SPM Mathematics Trial 2024 (Selangor Set 2), Paper 2 (Question 12)

Question 12:
(a) $$\text { It is given that matrix } H=\left(\begin{array}{cc} 2 & 6 \\ 1 & k \end{array}\right) \text {. }$$

(a) Find the value of k, if the inverse matrix of H does not exist. [2 marks]

(b)(i) In this year, sum of Puan Mariam’s age and her son’s age is 34 . Three years later, Puan Mariam’s age will be triple her son’s age.
By using matrix method, calculate the age of Puan Mariam and her son this year. [5 marks]

(b)(ii) Puan Mariam plans to retire at the age of 55 and she expects that her son will finish his education at the age of 25 . Based on your answer in 12(b)(i), will her son be able to finish his studies before she retires? Justify your answer. [2 marks]

Solution:
(a)
$$\begin{aligned} &\text { Given that matrix } H \text { does not have inverse matrix, }\\ &\begin{aligned} a d-b c & =0 \\ 2(k)-6(1) & =0 \\ 2 k-6 & =0 \\ 2 k & =6 \\ k & =3 \end{aligned} \end{aligned}$$


(b)(i)
Let, x = age of Puan Mariam
y = age of her son

Given that the sum of Puan Mariam’s age and her son’s age is 34.
x + y = 34 ….. (1)

3 years later, Puan Mariam’s age will be 3 times her son’s age.
$$\begin{aligned} x+3 & =3(y+3) \\ x & =3 y+9-3 \\ x-3 y & =6 \ldots (2)\\ {\left[\begin{array}{cc} 1 & 1 \\ 1 & -3 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right] } & =\left[\begin{array}{c} 34 \\ 6 \end{array}\right] \\ {\left[\begin{array}{l} x \\ y \end{array}\right] } & =\frac{1}{1(-3)-1(1)}\left[\begin{array}{cc} -3 & -1 \\ -1 & 1 \end{array}\right]\left[\begin{array}{c} 34 \\ 6 \end{array}\right] \\ & =-\frac{1}{4}\left[\begin{array}{l} -3(34)-1(6) \\ -1(34)+1(6) \end{array}\right] \\ & =-\frac{1}{4}\left[\begin{array}{c} -108 \\ -28 \end{array}\right] \\ & =\left[\begin{array}{c} 27 \\ 7 \end{array}\right] \end{aligned}$$

Age of Puan Mariam is 27 and her son is 7 this year.

(b)(ii)
Age of Puan Mariam when her son finishes her study
= 27 + (25 – 7)
= 45 years old

Yes, Puan Mariam’s age when her son finishes his studies is 45 years old (before her retirement).