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Date November 2017 Marks available 2 Reference code 17N.1.hl.TZ0.9
Level HL only Paper 1 Time zone TZ0
Command term Find Question number 9 Adapted from N/A

Question

In the following diagram, \(\overrightarrow {{\text{OA}}} \) = a, \(\overrightarrow {{\text{OB}}} \) = b. C is the midpoint of [OA] and \(\overrightarrow {{\text{OF}}}  = \frac{1}{6}\overrightarrow {{\text{FB}}} \).

N17/5/MATHL/HP1/ENG/TZ0/09

It is given also that \(\overrightarrow {{\text{AD}}}  = \lambda \overrightarrow {{\text{AF}}} \) and \(\overrightarrow {{\text{CD}}}  = \mu \overrightarrow {{\text{CB}}} \), where \(\lambda ,{\text{ }}\mu  \in \mathbb{R}\).

Find, in terms of a and \(\overrightarrow {{\text{OF}}} \).

[1]
a.i.

Find, in terms of a and \(\overrightarrow {{\text{AF}}} \).

[2]
a.ii.

Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of a, b and \(\lambda \);

[2]
b.i.

Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of a, b and \(\mu \).

[2]
b.ii.

Show that \(\mu  = \frac{1}{{13}}\), and find the value of \(\lambda \).

[4]
c.

Deduce an expression for \(\overrightarrow {{\text{CD}}} \) in terms of a and b only.

[2]
d.

Given that area \(\Delta {\text{OAB}} = k({\text{area }}\Delta {\text{CAD}})\), find the value of \(k\).

[5]
e.

Markscheme

\(\overrightarrow {{\text{OF}}}  = \frac{1}{7}\)b     A1

[1 mark]

a.i.

\(\overrightarrow {{\text{AF}}}  = \overrightarrow {{\text{OF}}}  - \overrightarrow {{\text{OA}}} \)     (M1)

\( = \frac{1}{7}\)ba     A1

[2 marks]

a.ii.

\(\overrightarrow {{\text{OD}}}  = \) a \( + \lambda \left( {\frac{1}{7}b -a} \right){\text{ }}\left( { = (1 - \lambda )a + \frac{\lambda }{7}b} \right)\)     M1A1

[2 marks]

b.i.

\(\overrightarrow {{\text{OD}}}  = \frac{1}{2}\) a \( + \mu \left( { - \frac{1}{2}a + b} \right){\text{ }}\left( { = \left( {\frac{1}{2} - \frac{\mu }{2}} \right)a + \mu b} \right)\)     M1A1

[2 marks]

b.ii.

equating coefficients:     M1

\(\frac{\lambda }{7} = \mu ,{\text{ }}1 - \lambda  = \frac{{1 - \mu }}{2}\)     A1

solving simultaneously:     M1

\(\lambda  = \frac{7}{{13}},{\text{ }}\mu  = \frac{1}{{13}}\)     A1AG

[4 marks]

c.

\(\overrightarrow {{\text{CD}}}  = \frac{1}{{13}}\overrightarrow {{\text{CB}}} \)

\( = \frac{1}{{13}}\left( {b - \frac{1}{2}a} \right){\text{ }}\left( { =  - \frac{1}{{26}}a + \frac{1}{{13}}b} \right)\)     M1A1

[2 marks]

d.

METHOD 1

\({\text{area }}\Delta {\text{ACD}} = \frac{1}{2}{\text{CD}} \times {\text{AC}} \times \sin {\rm{A\hat CB}}\)     (M1)

\({\text{area }}\Delta {\text{ACB}} = \frac{1}{2}{\text{CB}} \times {\text{AC}} \times \sin {\rm{A\hat CB}}\)     (M1)

\({\text{ratio }}\frac{{{\text{area }}\Delta {\text{ACD}}}}{{{\text{area }}\Delta {\text{ACB}}}} = \frac{{{\text{CD}}}}{{{\text{CB}}}} = \frac{1}{{13}}\)     A1

\(k = \frac{{{\text{area }}\Delta {\text{OAB}}}}{{{\text{area }}\Delta {\text{CAD}}}} = \frac{{13}}{{{\text{area }}\Delta {\text{CAB}}}} \times {\text{area }}\Delta {\text{OAB}}\)     (M1)

\( = 13 \times 2 = 26\)     A1

 

METHOD 2

\({\text{area }}\Delta {\text{OAB}} = \frac{1}{2}\left| {a \times b} \right|\)     A1

\({\text{area }}\Delta {\text{CAD}} = \frac{1}{2}\left| {\overrightarrow {{\text{CA}}}  \times \overrightarrow {{\text{CD}}} } \right|\) or \(\frac{1}{2}\left| {\overrightarrow {{\text{CA}}}  \times \overrightarrow {{\text{AD}}} } \right|\)     M1

\( = \frac{1}{2}\left| {\frac{1}{2}a \times \left( { - \frac{1}{{26}}a + \frac{1}{{13}}b} \right)} \right|\)

\( = \frac{1}{2}\left| {\frac{1}{2}a \times \left( { - \frac{1}{{26}}a} \right) + \frac{1}{2}a \times \frac{1}{{13}}b} \right|\)     (M1)

\( = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{{13}}\left| {a \times b} \right|{\text{ }}\left( { = \frac{1}{{52}}\left| {a \times b} \right|} \right)\)     A1

\({\text{area }}\Delta {\text{OAB}} = k({\text{area }}\Delta {\text{CAD}})\)

\(\frac{1}{2}\left| {a \times b} \right| = k\frac{1}{{52}}\left| {a \times b} \right|\)

\(k = 26\)     A1

[5 marks]

e.

Examiners report

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Syllabus sections

Topic 4 - Core: Vectors » 4.1 » Algebraic and geometric approaches to the sum and difference of two vectors.

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