angle between the direction of the line $l _ { 2 }$ and the direction of the $x$-axis.
\hfill \mbox{\textit{CAIE P3 2015 Q6 [9]}}