OCR Further Pure Core 1 2020 November — Question 6 5 marks

Exam BoardOCR
ModuleFurther Pure Core 1 (Further Pure Core 1)
Year2020
SessionNovember
Marks5
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Mark schemeDownload PDF ↗
TopicVectors: Cross Product & Distances
TypeShortest distance between two skew lines
DifficultyStandard +0.8 This is a Further Maths question requiring the skew lines distance formula involving cross product and scalar product. While the formula application is mechanical once known, it requires multiple vector operations (cross product, dot product, magnitude) and is beyond standard A-level content, placing it moderately above average difficulty.
Spec4.04i Shortest distance: between a point and a line
6 The equations of two non-intersecting lines, \(l _ { 1 }\) and \(l _ { 2 }\), are \(l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 1 \\ 2 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { c } 2 \\ 1 \\ - 2 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { c } 2 \\ 2 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { c } 1 \\ - 1 \\ 4 \end{array} \right)\).
Find the shortest distance between lines \(l _ { 1 }\) and \(l _ { 2 }\).
Question 6:
AnswerMarks
6 2   1   2 
     
n= 1 × −1 = −10 ,
     
     
 −2  4   −3 
n = 113
 1   2  −1
     
b−a= 2 − 2 = 0
     
     
 −1  −3  2 
−1  2 
   
0 . −10
   
   
 2   −3  8
⇒d = = =0.753 to 3sf
AnswerMarks
n 113M1
A1
A1
M1
AnswerMarks
A13.1a
1.1
1.1
1.1
AnswerMarks
1.1Cross product
Modulus
Using correct formula for d
Accept exact or correct to 3sf (0.75257669…)
FT an exact answer
[5]
Question 6:
6 |  2   1   2 
     
n= 1 × −1 = −10 ,
     
     
 −2  4   −3 
n = 113
 1   2  −1
     
b−a= 2 − 2 = 0
     
     
 −1  −3  2 
−1  2 
   
0 . −10
   
   
 2   −3  8
⇒d = = =0.753 to 3sf
n 113 | M1
A1
A1
M1
A1 | 3.1a
1.1
1.1
1.1
1.1 | Cross product
Modulus
Using correct formula for d
Accept exact or correct to 3sf (0.75257669…)
FT an exact answer
[5]
6 The equations of two non-intersecting lines, $l _ { 1 }$ and $l _ { 2 }$, are\\
$l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 1 \\ 2 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { c } 2 \\ 1 \\ - 2 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { c } 2 \\ 2 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { c } 1 \\ - 1 \\ 4 \end{array} \right)$.\\
Find the shortest distance between lines $l _ { 1 }$ and $l _ { 2 }$.

\hfill \mbox{\textit{OCR Further Pure Core 1 2020 Q6 [5]}}
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