| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Line intersection with plane |
| Difficulty | Standard +0.3 This is a standard three-part vectors question testing routine techniques: substituting a line equation into a plane (part i), using the angle formula between line and plane (part ii), and finding a plane through a line perpendicular to another plane using cross product (part iii). All methods are textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04d Angles: between planes and between line and plane4.04f Line-plane intersection: find point |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Express general point of \(l\) in component form e.g. \((1+\lambda,\ 3-2\lambda,\ -2+3\lambda)\) | B1 | |
| Substitute in equation of \(p\) and solve for \(\lambda\) | M1 | |
| Obtain final answer \(\dfrac{5}{3}\mathbf{i}+\dfrac{5}{3}\mathbf{j}\) from \(\lambda = \dfrac{2}{3}\) | A1 | OE; Accept \(1.67\mathbf{i}+1.67\mathbf{j}\) or better |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct method to evaluate a scalar product of relevant vectors e.g. \((\mathbf{i}-2\mathbf{j}+3\mathbf{k})\cdot(2\mathbf{i}+\mathbf{j}-3\mathbf{k})\) | M1 | |
| Using correct process for moduli, divide scalar product by product of moduli and evaluate inverse sine or cosine | M1 | \(\ |
| Obtain answer \(40.0°\) or \(0.698\) radians | A1 | AWRT |
| Alternative method: | ||
| Use correct method to evaluate a vector product e.g. \((\mathbf{i}-2\mathbf{j}+3\mathbf{k})\times(2\mathbf{i}+\mathbf{j}-3\mathbf{k})\) | M1 | |
| Using correct process for moduli, divide modulus of vector product by product of moduli and evaluate inverse sine or cosine | M1 | \(\cos\theta = \dfrac{\sqrt{115}}{14}\) |
| Obtain answer \(40.0°\) or \(0.698\) radians | A1 | AWRT |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(a-2b+3c=0\) or \(2a+b-3c=0\) | B1 | |
| Obtain two relevant equations and solve for one ratio e.g. \(a:b\) | M1 | Could use \(2a+b-3c=0\) and \(\begin{cases}a+3b-2c=d \\ \frac{5}{3}a+\frac{5}{3}b=d\end{cases}\); second M1 not scored until they solve for \(d\) |
| Obtain \(a:b:c = 3:9:5\) | A1 | OE |
| Substitute \(a\), \(b\), \(c\) and a relevant point in plane equation and evaluate \(d\) | M1 | Using their calculated normal and a relevant point |
| Obtain answer \(3x+9y+5z=20\) | A1 | OE |
| Alternative method (cross product): | ||
| Attempt \((\mathbf{i}-2\mathbf{j}+3\mathbf{k})\times(2\mathbf{i}+\mathbf{j}-3\mathbf{k})\) | M1 | |
| Obtain two correct components | A1 | |
| Obtain \(3\mathbf{i}+9\mathbf{j}+5\mathbf{k}\) | A1 | |
| Use product and relevant point to find \(d\) | M1 | Using their calculated normal and a relevant point |
| Obtain \(3x+9y+5z=20\) | A1 | OE |
| Alternative method (2-parameter equation): | ||
| Attempt 2-parameter equation with relevant vectors | M1 | |
| State correct equation e.g. \(\mathbf{r}=\mathbf{i}+3\mathbf{j}-2\mathbf{k}+\lambda(\mathbf{i}-2\mathbf{j}+3\mathbf{k})+\mu(2\mathbf{i}+\mathbf{j}-3\mathbf{k})\) | A1 | |
| State 3 equations in \(x\), \(y\), \(z\), \(\lambda\) and \(\mu\) | A1 | |
| Eliminate \(\lambda\) and \(\mu\) | M1 | |
| Obtain \(3x+9y+5z=20\) | A1 | OE |
