| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2017 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Line intersection verification |
| Difficulty | Standard +0.3 This is a standard textbook exercise on classifying line relationships in 3D. Students follow a routine algorithm: check if direction vectors are parallel (they're not), then solve simultaneous equations to test for intersection. The method is well-practiced and requires no geometric insight, just careful algebraic manipulation across 3 equations with 2 unknowns. Slightly easier than average due to its algorithmic nature. |
| Spec | 1.10h Vectors in kinematics: uniform acceleration in vector form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Not parallel as \(\begin{pmatrix}6\\3\\-1\end{pmatrix}\) is not equal (or a multiple of) \(\begin{pmatrix}3\\1\\2\end{pmatrix}\) | B1 | Valid reason e.g. \(6:3:-1\neq 3:1:2\), or \(6\div3=2\) and \(3\div1=3\) |
| Sets \(\begin{pmatrix}5\\-2\\4\end{pmatrix}+\lambda\begin{pmatrix}6\\3\\-1\end{pmatrix}=\begin{pmatrix}10\\5\\-3\end{pmatrix}+\mu\begin{pmatrix}3\\1\\2\end{pmatrix}\) giving three equations | M1 | Equates lines; evidence of two of three equations |
| Full method to solve any two equations: \(\mu=2, \lambda=3\) | M1, A1 | Correct values of \(\lambda\) or \(\mu\) for their two equations |
| Substitutes both values into third equation: \(5+6\times3\) and \(10+3\times2\) | M1 | Substitute both values into third equation |
| \(23\neq16\), states lines do not intersect (skew) | A1 cso | Requires all values correct and statement that lines do not intersect/are skew |
# Question 6:
| Answer/Working | Marks | Guidance |
|---|---|---|
| Not parallel as $\begin{pmatrix}6\\3\\-1\end{pmatrix}$ is not equal (or a multiple of) $\begin{pmatrix}3\\1\\2\end{pmatrix}$ | B1 | Valid reason e.g. $6:3:-1\neq 3:1:2$, or $6\div3=2$ and $3\div1=3$ |
| Sets $\begin{pmatrix}5\\-2\\4\end{pmatrix}+\lambda\begin{pmatrix}6\\3\\-1\end{pmatrix}=\begin{pmatrix}10\\5\\-3\end{pmatrix}+\mu\begin{pmatrix}3\\1\\2\end{pmatrix}$ giving three equations | M1 | Equates lines; evidence of two of three equations |
| Full method to solve any two equations: $\mu=2, \lambda=3$ | M1, A1 | Correct values of $\lambda$ or $\mu$ for their two equations |
| Substitutes both values into third equation: $5+6\times3$ **and** $10+3\times2$ | M1 | Substitute both values into third equation |
| $23\neq16$, states lines do not intersect (skew) | A1 cso | Requires all values correct and statement that lines do not intersect/are skew |\begin{enumerate}
\item The line $l _ { 1 }$ has vector equation $\mathbf { r } = \left( \begin{array} { r } 5 \\ - 2 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } 6 \\ 3 \\ - 1 \end{array} \right)$, where $\lambda$ is a scalar parameter. The line $l _ { 2 }$ has vector equation $\mathbf { r } = \left( \begin{array} { r } 10 \\ 5 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { l } 3 \\ 1 \\ 2 \end{array} \right)$, where $\mu$ is a scalar parameter.
\end{enumerate}
Justify, giving reasons in each case, whether the lines $l _ { 1 }$ and $l _ { 2 }$ are parallel, intersecting or skew.\\
(6)\\
\hfill \mbox{\textit{Edexcel C34 2017 Q6 [6]}}