AS October 2021 Paper 1 Q1

OCR ACurrent spec5 marks3D Lines & Planes

1 The lines \(l_1\) and \(l_2\) have the following equations.

\[\begin{aligned} l_1 &: \mathbf{r} = \begin{pmatrix} 8 \\ -11 \\ -2 \end{pmatrix} + \lambda\begin{pmatrix} -2 \\ 5 \\ 3 \end{pmatrix} \\ l_2 &: \mathbf{r} = \begin{pmatrix} -6 \\ 11 \\ 8 \end{pmatrix} + \mu\begin{pmatrix} -3 \\ 1 \\ -1 \end{pmatrix} \end{aligned}\]

(a) Show that \(l_1\) and \(l_2\) intersect. [4]

(b) Write down the point of intersection of \(l_1\) and \(l_2\). [1]

Mark scheme (a)

Scheme	Marks	AO
\(8 - 2\lambda = -6 - 3\mu\) and \(-11 + 5\lambda = 11 + \mu\)	B1	1.1a
\(8 - 2\lambda = -6 - 3\mu\) \(-33 + 15\lambda = 33 + 3\mu\) \(\Rightarrow -25 + 13\lambda = 27\)	M1	1.1
\(\lambda = 4,\ \mu = -2\)	A1	1.1
\(-2 + 3 \times 4 = 10\) and \(8 - -2 = 10\) so they do intersect	A1	2.4
	[4]

Notes

B1: Forming 2 correct equations in \(\lambda\) and \(\mu\).
Could be \(-2 + 3\lambda = 8 - \mu\)
Any two correct equations

M1: Attempt to solve (eg scaling one equation and adding or rewriting to a standard form for solution BC). Must reach an equation (possibly incorrect) with only one unknown.
\(-2\lambda + 3\mu = -14\)
\(5\lambda - \mu = 22\)
\(3\lambda + \mu = 10\)

A1: (1st) Both

A1: (2nd) Checking for consistency in 3^rd equation and conclusion. Equation must be correct and both sides must be evaluated
Allow eg \(\begin{aligned} 8 - 2 \times 4 &= -6 - 3 \times -2 \\ 0 &= 0 \end{aligned}\)
Might see \(\lambda = 4\) substituted into last equation and then \(\mu\) being found with this.
ie \(-2 + 3 \times 4 = 8 - -2\) alone is not sufficient for A1, need to see both sides becoming 10
\(x\): \(8 - 2 \times 4 = 0\) & \(-6 - 3 \times -2 = 0\)
\(y\): \(-11 + 5 \times 4 = 9\) & \(11 + -2 = 9\)

Scheme	Marks	AO
\((0, 9, 10)\)	B1	1.1
	[1]