A-LevelMathematicsVectors9709/12 · Oct/Nov 2010 · Q98 Marks
The diagram shows a pyramid OABCP in which the horizontal base OABC is a square of side 10 cm and the vertex P is 10cm vertically above O. The points D, E, F, G lie on OP, AP, BP, CP respectively and DEFG is a horizontal square of side 6 cm. The height of DEFG above the base is a cm. Unit vectors i, j and k are parallel to OA, OC and OD respectively. [Figure 9.1]
- (i)Show that a = 4.[2]
- (ii)Express the vector BG in terms of i, j and k.[2]
- (iii)Use a scalar product to find angle GBA.[4]
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Mark scheme
Marking points from the official mark scheme (Oct/Nov 2010). The mark or mark type is shown in brackets.
- [M1] (10-a) / 10 = 6 / 10 oe or PDE is isos hence PD = 6 (M1)
- [A1] a = 4 AG
- [B2,1] BG = -10j – 10i + 4k + 6j = -10i - 4j + 4k Any acceptable notation. Loses 1 for each error.
- [M1] BG.BA = 40 Use of x1x2 + y1y2 + z1z2
- [M1] cos GBA = 40 / (√132√100) Modulus worked correctly for either
- [DM1] All ok – must be using BG . BA
- [A1] GBA = 69.6° Must be the acute angle
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