Question 10:
--- 10(i) ---
10(i) | Equate at least two pairs of components of general points on l and m and solve for λ
or for µ | M1
Obtain correct answer for λ or µ, e.g. λ = 3 or µ = – 2 ; λ=0 or µ=−1;or
2
λ= 3 or µ=−7
2 2 | A1
Verify that not all three pairs of equations are satisfied and that the lines fail to
intersect | A1
3
--- 10(ii) ---
10(ii) | Carry out correct process for evaluating scalar product of direction vectors for l and m | *M1
Using the correct process for the moduli, divide the scalar product by the product of
the moduli and evaluate the inverse cosine of the result | DM1
Obtain answer 45° or 1π(0.785) radians
4 | A1
3
--- 10(iii) ---
10(iii) | EITHER: Use scalar product to obtain a relevant equation in a, b and c, e.g.
−a+b+4c=0 | B1
Obtain a second equation, e.g. 2a+b−2c=0and solve for one ratio,
e.g. a : b | M1
Obtain a : b : c = 2 : −2 : 1, or equivalent | A1
Substitute (3, −2, −1) and values of a, b and c in general equation and find d | M1
Obtain answer 2x−2y+z=9, or equivalent | A1
OR1: Attempt to calculate vector product of relevant vectors, e.g
(−i+j+4k)×(2i+j−2k) | (M1
Obtain two correct components | A1
Obtain correct answer, e.g. −6i+6j−3k | A1
Substitute (3, −2, −1) in −6x+6y−3z=d , or equivalent, and find d | M1
Obtain answer −2x+2y−z=−9, or equivalent | A1)
OR2: Using the relevant point and relevant vectors, form a 2-parameter equation
for the plane | (M1
State a correct equation, e.g. r=3i−2j−k+λ(−i+j+4k)+µ(2i+−j 2k) | A1
State three correct equations in x, y, z, λ and µ | A1
Eliminate λ and µ | M1
Question | Answer | Marks
Obtain answer 2x−2y+z=9, or equivalent | A1)
OR3: Using the relevant point and relevant vectors, form a determinant equation
for the plane | (M1
x−3 y+2 z+1
State a correct equation, e.g. −1 1 4 =0
2 1 −2 | A1
Attempt to expand the determinant | M1
Obtain two correct cofactors | A1
Obtain answer −2x+2y−z=−9, or equivalent | A1)
5