Question 9:
--- 9(i) ---
9(i) | EITHER:
uuur
FindAPfor a general point P on l with parameter λ , e.g.(8 + 3λ, – 3 − λ, 4 + 2λ) | (B1
uuur
Equate scalar product of APand direction vector of l to zero and solve for λ | M1
Obtainλ=− 5 and foot of perpendicular 3 i+ 3 j+3k
2 2 2 | A1
Carry out a complete method for finding the position vector of the reflection of A in l | M1
Obtain answer 2i+j+2k | A1)
Question | Answer | Marks
OR:
uuur
FindAPfor a general point P on l with parameter λ , e.g.(8 + 3λ, – 3 − λ, 4 + 2λ) | (B1
AP2
Differentiate and solve for λ at minimum | M1
Obtainλ=− 5 and foot of perpendicular 3 i+ 3 j+3k
2 2 2 | A1
Carry out a complete method for finding the position vector of the reflection of A in l | M1
Obtain answer 2i+j+2k | A1)
Total: | 5
--- 9(ii) ---
9(ii) | EITHER:
Use scalar product to obtain an equation in a, b and c, e.g. 3a − b + 2 c = 0 | (B1
Form a second relevant equation, e.g. 9a – b + 8c = 0 and solve for one ratio, e.g. a : b | M1
Obtain final answer a : b : c = 1 : 1 : – 1 and state plane equation x + y – z = 0 | A1)
OR1:
Attempt to calculate vector product of two relevant vectors, e.g. (3i−j+2k)×(9i−j+8k) | (M1
Obtain two correct components | A1
Obtain correct answer, e.g. −6i−6j+6k , and state plane equation −x−y+z=0 | A1)
OR2:
Using a relevant point and relevant vectors, attempt to form a 2-parameter equation for the
r=6i+6k+s(3i−j+2k)+t(9i−j+8k)
plane, e.g. | (M1
State 3 correct equations in x, y, z, s and t | A1
x+ y−z=0
Eliminate s and t and state plane equation , or equivalent | A1)
OR3:
Using a relevant point and relevant vectors, attempt to form a determinant equation for the
x−3 y−1 z−4
plane, e.g. 3 −1 2 =0
9 −1 8 | (M1
Expand a correct determinant and obtain two correct cofactors | A1
Obtain answer −6x−6y+6z=0, or equivalent | A1)
Total: | 3
Question | Answer | Marks
--- 9(iii) ---
9(iii) | EITHER:
uuur
Using the correct processes, divide the scalar product of OAand a normal to the plane by the
modulus of the normal or make a recognisable attempt to apply the perpendicular formula | (M1
1+2−4
Obtain a correct expression in any form, e.g. , or equivalent
2+1 2+(−1) 2
(1 ) | A1 FT
Obtain answer 1 3, or exact equivalent | A1)
OR1:
Obtain equation of the parallel plane through A, e.g. x + y – z = – 1
[The f.t. is on the plane found in part (ii).] | (B1 FT
Use correct method to find its distance from the origin | M1
Obtain answer 1 3, or exact equivalent | A1)
OR2:
Form equation for the intersection of the perpendicular through A and the plane
[FT on their n] | (B1 FT
Solve for λ | M1
1
λn =
3 | A1)
Total: | 3