O-LevelAdditional MathematicsVectors in two dimensions4037/21 · May/June 2010 · Q108 Marks
In this question, (1 / 0) is a unit vector due east and (0 / 1) is a unit vector due north. A lighthouse has position vector (27 / 48) km relative to an origin O. A boat moves in such a way that its position vector is given by (4+8t / 12+6t) km, where t is the time, in hours, after 1200.
- (i)Show that at 1400 the boat is 25 km from the lighthouse.[4]
- (ii)Find the length of time for which the boat is less than 25 km from the lighthouse.[4]
Mark scheme
Marking points from the official mark scheme (May/June 2010). The mark or mark type is shown in brackets.
- [B1] (i) OP = (20 / 24)
- [B1] (i) PL = (-7 / -24) or LP = (7 / 24)
- [M1A1] (i) √7² + 24² = 25
- [B1] (ii) PL = (8t-23 / 6t-36) or LP = (23-8t / 36-6t)
- [M1] (ii) (23-8t)²+(36-6t)² = 25²
- [M1] (ii) Solve 3 term quadratic [100(t² - 8t+12)=0]
- [A1] (ii) 6 - 2 = 4 hours
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