4MA0 Higher June 2017 Paper 3 Q21
21 The diagram shows a triangular prism with a horizontal base \(ABCD\).
Diagram NOT accurately drawn
\(M\) is the midpoint of \(AD\).
The vertex \(V\) is vertically above \(M\).
\(DC = 18\) cm, \(BC = 10\) cm, \(MV = 7\) cm.
Calculate the size of the angle between \(VC\) and the plane \(ABCD\).
Give your answer correct to 3 significant figures.
(4)
| Scheme | Marks |
|---|---|
| B1 | |
\((MC =)\;\sqrt{5^2 + 18^2}\) or \(\sqrt{349}\) or 18.6(8154....) \((VC =)\;\sqrt{5^2 + 7^2 + 18^2}\) or \(\sqrt{398}\) or 19.9(499..) | M1 |
\((VCM =)\tan^{-1}\left(\dfrac{7}{\sqrt{349}}\right)\) or \((VCM =)\sin^{-1}\left(\dfrac{7}{\sqrt{398}}\right)\) or \((VCM =)\cos^{-1}\left(\dfrac{\sqrt{349}}{\sqrt{398}}\right)\) | M1 |
| 20.5 | A1 |
| (4) | |
| (4 marks) |
Notes
B1: for identifying the correct angle on the diagram (may be implied by a correct trig statement)
M1: for a correct method to find \(MC\) or \(VC\)
Accept 18.6(8154....) rounded or truncated to at least 3sf.
Accept 19.9(4993..) rounded or truncated to at least 3 sf
M1: dep M1
for a complete method to find angle \(VCM\) (could be use of sine or cosine rule)
e.g. \(90 - \tan^{-1}\left(\dfrac{\sqrt{349}}{7}\right)\)
A1: accept 20.5 – 20.62