4MA0 Higher June 2016 Paper 4R Q17
17 The diagram shows a prism.
Diagram NOT accurately drawn
Triangle \(PQR\) is a cross section of the prism.
\(PR\) = 20 cm
\(MP\) = 12 cm
Angle \(PRQ\) = 30°
Angle \(PQR\) = 90°
Calculate the size of the angle that the line \(MR\) makes with the plane \(RQLN\).
Give your answer correct to 1 decimal place.
(5)
| Scheme | Marks |
|---|---|
| B1 | |
\(PQ(ML) = 20\sin 30^\circ\) (=10) or \(MR = \sqrt{12^2 + 20^2} = \sqrt{544} = 4\sqrt{34}\) =23.32..) \(LR = \sqrt{12^2 + (RQ)^2} = \sqrt{12^2 + (10\sqrt{3})^2} = \sqrt{444} = 2\sqrt{111} = 21.07..\) | M2 |
\(\sin MRL = \dfrac{10}{4\sqrt{34}} \left(\dfrac{ML}{MR}\right)\) or \(\cos MRL = \dfrac{2\sqrt{111}}{4\sqrt{34}} \left(\dfrac{LR}{MR}\right)\) or \(\tan MRL = \dfrac{10}{2\sqrt{111}} \left(\dfrac{ML}{LR}\right)\) | M1 |
| 25.4 | A1 |
| (5) |
Notes
B1: Recognition of angle \(LRM\) as required angle either drawn on diagram or from working
M2: For a correct method to calculate \(PQ(ML)\) & \(MR\) or \(MR\) & \(LR\) or \(PQ(ML)\) & \(LR\) (NB: \(LR\) requires use of \(RQ = \sqrt{20^2 - 10^2}\) or \(20\cos 30 = \sqrt{300} = 10\sqrt{3} = 17.32..\))
Or M1 for a correct method to calculate one of the sides \(PQ\) or \(MR\) or \(LR\)
M1: (Dep on M2) Use of a correct trig ratio to find angle \(MRL\)
A1: 25.38 - 25.5