Mathematics · Oct / Nov 2023 · Paper 51 · Question 6
YMA01
As printed in the paper
The question as text
6. [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.] At 12:00, a ship P sets sail from a harbour with position vector (15i + 36j) km. At 12:30, P is at the point with position vector (20i + 34j) km. Given that P moves with constant velocity,
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6(a)[2]
i
j
i
j oe
0.5
(10
−
4 )
j *
i
(15
+
36 )
+
(10
−
4 )
j
6(b)[2]
i
j
t
i
Verify using
t =
1.5
in p or q
6(c)(i)
j
p
=
(15
i
+
36 )
j
+
1.5(10
i
−
4 )
j
=
30
i
+
30
=
(42
−×
8 1.5)
+
(9 14 1.5)
+
×
=
30
+
30
j
q
i
j
i
j
30
+
30
(ii)
i
N.B. The A mark for (ii) is now to be treated as a B mark.
ALT1 (i)
Find t by equating i or j components of p and q
Equate i’s 15 10
+
t
=
42
−
8
t
→ t = 1.5
j’s 36
−
4
t
=
9 14
+
t
→t = 1.5
30
j
(ii)
i
+
30
ALT2 (i)
Uses ratio: 15 10
+
t
42
−
8
t
=
36
−
4
9 14
+
t
t
→t = 1.5 or −8.5
verifies that components are both 30 at t = 1.5
30
30
j
(ii)
i
+
Position of P at 14:30 is 40
j
6(c)[4]
i
j
i
j oe
0.5
(10
−
4 )
j *
i
(15
+
36 )
+
(10
−
4 )
j
i
j
t
i
Verify using
t =
1.5
in p or q
6(c)(i)
j
p
=
(15
i
+
36 )
j
+
1.5(10
i
−
4 )
j
=
30
i
+
30
=
(42
−×
8 1.5)
+
(9 14 1.5)
+
×
=
30
+
30
j
q
i
j
i
j
30
+
30
(ii)
i
N.B. The A mark for (ii) is now to be treated as a B mark.
ALT1 (i)
Find t by equating i or j components of p and q
Equate i’s 15 10
+
t
=
42
−
8
t
→ t = 1.5
j’s 36
−
4
t
=
9 14
+
t
→t = 1.5
30
j
(ii)
i
+
30
ALT2 (i)
Uses ratio: 15 10
+
t
42
−
8
t
=
36
−
4
9 14
+
t
t
→t = 1.5 or −8.5
verifies that components are both 30 at t = 1.5
30
30
j
(ii)
i
+
Position of P at 14:30 is 40
j
i
+
26
Position of Q when t = 0.5
=
(42
−×
8 0.5)
+
(9 14 0.5)
+
×
j
q
i
(=(38
+
16 )
j )
i
15j seen or implied
New position of Q at time 14:30
j
q
=
(38
i
+
16 )
j
+
2(15 )
N.B. M0 if 2.5 is used.
38
46
j
q
=
i
+
| PQ | =
(40
−
38)
2
+
(26
−
46)
2
= 404 or 2 101 (km)
Notes for question 6
N.B. Allow use of column vectors throughout apart from the
A marks in (a) and (b).
(a)
Complete method to find an expression for the velocity.
M1
Allow if they use 30 minutes.
Give M1A0, if there are missing brackets.
Reaches the given answer from fully correct working.
A1*
(b)
M1
Finds an expression for p in terms of t with the correct
structure
Correct answer in terms of i, j and t.
A1
(c)
(i) M1
Substitutes t = 1.5 into the given q or their p
P equation correct
A1
Q equation correct
A1
N.B. p or q = 30
30
i
+
jalone can imply a correct equation in
each case.
(ii) A1(B1)
30
+
30
j seen
i
ALT 1
Equates coefficients of i or j using the given q and their p
(i) M1
A1
Correct equation for i leading to t = 1.5
Correct equation for j leading to t = 1.5
A1
N.B. Allow both A marks if they only write t = 1.5 once.
(ii) A1(B1)
30
+
30
j seen
i
ALT 2
Uses ratio of components to form equation.
(i) M1
A1
Two t values
Verifies that components are both 30 at t = 1.5
A1
j seen
(ii) A1(B1)
30
i
+
30
(d)
B1
Position of P at 14:30
Use t = 0.5 to find the new position of Q at 12:30
M1
B1
Correct expression seen for new velocity of Q
Complete method to find the new position of Q at 14:30,
M1
using their new v for Q.
A1
Correct position,
Use of Pythagoras to find the distance. Dependent on both
dM1
previous M marks.
Correct surd answer
A1
6(d)[7]
i
+
26
Position of Q when t = 0.5
=
(42
−×
8 0.5)
+
(9 14 0.5)
+
×
j
q
i
(=(38
+
16 )
j )
i
15j seen or implied
New position of Q at time 14:30
j
q
=
(38
i
+
16 )
j
+
2(15 )
N.B. M0 if 2.5 is used.
38
46
j
q
=
i
+
| PQ | =
(40
−
38)
2
+
(26
−
46)
2
= 404 or 2 101 (km)
Notes for question 6
N.B. Allow use of column vectors throughout apart from the
A marks in (a) and (b).
(a)
Complete method to find an expression for the velocity.
M1
Allow if they use 30 minutes.
Give M1A0, if there are missing brackets.
Reaches the given answer from fully correct working.
A1*
(b)
M1
Finds an expression for p in terms of t with the correct
structure
Correct answer in terms of i, j and t.
A1
(c)
(i) M1
Substitutes t = 1.5 into the given q or their p
P equation correct
A1
Q equation correct
A1
N.B. p or q = 30
30
i
+
jalone can imply a correct equation in
each case.
(ii) A1(B1)
30
+
30
j seen
i
ALT 1
Equates coefficients of i or j using the given q and their p
(i) M1
A1
Correct equation for i leading to t = 1.5
Correct equation for j leading to t = 1.5
A1
N.B. Allow both A marks if they only write t = 1.5 once.
(ii) A1(B1)
30
+
30
j seen
i
ALT 2
Uses ratio of components to form equation.
(i) M1
A1
Two t values
Verifies that components are both 30 at t = 1.5
A1
j seen
(ii) A1(B1)
30
i
+
30
(d)
B1
Position of P at 14:30
Use t = 0.5 to find the new position of Q at 12:30
M1
B1
Correct expression seen for new velocity of Q
Complete method to find the new position of Q at 14:30,
M1
using their new v for Q.
A1
Correct position,
Use of Pythagoras to find the distance. Dependent on both
dM1
previous M marks.
Correct surd answer
A1
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