Question 11:
11 | (a) | 2  8 + ( − 1 )  4 + 2  5 − 4
d =
2 2 + ( − 1 ) 2 + 2 2
= 6 | M1
A1 | 1.1
1.1 | Use of distance from point to plane formula, must be
correct. Condone missing modulus.
Alternative method
Perpendicular from P to plane intersects plane at (4,6,1)
𝑑 = √(8−4)2+(4−6)2+(5−1)2 | M1 | No slips allowed
= 6 | A1
[2]
11 | (b) | 8 − 2 4 5 + 3
= = [ = 2 ]
3 2 4
so P(8, 4, 5) lies on the line | M1
A1 | 2.1
2.2a | Substituting coordinates into line equation
Conclusion
Alternative method
x = 2 + 3, y = 2, z = −3 + 4
2 + 3 = 8
  = 2 | M1 | A method to find  from one coordinate
y = 22 = 4, z = −3 + 42 = 5
so P(8, 4, 5) lies on the line | so P(8, 4, 5) lies on the line | A1 | A1 | Must show substitution for all three coordinates and
conclude
[2]
Alternative method | M1 | For an equation of the plane in only one of 𝑥, 𝑦 or 𝑧.
Allow a slip.
3𝑦+4 3𝑦+4
𝑥 = and 𝑧 = 2𝑦−3 ⇒ 2( )−𝑦+2(2𝑦−3) = 4
2 2
𝑦 = 1 | A1 | Solving for the correct coordinate
7
Point of intersection is ( ,1,−1)
2 | A1 | 7
Accept 𝑥 = , 𝑦 = 1, 𝑧 = −1
2
Do not accept a position vector
[3]
For an equation of the plane in only one of 𝑥, 𝑦 or 𝑧.
Allow a slip.
11 | (d) | Let angle between line and normal = 
(2𝐢−𝐣+2𝐤).(3𝐢+2𝐣+4𝐤)
cos𝜃 =
√22+(−1)2+22×√32+22+42
1 2
=
9 2 9
 = 42.03
90−42.03
angle between line and plane =
= 48.0 | M1
A1
M1
A1 | 1.1
1.1
1.1
1.1 | Must see a dot product or sum of three terms
or 0.73[358…] rad
Allow their angle 𝜃−90 if their 𝜃 obtuse. Can be implied
by correct final answer if correct 𝜃 found.
or 0.837… rad. www.
Alternative method 1
Let angle between line and plane = 𝜙
(2𝐢−𝐣+2𝐤).(3𝐢+2𝐣+4𝐤)
sin𝜙 =
√22+(−1)2+22×√32+22+42 | (2𝐢−𝐣+2𝐤).(3𝐢+2𝐣+4𝐤)
sin𝜙 =
√22+(−1)2+22×√32+22+42 | M2
Must see a dot product or sum of three terms
1 2
=
9 2 9
𝜙 = 48.0° | A2 | or 0.837… rad
Alternative method 2
Let angle between line and normal = 𝜃
|(2𝐢−𝐣+2𝐤)×(3𝐢+2𝐣+4𝐤)|
sin𝜃 =
√22+(−1)2+22×√32+22+42 | M1 | Complete method using vector product. Do not condone
missing modulus unless a correct value for sin𝜃 is given
√117
=
√9√29
 = 42.03 | A1 | or 0.73[358…] rad
90−42.03
angle between line and plane = | 90−42.03
angle between line and plane = | M1 | M1 | 1.1 | 1.1 | Allow their angle 𝜃−90 if their 𝜃 obtuse. Can be implied
by correct final answer if correct 𝜃 found.
= 48.0 | A1 | 1.1 | or 0.837… rad. www.
[4]
M2
Complete method using vector product. Do not condone
missing modulus unless a correct value for sin𝜃 is given
11 | (e) | 7
P to intersection = √(8− )2+(4−1)2+(5+1)2
2
3 2 9
=
2
3√29 12
× = 6
2 3√29 | M1
A1
A1
[3] | 3.1a
1.1
3.2a | FT their coordinates from (c)
or 8.0777... Do not FT.
Condone using approximate values, e.g sin 48 or cos 42
12
for , in which case condone ≈ for =
3√29
FT their angle labels from previous parts