## Question 10(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Express general point of $l$ in component form e.g. $(1+\lambda,\ 3-2\lambda,\ -2+3\lambda)$ | B1 | |
| Substitute in equation of $p$ and solve for $\lambda$ | M1 | |
| Obtain final answer $\dfrac{5}{3}\mathbf{i}+\dfrac{5}{3}\mathbf{j}$ from $\lambda = \dfrac{2}{3}$ | A1 | OE; Accept $1.67\mathbf{i}+1.67\mathbf{j}$ or better |
---
## Question 10(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct method to evaluate a scalar product of relevant vectors e.g. $(\mathbf{i}-2\mathbf{j}+3\mathbf{k})\cdot(2\mathbf{i}+\mathbf{j}-3\mathbf{k})$ | M1 | |
| Using correct process for moduli, divide scalar product by product of moduli and evaluate inverse sine or cosine | M1 | $\|\sin\theta\| = \dfrac{9}{14}$ |
| Obtain answer $40.0°$ or $0.698$ radians | A1 | AWRT |
| **Alternative method:** | | |
| Use correct method to evaluate a vector product e.g. $(\mathbf{i}-2\mathbf{j}+3\mathbf{k})\times(2\mathbf{i}+\mathbf{j}-3\mathbf{k})$ | M1 | |
| Using correct process for moduli, divide modulus of vector product by product of moduli and evaluate inverse sine or cosine | M1 | $\cos\theta = \dfrac{\sqrt{115}}{14}$ |
| Obtain answer $40.0°$ or $0.698$ radians | A1 | AWRT |
---
## Question 10(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $a-2b+3c=0$ or $2a+b-3c=0$ | B1 | |
| Obtain two relevant equations and solve for one ratio e.g. $a:b$ | M1 | Could use $2a+b-3c=0$ and $\begin{cases}a+3b-2c=d \\ \frac{5}{3}a+\frac{5}{3}b=d\end{cases}$; second M1 not scored until they solve for $d$ |
| Obtain $a:b:c = 3:9:5$ | A1 | OE |
| Substitute $a$, $b$, $c$ and a relevant point in plane equation and evaluate $d$ | M1 | Using their calculated normal and a relevant point |
| Obtain answer $3x+9y+5z=20$ | A1 | OE |
| **Alternative method (cross product):** | | |
| Attempt $(\mathbf{i}-2\mathbf{j}+3\mathbf{k})\times(2\mathbf{i}+\mathbf{j}-3\mathbf{k})$ | M1 | |
| Obtain two correct components | A1 | |
| Obtain $3\mathbf{i}+9\mathbf{j}+5\mathbf{k}$ | A1 | |
| Use product and relevant point to find $d$ | M1 | Using their calculated normal and a relevant point |
| Obtain $3x+9y+5z=20$ | A1 | OE |
| **Alternative method (2-parameter equation):** | | |
| Attempt 2-parameter equation with relevant vectors | M1 | |
| State correct equation e.g. $\mathbf{r}=\mathbf{i}+3\mathbf{j}-2\mathbf{k}+\lambda(\mathbf{i}-2\mathbf{j}+3\mathbf{k})+\mu(2\mathbf{i}+\mathbf{j}-3\mathbf{k})$ | A1 | |
| State 3 equations in $x$, $y$, $z$, $\lambda$ and $\mu$ | A1 | |
| Eliminate $\lambda$ and $\mu$ | M1 | |
| Obtain $3x+9y+5z=20$ | A1 | OE |10 The line $l$ has equation $\mathbf { r } = \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } )$. The plane $p$ has equation $2 x + y - 3 z = 5$.\\
(i) Find the position vector of the point of intersection of $l$ and $p$.\\
(ii) Calculate the acute angle between $l$ and $p$.\\
(iii) A second plane $q$ is perpendicular to the plane $p$ and contains the line $l$. Find the equation of $q$, giving your answer in the form $a x + b y + c z = d$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE P3 2019 Q10 [11]}}