OCR MEI Further Pure Core 2020 November — Question 8 9 marks

Exam BoardOCR MEI
ModuleFurther Pure Core (Further Pure Core)
Year2020
SessionNovember
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicVectors: Lines & Planes
TypeAngle between two lines
DifficultyStandard +0.3 Part (a) requires setting equal the parametric equations and solving a system to find k (standard technique). Part (b) is a direct application of the dot product formula for angle between direction vectors. This is a routine Further Maths question testing standard vector line techniques with no novel insight required, making it slightly easier than average.
Spec4.04c Scalar product: calculate and use for angles4.04e Line intersections: parallel, skew, or intersecting
8
  1. Given that the lines \(\mathbf { r } = \left( \begin{array} { l } 0 \\ 2 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 1 \\ 3 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { r } - 1 \\ 2 \\ k \end{array} \right) + \mu \left( \begin{array} { l } 2 \\ 3 \\ 4 \end{array} \right)\) meet, determine \(k\).
  2. In this question you must show detailed reasoning. Find the acute angle between the two lines.
Question 8:
AnswerMarks Guidance
8(a) −λ = −1 + 2 , 2 + λ = 2 + 3 , 2 + 3λ = k +
​ ​ µ​ ​ ​ ​ µ​ ​ ​ ​ ​​
4
µ​
λ = 3 − −
​ µ​ ​,​ ​3µ​ ​=​ ​1+2µ​
⇒ = 1/5, λ = 3/5
µ​ ​ ​ ​
2 + 9/5 = k + 4/5 ⇒ k = 3
AnswerMarks
​​ ​​M1
M1
A1A1
A1
AnswerMarks
[5]3.1a
1.1
1.1,1.1
1.1
AnswerMarks Guidance
8(b) DR
⇒ θ = 43.3
AnswerMarks
​ ​ ​°M1A1
B1
A1
AnswerMarks
[4]1.1,1.1
1.1
AnswerMarks Guidance
1.1soi accept 0.756 rad 
Question 8:
8 | (a) | −λ = −1 + 2 , 2 + λ = 2 + 3 , 2 + 3λ = k +
​ ​ µ​ ​ ​ ​ µ​ ​ ​ ​ ​​
4
µ​
λ = 3 − −
​ µ​ ​,​ ​3µ​ ​=​ ​1+2µ​
⇒ = 1/5, λ = 3/5
µ​ ​ ​ ​
2 + 9/5 = k + 4/5 ⇒ k = 3
​​ ​​ | M1
M1
A1A1
A1
[5] | 3.1a
1.1
1.1,1.1
1.1
8 | (b) | DR
⇒ θ = 43.3
​ ​ ​° | M1A1
B1
A1
[4] | 1.1,1.1
1.1
1.1 | soi | accept 0.756 rad
8
\begin{enumerate}[label=(\alph*)]
\item Given that the lines $\mathbf { r } = \left( \begin{array} { l } 0 \\ 2 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 1 \\ 3 \end{array} \right)$ and $\mathbf { r } = \left( \begin{array} { r } - 1 \\ 2 \\ k \end{array} \right) + \mu \left( \begin{array} { l } 2 \\ 3 \\ 4 \end{array} \right)$ meet, determine $k$.
\item In this question you must show detailed reasoning.

Find the acute angle between the two lines.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Pure Core 2020 Q8 [9]}}
This paper (16 questions)
View full paper
Q1 6 Q2 6 Q3 4 Q4 8 Q5 8 Q6 4 Q7 6 Q8 9 Q9 8 Q10 7 Q11 8 Q12 8 Q13 9 Q14 11 Q15 17 Q16 25