Mathematics · Oct / Nov 2022 · Paper 51 · Question 8
YMA01
As printed in the paper
The question as text
8. [In this question, i and j are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin O.] Two ships, A and B, are moving with constant velocities. The velocity of A is (3i + 12j) km h–1 and the velocity of B is (pi + qj) km h–1
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
8(a)[2]
3
+
12
153 ,3 17 , 12 or better (
km h−)
1
(
−
9
i
+
6
j
)
+
t
(3
i
+
1
2
j
)
8(b)[7]
(16
6
)
(
)
i
+
j
+
t
p
i
+
q
j
(
)
=
−
=
(16
+
6
)
+
(
+
)
−
( 9
−
+
6
)
+
(
3
+
1
2
)
A
B
b
a
i
j
t p
i
q
j
i
j
t
i
j
[
]
=
25
+
(
−
3)]
+
(
−
12)
j
t p
i
t q
Compare with: [
]
(
25
−
12t
)
i
−
9
t
j or e.g. use
b
=
AB
+
a to obtain an
equation in p only and an equation in q only. May be implied by correct
answers only.
( 12
−
=
−
3 and
−
9
=
−
12
)
p
q
N.B. This mark may not be available if they go wrong and the t’s don’t
cancel.
= −
9 ,
=
3
p
q
(25 12 )
−
2
+ −
( 9 )
2
=
15
2
(
225
2
−
600
+
400
=
0
)
8(c)[7]
t
t
t
t
t = 4
3
(9
1
2 )
±
i
−
j Note that this a method mark.
9
tan
θ=
12
θ=
37o
Bearing is 323o to nearest degree
Notes for question 8
8(a)
M1 Use of Pythagoras with square root
A1 cao
M1 Correct structure for either
8(b)
A1 cao
A1 cao
M1 Allow a − b
A1 for a correct unsimplified expression for either b – a or a – b
M1 for an equation in p only and an equation in q only
A1 cao
M1 Use of Pythagoras to give an equation in t only
8(c)
Allow with a square root.
A1 Correct unsimplified quadratic equation
A1 t = 1.3 or better
DM1 Use of their t to find
or
, dependent on previous M. May
AB
BA
be implied. Allow if they use one of their two incorrect t values.
M1 For an equation in a relevant angle for their AB. Could be implied
by a relevant angle seen on a diagram which could need checking with a
calculator
o etc or better
A1 Correct relevant angle e.g 37 ,53 ,127 ,
o
o
A1 cao
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