Mathematics · May / June 2022 · Paper 41 · Question 6
YMA01
As printed in the paper
The question as text
6. Relative to a fixed origin O, • the point A has position vector i – 4j + 3k • the point B has position vector 5i + 3j – 2k • the point C has position vector 3i + pj – k where p is a constant. The line l passes through A and B.
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
6(a)[3]
5 1
−
4
AB
3
4
7
4
7
5
=
−−
=
=
i
+
j
−
k
−−
2
3
−
5
e.g.
r
= −
i
4
j
+
3
k
+
λ
(4
i
+
7
j
−
5 )
k or
r
=
5
i
+
3
j
−
2
k
+
λ
(4
i
+
7
j
−
5 )
k
6(b)[3]
3 1
−
2
AC
=
p
−−
4
=
p
+
4
=
2
i
+
(
p
+
4)
j
−
4
k
−−
1 3
−
4
2
4
p
+
4
7
=
8
+
7
p
+
28
+
20
=
0
p
= −
8
−
4
−
5
6(c)[3]
AB =
4
2
+
7
2
+ −
( 5)
2
=
90
or
AC =
2
2
+ −
( 4)
2
+ −
( 4)
2
=
6
Area 1
2 ×
" 90" "6"
×
=
9 10
Notes
Accept either vector form throughout but extra i, j k in column vectors will lose A mark in (a).
(a) This is now being marked MMA
M1: Attempts to find AB
. Score for subtracting either way round. Implied by 2 out of 3 correct coordinates.
M1: Attempts equation for the line, score for
OA
+
λ
×
their
AB
or
OB
+
λ
×
their
AB
No need for r = for
this mark.
A1: Any correct equation. Must be r = … (l = .. is A0)
(b)
M1: Attempts to find AC
. Score for subtracting either way round. Implied by 2 out of 3 correct coordinates.
M1: Takes scalar product of their AB
and their AC
to form and solve a linear equation in p
p = −
8
A1:
(c)
M1: Attempts to find the magnitude of either their AB
or their AC
using their p
dM1: Attempts to find the exact area of the triangle ABC. It is dependent on the previous method mark. There
most common method will be 1
2 AB AC
as in scheme but other methods are possible. E.g.
BA BC
.
1
c
os
∠
ABC
=
A
=
BA BC
s
i
n
∠
ABC
. Such a method must be complete, including use of
2
BA BC
Pythagorean identity to find sin
∠
ABC
. Other more advanced methods (such as cross products) are also
possible. If you see something you feel is worthy of some credit but does not fit the scheme, send to Review.
A1: 9 10
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