A-LevelMathematicsVectors9709/12 · Oct/Nov 2015 · Q79 Marks
Relative to an origin O, the position vectors of points A, B and C are given by `OA = (0, 2, -3)` `OB = (2, 5, -2)` `OC = (3, p, q)`
- (i)In the case where ABC is a straight line, find the values of p and q.[4]
- (ii)In the case where angle BAC is 90°, express q in terms of p.[2]
- (iii)In the case where p = 3 and the lengths of AB and AC are equal, find the possible values of q.[3]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2015). The mark or mark type is shown in brackets.
- [B1 B1] AB = (2, 3, 1), AC = (3, p-2, q+3), BC = (1, p-5, q+2) (Any 2 of 3 relevant vectors)
- [B1 B1] → p = 6½ and q = -1½
- [M1] 6 + 3p - 6 + q + 3 = 0 (Use of x₁x₂ + y₁y₂ + z₁z₂ = 0 for dot product)
- [A1] → q = -3p - 3
- [M1] AB² = 4 + 9 + 1, AC² = 9 + 1 + (q+3)² (For attempt at either)
- [A1 A1] → (q+3)² = 4 → q = -1 or -5
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