Mathematics · Oct / Nov 2025 · Paper 41 · Question 5
YMA01
As printed in the paper
The question as text
5. The line l1 has equation 2 1 r λ 1 4 where λ is a scalar parameter 3 2 The line l2 has equation 1 2 r 1 where β is a constant and μ is a scalar parameter 6 7 Given that l1 and l2 intersect at the point P (i) find the value of β (ii) find the coordinates of P (6)
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
1
2
+ λ=
2
6
7
Correct solution
λ= −
μ
= −
9,
Substitutes their
λ= −
μ
= −
into 1
−+ λ=
β
−
μ
9,
4
β
= −
40
2
1
E.g. Substitutes their
λ= −into
9
= −
+ λ
(ii)
r
1
4
3
2
(
)
−
7,
−
37, 15
−
(i)
M1:
Attempts to write down and solve equations (1) and (3) to find the
value of either λ or μ
Do not be concerned by the mechanics of the rearrangement in solving.
May be implied by
λ= −
9 or
μ
= −
.
A1:
Correct solution for both
λ= −
μ
= −
9,
dM1: Substitutes their values for λ and μinto equation (2) and solves to find a
value for β
Do not be concerned by the mechanics of the rearrangement in solving. It
is dependent on the previous method mark.
A1:
β
= −
40
(ii)
2
1
dM1: Scored for either substituting their
λ= − into
or
9
r
= −
1
+ λ
4
3
2
−
1
2
β
= −
substituting their
μ
= −
and
40
into
r
=
β
+
μ
−
1
and
6
7
proceeding to find coordinates for P. Do not allow this mark to be scored
if they then do some further manipulation.
It is dependent upon the attempt to find either λ or both of μand β. May
be implied by 2 correct coordinates for their λ or both of μand β.
−
7
A1:
(
)
or position vector e.g. 7 37 15
k or
condone
−
7,
−
37, 15
−
i
j
−
37
−
15
(
)
−
7 ,
i
−
37 , 15
j
−
k
Allow
= −
= −
= −
or condone 7,
−
−
−
. Do not isw if they
x
7,
y
37,
z
15
37,
15
calculate two different sets of coordinates using μand β for one and
then λ for the other unless they select the correct answer.
d
u
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