SPM Mathematics Trial 2021 (Kelantan), Paper 2 (Question 17)

Question 17 (15 Marks):
(a) Diagram 7.1 shows the shape of the cake that Cikgu Alam wants to order to celebrate her class’s cheerful victory. The two-tiered cake consists of a combination of a cuboid and a cylinder with a radius of 14 cm.


The total volume of the cake in Figure 7.1 is 34 998 cm³. Cikgu Alam then wants to change the cylindrical cake to a cube-shaped cake as in Figure 7.2 without changing the original volume of the cake.
[use π = 22/7 ]

(i) Calculate the height, in cm, of the cylindrical cake in Figure 7.1. Round the correct answer to 3 significant figures. [4 marks]

(ii) Calculate the length, in cm, of the sides of the cube -shaped cake. Round off the correct answer to 2 significant figures. [3 marks]


Solution:
(a)(i)

$\begin{array}{l}\text{The volume of cake in Diagram 7}\text{.1}=34998\\ \text{The volume of cuboid + volume of cylinder}=34998\\ 36\times 30\times 27+\pi r^{2}h=34998\\ 29160+\left(\frac{22}{7}\times {14}^2\times h\right)=34998\\ 29160+616h=34998\\ 616h=5838\\ h=\frac{5838}{616}\\ h=9.48\text{ cm}\end{array}$


(a)(ii)

$\begin{array}{l}\text{Let the side of the cube is }x\\ \text{The volume of cake in Diagram 7}\text{.2}=34998\\ \text{The volume of cuboid + volume of cube}=34998\\ 29160+x^3=34998\\ x^3=34998-29160\\ x^3=5838\\ x=\sqrt[3]{5838}\\ x=18\text{ cm}\end{array}$

(b):
Cikgu Alam decides to order a cake as shown in Figure 7.2. During the ceremony, Cikgu Alam cut the first part of the cube -shaped cake. The remainder of the cake after the first cut is shown in Figure 7.3.


Diagram 7.3

The edges of RK and SL are perpendicular. Given GF = ML = 12 cm. By using a scale of 1 : 3, draw the elevation of the remaining cake on a vertical plane parallel to BC, as viewed from Y. [4 marks]


Solution:

(c):
The probability that the ordered cake cannot be eaten after a week is 1/8. Two cakes are chosen at random.
(i) Complete the tree diagram in the answer space to show all possible outcomes. [2 marks]
(ii) Calculate the probability that at least one of the selected cakes cannot be eaten after a week. [2 marks]

Answer:


Solution:
(c)(i)


(c)(ii)

$\begin{array}{l}\text{Probability that at least one of the selected cakes}\\ \text{cannot be eaten after a week}\\ =P\left(YN\right)+P\left(NY\right)+P\left(NN\right)\\ =\left(\frac{7}{8}\times \frac{1}{8}\right)+\left(\frac{1}{8}\times \frac{7}{8}\right)+\left(\frac{1}{8}\times \frac{1}{8}\right)\\ =\frac{7+7+1}{64}\\ =\frac{15}{64}\\ \\ \text{Alternative method}\\ =1-P\left(YY\right)\\ =1-\left(\frac{7}{8}\times \frac{7}{8}\right)\\ =1-\frac{49}{64}\\ =\frac{15}{64}\end{array}$