| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Line intersection with line |
| Difficulty | Standard +0.3 This is a standard three-part vector lines question requiring routine techniques: finding a unit vector (normalize the direction vector), finding a line equation through two points (subtract position vectors), and checking if lines intersect (equate and solve system). All are textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.10h Vectors in kinematics: uniform acceleration in vector form4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct process for modulus on direction vector of \(l\), e.g. \(\sqrt{(-1)^2+1^2+2^2}\) | M1 | SOI. Allow \(-1^2\). Allow \(\sqrt{(-\lambda)^2+\lambda^2+(2\lambda)^2}\) |
| \(\left[\pm\right]\frac{1}{\sqrt{6}}(-\mathbf{i}+\mathbf{j}+2\mathbf{k})\) | A1 | OE. Allow coordinates as row or column, but not row or column with i, j and k included. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use a correct method to form an equation for line \(m\) | M1 | Allow even if all signs of point incorrect, namely use \(+2\mathbf{i}-2\mathbf{j}+\mathbf{k}\) or \(-3\mathbf{i}+\mathbf{j}-\mathbf{k}\) |
| Obtain \(\mathbf{r}=-2\mathbf{i}+2\mathbf{j}-\mathbf{k}+\mu_1(-5\mathbf{i}+3\mathbf{j}-2\mathbf{k})\) | A1 | OE, e.g. \(\mathbf{r}=3\mathbf{i}-\mathbf{j}+\mathbf{k}+\mu_2(-5\mathbf{i}+3\mathbf{j}-2\mathbf{k})\). Must have \(\mathbf{r}=\ldots\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Justify lines are not parallel | B1 | \((-5,3,-2)\neq d(-1,1,2)\) or \((-5,3,-2)\times(-1,1,2)\neq 0\). Can find angle (\(105°\), \(74.6°\), \(1.84^c\) or \(1.3(0)^c\)) instead but if incorrect B0 and A0 at end. Accept direction vectors don't have common factor but not that direction vectors are different or scalar product \(\neq 0\). |
| Express \(l\) or \(m\) in component form, e.g. \((-2-5\mu_1,\,2+3\mu_1,\,-1-2\mu_1)\) or \((3-5\mu_2,\,-1+3\mu_2,\,1-2\mu_2)\) or \((1-\lambda,\,-2+\lambda,\,-3+2\lambda)\) | B1 | |
| Equate two pairs of components of general points on \(l\) and their \(m\) and solve simultaneously for \(\lambda\) or \(\mu\) | M1 | |
| Obtain correct answer for \(\lambda\) or \(\mu\), e.g. \(\lambda=\frac{11}{2}\), \(\mu_1=\frac{1}{2}\) | A1 | |
| Determine that all three equations are not satisfied and lines fail to intersect, conclude lines are skew. Conclusion needs to follow correct working. | A1 | Dependent on 4 previous marks gained. Table shows: for pair ij: \(\lambda=11/2\), \(\mu_1=1/2\) gives \(8\neq -2\); pair ik: \(\lambda=4/3\), \(\mu_1=-1/3\) gives \(-2/3\neq 1\); pair jk: \(\lambda=7/4\), \(\mu_1=-3/4\) gives \(-3/4\neq 7/4\) |
## Question 11(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct process for modulus on direction vector of $l$, e.g. $\sqrt{(-1)^2+1^2+2^2}$ | M1 | SOI. Allow $-1^2$. Allow $\sqrt{(-\lambda)^2+\lambda^2+(2\lambda)^2}$ |
| $\left[\pm\right]\frac{1}{\sqrt{6}}(-\mathbf{i}+\mathbf{j}+2\mathbf{k})$ | A1 | OE. Allow coordinates as row or column, but not row or column with **i**, **j** and **k** included. |
**Total: 2 marks**
---
## Question 11(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use a correct method to form an equation for line $m$ | M1 | Allow even if all signs of point incorrect, namely use $+2\mathbf{i}-2\mathbf{j}+\mathbf{k}$ or $-3\mathbf{i}+\mathbf{j}-\mathbf{k}$ |
| Obtain $\mathbf{r}=-2\mathbf{i}+2\mathbf{j}-\mathbf{k}+\mu_1(-5\mathbf{i}+3\mathbf{j}-2\mathbf{k})$ | A1 | OE, e.g. $\mathbf{r}=3\mathbf{i}-\mathbf{j}+\mathbf{k}+\mu_2(-5\mathbf{i}+3\mathbf{j}-2\mathbf{k})$. Must have $\mathbf{r}=\ldots$ |
**Total: 2 marks**
---
## Question 11(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Justify lines are not parallel | B1 | $(-5,3,-2)\neq d(-1,1,2)$ or $(-5,3,-2)\times(-1,1,2)\neq 0$. Can find angle ($105°$, $74.6°$, $1.84^c$ or $1.3(0)^c$) instead but if incorrect B0 and A0 at end. Accept direction vectors don't have common factor but not that direction vectors are different or scalar product $\neq 0$. |
| Express $l$ or $m$ in component form, e.g. $(-2-5\mu_1,\,2+3\mu_1,\,-1-2\mu_1)$ or $(3-5\mu_2,\,-1+3\mu_2,\,1-2\mu_2)$ or $(1-\lambda,\,-2+\lambda,\,-3+2\lambda)$ | B1 | |
| Equate two pairs of components of general points on $l$ and their $m$ and solve simultaneously for $\lambda$ or $\mu$ | M1 | |
| Obtain correct answer for $\lambda$ or $\mu$, e.g. $\lambda=\frac{11}{2}$, $\mu_1=\frac{1}{2}$ | A1 | |
| Determine that all three equations are not satisfied and lines fail to intersect, conclude lines are skew. Conclusion needs to follow correct working. | A1 | Dependent on 4 previous marks gained. Table shows: for pair **ij**: $\lambda=11/2$, $\mu_1=1/2$ gives $8\neq -2$; pair **ik**: $\lambda=4/3$, $\mu_1=-1/3$ gives $-2/3\neq 1$; pair **jk**: $\lambda=7/4$, $\mu_1=-3/4$ gives $-3/4\neq 7/4$ |
**Total: 5 marks**11 The line $l$ has equation $\mathbf { r } = \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k } + \lambda ( - \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )$. The points $A$ and $B$ have position vectors $- 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$ and $3 \mathbf { i } - \mathbf { j } + \mathbf { k }$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Find a unit vector in the direction of $l$.\\
The line $m$ passes through the points $A$ and $B$.
\item Find a vector equation for $m$.
\item Determine whether lines $l$ and $m$ are parallel, intersect or are skew.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2023 Q11 [9]}}