The diagram shows a pyramid OABC with a horizontal triangular base OAB and vertical height OC. Angles AOB, BOC and AOC are each right angles. Unit vectors i, j and k are parallel to OA, OB and OC respectively, with OA = 4 units, OB = 2.4 units and OC = 3 units. The point P on CA is such that CP = 3 units. [Figure 7]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2016). The mark or mark type is shown in brackets.

1. [M1] CP = 3/5 CA soi
2. [A1] CA = OA - OC = 4i - 3k, so CP = 3/5 (4i - 3k) = 2.4i - 1.8k AG
3. [B1] OP = OC + CP = 3k + (2.4i - 1.8k) = 2.4i + 1.2k
4. [B1] BP = OP - OB = (2.4i + 1.2k) - 2.4j = 2.4i - 2.4j + 1.2k
5. [M1] BP ⋅ CP = (2.4i - 2.4j + 1.2k) ⋅ (2.4i - 1.8k) = 5.76 - 2.16 = 3.6 (Use of x₁x₂ + y₁y₂ + z₁z₂)
6. [M1] |BP||CP| = √(2.4² + (-2.4)² + 1.2²) √(2.4² + (-1.8)²) = √(12.96)√(9) (Product of moduli)
7. [M1] cos BPC = (BP ⋅ CP) / (|BP||CP|) = 3.6 / (√12.96 * 3) = 1/3 (All linked correctly)
8. [A1] Angle BPC = 70.5° (or 1.23 rads) cao