Peperiksaan Akhir Tahun Matematik Tingkatan 4 (Kertas 1 & Kertas 2)
ARAHAN KEPADA CALON / INSTRUCTIONS TO CANDIDATES:
1. This paper consists of Paper 1 (10 Objective Questions) and Paper 2 (Section A & Section B).
2. Answer all questions in Paper 1 and Paper 2. Write your workings clearly in the spaces provided.
3. Non-programmable scientific calculators may be used. Diagrams provided are not drawn to scale unless stated.
Each question is followed by four options, A, B, C and D. Choose the correct answer.
Which of the following quadratic functions has an axis of symmetry given by the line $x = 3$?
Given that $11011_2 + x_5 = 45_{10}$, find the value of $x$.
Consider the statement: "If $x$ is a multiple of 6, then $x$ is a multiple of 3."
Which of the following is the contrapositive of this statement?
Given the universal set $\xi = \{x : 1 \le x \le 20, x \text{ is an integer}\}$, set $P = \{x : x \text{ is a prime number}\}$, and set $Q = \{x : x \text{ is an odd number}\}$. Find $n(P \cap Q')$.
A tree has 8 vertices. How many edges does it contain, and what is the sum of degrees of all vertices?
Which point satisfies the simultaneous linear inequalities $y \le 2x + 1$, $x + y > 3$, and $x \ge 1$?
In a speed-time graph, a motorcycle accelerates uniformly from $12\text{ m s}^{-1}$ to $28\text{ m s}^{-1}$ in 4 seconds. What is its acceleration and the distance traveled during this interval?
A dataset has a standard deviation of 3.2. If every value in the dataset is multiplied by 3 and then increased by 5, what is the new standard deviation?
Event $A$ and event $B$ are two independent events such that $P(A) = \frac{2}{5}$ and $P(B) = \frac{1}{3}$. Find $P(A \cup B)$.
Which of the following statements represents a SMART financial goal?
Answer all questions in this section. Show all working steps clearly.
A rectangular garden has a length of $(3x - 1)\text{ m}$ and a width of $(x + 4)\text{ m}$. Given that the total area of the garden is $84\text{ m}^2$:
(a) Form a quadratic equation in the general form $ax^2 + bx + c = 0$. [2 marks]
(b) Solve the equation to find the positive value of $x$, and hence calculate the perimeter of the garden. [2 marks]
Given that set $M = \{x : x \text{ is a digit in the number } 24305_7\}$ and set $N = \{y : y \text{ is a prime factor of } 140_{10}\}$.
(a) List all the elements of set $M$ and set $N$. [2 marks]
(b) Determine $M \cap N$ and hence find $n(M \cup N)'$ given universal set $\xi = \{0, 1, 2, 3, 4, 5, 6, 7\}$. [2 marks]
(a) Complete the premise in the following deductive argument:
Premise 1: All regular octagons have 8 axes of symmetry.
Premise 2: ..........................................................................................................
Conclusion: Polygon $P$ has 8 axes of symmetry. [1 mark]
(b) Make a general inductive conclusion for the following sequence of numbers:
$7 = 2(1)^2 + 5$
$13 = 2(2)^2 + 5$
$23 = 2(3)^2 + 5$
$37 = 2(4)^2 + 5$
$\dots$ [2 marks]
(c) State whether the inductive argument in (b) is strong or weak, and cogent or not cogent. [1 mark]
The diagram below describes the speed-time graph of a commuter express bus for a duration of $T$ seconds:
(a) State the uniform speed of the bus and the duration, in seconds, for which the bus moves at this uniform speed. [2 marks]
(b) Given that the total distance traveled during the entire journey is $1,140\text{ m}$, calculate the value of $T$. [2 marks]
The marks obtained by 5 participants in a robotics coding trial are: $72, 75, 80, 85, 88$.
(a) Calculate the mean mark, $\bar{x}$. [1 mark]
(b) Calculate the variance, $\sigma^2$, and the standard deviation, $\sigma$. [2 marks]
(c) Another candidate scoring 20 marks is added to the group. Predict the effect on the mean and standard deviation of the group. [1 mark]
Comprehensive HOTS / KBAT questions. Answer all 3 questions.
A cooperative intends to produce $x$ units of souvenir mug $A$ and $y$ units of souvenir mug $B$ for the school carnival based on the following constraints:
(a) Write three linear inequalities other than $x \ge 0$ and $y \ge 0$ which satisfy all the constraints. [3 marks]
(b) Identify whether the production of 40 units of mug $A$ and 50 units of mug $B$ satisfies the constraints. Show your mathematical substitution. [2 marks]
(c) The profit from the sale of mug $A$ is RM 4 and mug $B$ is RM 6. By identifying the coordinates of the feasible vertices, find:
(i) the maximum number of mug $B$ that can be produced if exactly 20 units of mug $A$ are produced. [2 marks]
(ii) the combination of $(x, y)$ that maximizes the cooperative's total profit and calculate the maximum profit. [3 marks]
The table below shows the distance (in km) of proposed high-speed underground fiber-optic links connecting six administrative hubs $A, B, C, D, E$ and $F$:
| Edge | $A-B$ | $A-C$ | $B-C$ | $B-D$ | $C-D$ | $C-E$ | $D-E$ | $D-F$ | $E-F$ |
|---|---|---|---|---|---|---|---|---|---|
| Distance (km) | 14 | 9 | 11 | 18 | 15 | 12 | 8 | 16 | 10 |
(a) State the number of vertices and the number of edges in a tree that connects all 6 hubs. [2 marks]
(b) Using Kruskal's or Prim's algorithm, select the edges to form a minimum spanning tree (MST). List the sequence of edges selected and their weights. [4 marks]
(c) Calculate the total minimum distance of fiber optic cable required. [2 marks]
(d) If the installation cost is RM 45,000 per kilometer, calculate the total cost saved compared to connecting every possible direct link in the table. [2 marks]
A supermarket runs an anniversary customer appreciation event. Every shopper spending over RM 100 draws two discount cards without replacement from a bag containing 3 Gold cards (RM 50 voucher), 5 Silver cards (RM 20 voucher), and 7 Bronze cards (RM 10 voucher).
(a) Draw a probability tree diagram representing all possible outcomes and their probabilities. [3 marks]
(b) Calculate the probability that a customer wins:
(i) Exactly two Gold cards. [2 marks]
(ii) A total voucher value of at least RM 60. [3 marks]
(c) Encik Razif wins RM 70 in vouchers. He has a monthly net salary of RM 3,800, fixed expenses of RM 2,200, and variable expenses of RM 1,400. He plans to save RM 3,600 over 12 months for an emergency medical insurance rider.
Evaluate whether his current financial situation allows him to achieve this goal without relying on the vouchers. [2 marks]