Standard +0.3 This is a straightforward Further Maths question on 3D vector geometry requiring standard techniques: finding angle between line and plane using dot product formula, and checking if a line lies in/is parallel to/intersects a plane. Both parts are routine applications of formulas with no novel problem-solving required, though slightly above average difficulty due to being Further Maths content.
The equation of a plane \(\Pi\) is \(x-2y-z=30\).
\begin{enumerate}[label=(\roman*)]
\item Find the acute angle between the line \(\mathbf{r} = \begin{pmatrix} 3 \\ 2 \\ -5 \end{pmatrix} + \lambda \begin{pmatrix} -5 \\ 3 \\ 2 \end{pmatrix}\) and \(\Pi\). [4]
\item Determine the geometrical relationship between the line \(\mathbf{r} = \begin{pmatrix} 1 \\ 4 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ -1 \\ 5 \end{pmatrix}\) and \(\Pi\). [4]
Question 6:
6 | (i) | (cid:16)5(cid:16)6(cid:16)2(cid:32) 1(cid:14)4(cid:14)1 25(cid:14)9(cid:14)4cos(cid:84)oe
(cid:16)13
(cid:84)(cid:32)cos(cid:16)1 oe
6(cid:117)38
149.4 (cid:16)90
59.4 | *M1
dep*M1
M1
A1
[4] | 1.1a
1.1
2.2a
1.1 | Allow one slip e.g. sign error
Using angle with normal to find
angle with plane
1.04 in radians
6 | (ii) | 1(cid:117)3(cid:14)((cid:16)2)(cid:117)((cid:16)1)(cid:14)((cid:16)1)(cid:117)5= 0
The line is in or parallel to the plane
1(cid:117)1(cid:16)2(cid:117)4(cid:16)1(cid:117)(2)(cid:122)30
Point on line not in plane, so line is parallel to plane | M1
E1
M1
E1
[4] | 3.1a
2.2a
m
3.1a
i
2.2a | n
e
Substitution of e.g. (1, 4, 2) in
equation of plane
oe | Alternative method
1(cid:14)3(cid:80)(cid:16)2(cid:11)4(cid:16)(cid:80)(cid:12)(cid:16)(cid:11)2(cid:14)5(cid:80)(cid:12)(cid:32)(cid:16)9
M1A1
–9 is not 30 E1
Conclusion E1
The equation of a plane $\Pi$ is $x-2y-z=30$.
\begin{enumerate}[label=(\roman*)]
\item Find the acute angle between the line $\mathbf{r} = \begin{pmatrix} 3 \\ 2 \\ -5 \end{pmatrix} + \lambda \begin{pmatrix} -5 \\ 3 \\ 2 \end{pmatrix}$ and $\Pi$. [4]
\item Determine the geometrical relationship between the line $\mathbf{r} = \begin{pmatrix} 1 \\ 4 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ -1 \\ 5 \end{pmatrix}$ and $\Pi$. [4]
</end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 Q6 [8]}}