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Date May 2022 Marks available 5 Reference code 22M.2.AHL.TZ1.7
Level Additional Higher Level Paper Paper 2 Time zone Time zone 1
Command term Find Question number 7 Adapted from N/A

Question

Consider the vectors a and b such that a=(12-5) and |b|=15.

Consider the vector p such that p=a+b.

Consider the vector q such that q=(xy), where x, y+.

Find the possible range of values for |a+b|.

[2]
a.

Given that |a+b| is a minimum, find p.

[2]
b.

Find q such that |q|=|b| and q is perpendicular to a.

[5]
c.

Markscheme

|a|=122+(-5)2(=13)            (A1)

2|a+b|28  (accept min 2 and max 28)           A1

 

Note: Award (A1)A0 for 2 and 28 seen with no indication that they are the endpoints of an interval.

 

[2 marks]

a.

recognition that p or b is a negative multiple of a            (M1)

p=-2ˆa  OR  b=-1513a(=-1513(12-5))

p=-213(12-5)(=(-1.850.769))           A1

 

 

[2 marks]

b.

METHOD 1

q is perpendicular to (12-5)

q is in the direction (512)            (M1)

q=k(512)            (A1)

(|q|=)(5k)2+(12k)2=15            (M1)

k=1513            (A1)

q=1513(512)(=(751318013)=(5.7713.8))           A1

 

METHOD 2

q is perpendicular to (12-5)

attempt to set scalar product q.a=0  OR  product of gradients =-1            (M1)

12x-5y=0            (A1)

(|q|=)x2+y2=15

attempt to solve simultaneously to find a quadratic in x or y            (M1)

x2+(12x5)2=152  OR  (5y12)2+y2=152

q=(751318013)(=(5.7713.8))           A1A1

Note: Award A1 independently for each value. Accept values given as x=7513 and y=18013 or equivalent.

 

[5 marks]

c.

Examiners report

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c.

Syllabus sections

Topic 3— Geometry and trigonometry » AHL 3.13—Scalar (dot) product
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Topic 3— Geometry and trigonometry

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