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∈ℝ+.
Find the possible range of values for |a+b|.
Given that |a+b| is a minimum, find p.
Find q such that |q|=|b| and q is perpendicular to a.
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]
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]
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]