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Date November 2017 Marks available 5 Reference code 17N.1.AHL.TZ0.H_9
Level Additional Higher Level Paper Paper 1 Time zone Time zone 0
Command term Find Question number H_9 Adapted from N/A

Question

In the following diagram, OA = a, OB = b. C is the midpoint of [OA] and OF = 1 6 FB .

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

It is given also that AD = λ AF and CD = μ CB , where λ ,   μ R .

Find, in terms of a and OF .

[1]
a.i.

Find, in terms of a and AF .

[2]
a.ii.

Find an expression for  OD in terms of a, b and λ ;

[2]
b.i.

Find an expression for OD in terms of a, b and μ .

[2]
b.ii.

Show that μ = 1 13 , and find the value of λ .

[4]
c.

Deduce an expression for CD in terms of a and b only.

[2]
d.

Given that area Δ OAB = k ( area  Δ CAD ) , find the value of k .

[5]
e.

Markscheme

OF = 1 7 b     A1

[1 mark]

a.i.

AF = OF OA     (M1)

= 1 7 ba     A1

[2 marks]

a.ii.

OD = a + λ ( 1 7 b a )   ( = ( 1 λ ) a + λ 7 b )     M1A1

[2 marks]

b.i.

OD = 1 2 a + μ ( 1 2 a + b )   ( = ( 1 2 μ 2 ) a + μ b )     M1A1

[2 marks]

b.ii.

equating coefficients:     M1

λ 7 = μ ,   1 λ = 1 μ 2     A1

solving simultaneously:     M1

λ = 7 13 ,   μ = 1 13     A1AG

[4 marks]

c.

CD = 1 13 CB

= 1 13 ( b 1 2 a )   ( = 1 26 a + 1 13 b )     M1A1

[2 marks]

d.

METHOD 1

area  Δ ACD = 1 2 CD × AC × sin A C ^ B     (M1)

area  Δ ACB = 1 2 CB × AC × sin A C ^ B     (M1)

ratio  area  Δ ACD area  Δ ACB = CD CB = 1 13     A1

k = area  Δ OAB area  Δ CAD = 13 area  Δ CAB × area  Δ OAB     (M1)

= 13 × 2 = 26     A1

 

METHOD 2

area  Δ OAB = 1 2 | a × b |     A1

area  Δ CAD = 1 2 | CA × CD | or 1 2 | CA × AD |     M1

= 1 2 | 1 2 a × ( 1 26 a + 1 13 b ) |

= 1 2 | 1 2 a × ( 1 26 a ) + 1 2 a × 1 13 b |     (M1)

= 1 2 × 1 2 × 1 13 | a × b |   ( = 1 52 | a × b | )     A1

area  Δ OAB = k ( area  Δ CAD )

1 2 | a × b | = k 1 52 | a × b |

k = 26     A1

[5 marks]

e.

Examiners report

[N/A]
a.i.
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a.ii.
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b.i.
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c.
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d.
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e.

Syllabus sections

Topic 3—Geometry and trigonometry » AHL 3.10—Vector definitions
Topic 3—Geometry and trigonometry

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