SPM Mathematics 2017, Paper 2 (Questions 3 & 4)

Question 3 (4 marks):
Diagram 2 shows a composite formed by joining a quarter cylinder and a right prism at the rectangular plane EFGH. The trapezium ABGF and the quarter circle FGK are the uniform cross sections of the solid.

Diagram 2

Using  $\pi =\frac{22}{7}$ , calculate the volume, in  cm³, of the composite solid.


Solution:
$\begin{array}{l}\text{Volume of }FGK\\ =\frac{1}{4}\left(\pi r^{2}h\right)\\ =\frac{1}{4}\left(\frac{22}{7}\times 5^2\times 7\right)\\ =\frac{1}{4}\left(\frac{22}{7}\times 25\times 7\right)\\ =137\frac{1}{2}{\text{ cm}}^3\\ \\ \text{Volume of }ABGF\\ =\left[\frac{1}{2}\left(10+5\right)5\right]\times 7\\ =\frac{5}{2}\left(15\right)\times 7\\ =\frac{525}{2}\\ =262\frac{1}{2}{\text{ cm}}^3\\ \\ \text{Total volume of solid}\\ =137\frac{1}{2}{\text{ cm}}^3+262\frac{1}{2}{\text{ cm}}^3\\ =400{\text{ cm}}^3\end{array}$

Question 4 (5 marks):
Solution by matrix method is not allowed to answer this question.
Diagram 3 shows a bicycle and a tricycle.

Diagram 3

Calculate the number of bicycles and the number of tricycles if there are 64 pedals and 74 tyres.


Solution:
Let bicycle = x and tricycle = y
So,
2x + 2y = 64 …………. (1)
2x + 3y = 74 …………. (2)
(2) – (1)
3y – 2y = 74 – 64
y = 10

When y = 10
From (2)
2x + 3(10) = 74
2x = 74 – 30
2x = 44
x = 22

Thus, the number of bicycles are 22 and the number of tricycles are 10.