Mathematics · Oct / Nov 2022 · Paper 41 · Question 9
YMA01
As printed in the paper
The question as text
9. With respect to a fixed origin O, the equations of lines l1 and l2 are given by 2 − 1 l1 : r = 8 + λ 2 10 3 − 4 5 l2 : r = + μ − 1 4 2 8 where λ and μ are scalar parameters. Prove that lines l1 and l2 are skew. (5)
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
8
+
2
λ
=
−+
1
4
μ
10
+
3
λ
2
+
8
μ
Attempts to solve any two of the three equations
✓
Either (1) and (2) 2
−λ= −+ μλ= −
4
5
3
3
,
μ
=
8
+ λ= −+ μ
2
1
4
2
2
✓
(1) and (3) 2
−λ= −+ μλ=
4
5
8
26
,
μ
=
10
+ λ=
3
2
+ μ
8
23
23
✓
+ λ= −+
μ
(2) and (3) 8
2
1
4
λ= −
11
10,
μ
= −
10
+ λ=
3
2
+
8
μ
4
Substitutes their values of λ and μ into both sides of the ''third'' equation
11
3
3
+
λ
=
E.g.
λ= −
into
10
3
and
μ
=
into 2
+ μ
8
=14
2
2
2
Concludes that lines don't intersect with correct calculations and minimal
reason
−
1
5
Additionally states that
2
is not parallel to
4
with a minimal reason
3
8
So lines are skew CSO *
Notes:
Main method seen
M1: Attempts to solve two of the three equations.
Accept as an attempt, writing down two of the three equations (condoning slips) followed by
values for
both λ and μ
A1: Solves two of the three equations to find correct values for both λ and μ, Allow equivalent
fractions
dM1: Either: Substitutes their values of λ and μ into both sides of the third equation....or into the
equations of both lines to find both coordinates
A1: Having achieved correct values for λ and μ, the values for the third equation are found to
enable a comparison to be made. E.g. solving equations (1) and (2) and using equation (3) stating
3
3
+ ×−
≠
+ ×
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