Mathematics · May / June 2024 · Paper 51 · Question 7
YMA01
As printed in the paper
The question as text
7. [In this question, the horizontal unit vectors i and j are directed due east and due north respectively and position vectors are given relative to a fixed origin O.] Two speedboats, A and B, are each moving with constant velocity. • the velocity of A is 40 km h –1 due east –1 on a bearing of angle α (0° < α < 90°), where tan 4 • the velocity of B is 20 km h 3 The boats are modelled as particles.
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
7(a)[2]
B
to an answer
j ( km h1)
=
16
i
+
12
7(b)[2]
(
s
=
) (10
i
+
5 ) + (16
j
i
+
12 )
j or (10 16 )
+
i
+
(5 12 )
+
j
t
t
t
s
r
i
j
i
j
[
j
i
]
7(c)[2]
AB
= −=
(10
+
5 ) + (16
t
+
12 )
−
20
+
40
t
[
]
=
(10
−
24 )
i
+
(12
−
15) km
j
*
AB
t
t
7(d)[3]
10
−
24
=
0 and 12
−
15
=
0
OR 40t =10 + 16t and 20 = 5 + 12t
t
t
5
5
t =
and
or one correct t value which is then used in the other
12
4
equation correctly to show that the equation is not true.
Different t values oe so never collide*
ALT 1:
(10
−
24 )
t
2
+
(12
t
−
15)
2
=
0
(i.e.
720
t
2
−
840
t
+
325
=
0
) M1
×
( 840)
−
2
−
4 720 325 (
×
= −
230,400)
<
0
A1
Or roots 7
4
±
i
(calculator)
12
No real roots oe so never collide* A1*
N.B. Must see justification for ‘no real roots’ to score either of the A
marks.
ALT 2:
Finds minimum value of
720
2
−
840
+
325
or its square M1
t
t
root using derivative or completing the square or calculator
(at t = 7
80 or 80 or
AB = −
4
i
−
8
j
12 ) A1
so never collide* A1*
10
24
12
15
7(e)[4]
−
t
=
t
−
oe
25
t =
or 0.69 or better
36
25
25
AB
=
(10
−
24
×
)
i
+
(12
×
−
15)
j
(km)
36
36
2
AB =
20 3
, 9.4 or better (km)
Notes
M1: Condone sign errors and sin/cos confusion but both components
3
must be resolved. Allow if they use
cos( ) or similar.
7(a)
5
If 12
i
+
16
j appears without working, award M1A0.
A1: cao
7(b)
M1: Correct structure, condone slips
A1ft: ft on their answer to (a)
7(c)
M1: Allow −
r
s . r and s must be substituted.
A1*: Correct given answer, correctly obtained
N.B. Need to see AB
at the start or finish for the A1* and answer must
be exactly as printed, ignoring [] and km.
7(d)
M1: They may use
r
s with both i and j cpts equated.
=
A1: Need both t values. Accept 0.42 or better and 1.25.
A1*: Correct conclusion
7(e)
M1: Correct method
A1: cao
M1: Sub their calculated t value into AB
or BA
, seen or implied, oe.
Note that this is an independent M mark.
A1: cao
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