Question 3 - A-Level Maths

WJEC Further Unit 1 2022 June — Question 3 5 marks

Exam Board	WJEC
Module	Further Unit 1 (Further Unit 1)
Year	2022
Session	June
Marks	5
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Vectors: Lines & Planes
Type	Line intersection with plane
Difficulty	Standard +0.3 This is a straightforward application of standard techniques: substitute the parametric equations of the line into the plane equation, solve for λ, then find coordinates. It requires no novel insight and is a routine textbook exercise in Further Maths vectors, making it slightly easier than average overall but typical for this topic.
Spec	4.04f Line-plane intersection: find point

3. The vector equation of the line $L$ is given by $$\mathbf { r } = - \mathbf { i } + 2 \mathbf { j } - 6 \mathbf { k } + \lambda ( 4 \mathbf { i } - 2 \mathbf { j } + 7 \mathbf { k } ) .$$ The Cartesian equation of the plane $\Pi$ is given by $$3 x + 8 y - 9 z = 0$$ Find the Cartesian coordinates of the point of intersection of $L$ and $\Pi$.

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Question 3:

Answer	Marks	Guidance
Working	Mark	Guidance
$x = -1+4\lambda$, $y = 2-2\lambda$, $z = -6+7\lambda$	B1	si
Substituting: $-3+12\lambda+16-16\lambda+54-63\lambda = 0$	M1
$67-67\lambda = 0$	A1
$\lambda = 1$	A1
$\therefore x=3$, $y=0$, $z=1$, $\Rightarrow (3,0,1)$	B1	FT their $\lambda$ and their $x,y,z$ provided at least 2 correct
[5]

# Question 3:

| Working | Mark | Guidance |
|---------|------|----------|
| $x = -1+4\lambda$, $y = 2-2\lambda$, $z = -6+7\lambda$ | B1 | si |
| Substituting: $-3+12\lambda+16-16\lambda+54-63\lambda = 0$ | M1 | |
| $67-67\lambda = 0$ | A1 | |
| $\lambda = 1$ | A1 | |
| $\therefore x=3$, $y=0$, $z=1$, $\Rightarrow (3,0,1)$ | B1 | FT their $\lambda$ and their $x,y,z$ provided at least 2 correct |
| | [5] | |

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3. The vector equation of the line $L$ is given by

$$\mathbf { r } = - \mathbf { i } + 2 \mathbf { j } - 6 \mathbf { k } + \lambda ( 4 \mathbf { i } - 2 \mathbf { j } + 7 \mathbf { k } ) .$$

The Cartesian equation of the plane $\Pi$ is given by

$$3 x + 8 y - 9 z = 0$$

Find the Cartesian coordinates of the point of intersection of $L$ and $\Pi$.\\

\hfill \mbox{\textit{WJEC Further Unit 1 2022 Q3 [5]}}

This paper (9 questions)

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Q1 12 Q2 11 Q3 5 Q4 7 Q5 4 Q6 8 Q7 7 Q8 7 Q9 9

Answer	Marks	Guidance
Working	Mark	Guidance
\(x = -1+4\lambda\), \(y = 2-2\lambda\), \(z = -6+7\lambda\)	B1	si
Substituting: \(-3+12\lambda+16-16\lambda+54-63\lambda = 0\)	M1
\(67-67\lambda = 0\)	A1
\(\lambda = 1\)	A1
\(\therefore x=3\), \(y=0\), \(z=1\), \(\Rightarrow (3,0,1)\)	B1	FT their \(\lambda\) and their \(x,y,z\) provided at least 2 correct
[5]