Matematik Tingkatan 4 Kertas 2Mathematics Form 4 Paper 2

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1

Question 1

Jawab soalan berikut tentang persamaan kuadratik.Answer the following about quadratic equations.

[2 marks] (a)Tulis 3x(x − 4) = 5 − 2x dalam bentuk am ax² + bx + c = 0.Write 3x(x − 4) = 5 − 2x in the general form ax² + bx + c = 0.

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• 3x² − 12x = 5 − 2x [1]1
• 3x² − 10x − 5 = 0 [1]1

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[2 marks] (b)Selesaikan 2x² − 5x − 12 = 0 dengan pemfaktoran.Solve 2x² − 5x − 12 = 0 by factorisation.

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• (2x + 3)(x − 4) = 0 [1]1
• x = −3/2 or x = 4 [1]1

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For the teacher: Applying, medium difficulty, Chapter 1, standard 1.1.8

Textbook p. 24

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Question 2

Jawab soalan berikut tentang asas nombor.Answer the following about number bases.

[2 marks] (a)Tukarkan 1011101₂ kepada nombor dalam asas lapan.Convert 1011101₂ to a number in base eight.

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• Group in threes from the right: 001 011 101 [1]1
• 135₈ [1]1

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[2 marks] (b)Hitung 5024₆ − 1345₆. Beri jawapan dalam asas enam.Calculate 5024₆ − 1345₆. Give the answer in base six.

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• Convert to base ten: 1096 − 353 = 743 (or subtract column by column in base six) [1]1
• 3235₆ [1]1

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For the teacher: Applying, low difficulty, Chapter 2, standard 2.1.4

Textbook p. 49

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Question 3

Jawab soalan berikut tentang penaakulan logik.Answer the following about logical reasoning.

[2 marks] (a)Tulis akas bagi implikasi, "Jika x ialah gandaan 4, maka x ialah gandaan 2". Nyatakan sama ada akas itu benar atau palsu. Beri satu contoh penyangkal jika akas itu palsu.Write the converse of the implication, "If x is a multiple of 4, then x is a multiple of 2". State whether the converse is true or false. Give a counter-example if it is false.

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• Converse: If x is a multiple of 2, then x is a multiple of 4 [1]1
• False; counter-example x = 6 (or 2, 10) [1]1

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[1 mark] (b)Lengkapkan hujah berikut.
Premis 1: Semua gandaan 9 ialah gandaan 3.
Premis 2: 63 ialah gandaan 9.
Kesimpulan: ______Complete the following argument.
Premise 1: All multiples of 9 are multiples of 3.
Premise 2: 63 is a multiple of 9.
Conclusion: ______

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• 63 is a multiple of 3 [1]1

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[2 marks] (c)Perhatikan corak nombor 3 = 1 × 3, 8 = 2 × 4, 15 = 3 × 5, 24 = 4 × 6. Dengan penaakulan induktif, tentukan sebutan keenam.Observe the number pattern 3 = 1 × 3, 8 = 2 × 4, 15 = 3 × 5, 24 = 4 × 6. Using inductive reasoning, find the sixth term.

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• The n-th term = n(n + 2) [1]1
• 6 × 8 = 48 [1]1

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For the teacher: Applying, medium difficulty, Chapter 3, standard 3.2.4

Textbook p. 81

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Question 4

ξ = {x : x ialah integer, 1 ≤ x ≤ 15}, set A = {x : x ialah gandaan 3} dan set B = {x : x ialah faktor bagi 12}.ξ = {x : x is an integer, 1 ≤ x ≤ 15}, set A = {x : x is a multiple of 3} and set B = {x : x is a factor of 12}.

[1 mark] (a)Senaraikan semua unsur bagi A ∩ B.List all the elements of A ∩ B.

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• {3, 6, 12} [1]1

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[2 marks] (b)Cari n((A ∪ B)′).Find n((A ∪ B)′).

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• A ∪ B = {1, 2, 3, 4, 6, 9, 12, 15}, so n(A ∪ B) = 8 [1]1
• 15 − 8 = 7 [1]1

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For the teacher: Applying, low difficulty, Chapter 4, standard 4.2.2

Textbook p. 110

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Question 5

Rajah 1 menunjukkan graf yang mewakili lima pondok pengawal, P, Q, R, S dan T, serta lorong yang menyambungkannya.Diagram 1 shows a graph that represents five guard posts, P, Q, R, S and T, and the lanes that join them.

[2 marks] (a)Nyatakan darjah bagi bucu Q dan bilangan tepi graf itu.State the degree of vertex Q and the number of edges of the graph.

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• Degree of Q = 3 [1]1
• 6 edges [1]1

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[1 mark] (b)Adakah graf itu sebuah pokok? Beri satu sebab.Is the graph a tree? Give a reason.

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• No; it has a cycle (P, Q, R) [1]1

A tree is connected with no cycle.

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[1 mark] (c)Berapakah bilangan tepi yang perlu dibuang untuk membentuk satu pokok yang merangkumi semua bucu?How many edges must be removed to form a tree that includes all the vertices?

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• A tree on 5 vertices has 4 edges, so remove 2 [1]1

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For the teacher: Understanding, low difficulty, Chapter 5, standard 5.1.1

Textbook p. 132

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Question 6

Sebuah gerai menjual x keping kek dan y cawan teh. Jumlah kek dan teh yang dijual tidak melebihi 40. Bilangan cawan teh sekurang-kurangnya dua kali bilangan keping kek.A stall sells x slices of cake and y cups of tea. The total number of cakes and cups of tea sold is not more than 40. The number of cups of tea is at least twice the number of slices of cake.

[2 marks] (a)Tulis dua ketaksamaan linear yang mewakili maklumat di atas.Write two linear inequalities that represent the information above.

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• x + y ≤ 40 [1]1
• y ≥ 2x [1]1

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[2 marks] (b)Adakah (10, 22) memuaskan kedua-dua ketaksamaan itu? Tunjukkan pengiraan anda.Does (10, 22) satisfy both inequalities? Show your working.

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• 10 + 22 = 32 ≤ 40 and 22 ≥ 2(10) = 20 [1]1
• Yes, it satisfies both [1]1

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For the teacher: Applying, low difficulty, Chapter 6, standard 6.1.2

Textbook p. 160

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Question 7

Rajah 2 menunjukkan graf laju-masa bagi sebuah bas dalam tempoh 18 saat.Diagram 2 shows the speed-time graph of a bus for 18 seconds.

[1 mark] (a)Hitung pecutan bas itu dalam 6 saat pertama.Calculate the acceleration of the bus in the first 6 seconds.

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• 15 ÷ 6 = 2.5 m/s² [1]1

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[1 mark] (b)Hitung nyahpecutan bas itu dalam 4 saat terakhir.Calculate the deceleration of the bus in the last 4 seconds.

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• (15 − 3) ÷ 4 = 3 m/s² [1]1

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[2 marks] (c)Hitung jumlah jarak yang dilalui oleh bas itu.Calculate the total distance travelled by the bus.

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• Area = ½(6)(15) + 8(15) + ½(15 + 3)(4) [1]1
• 45 + 120 + 36 = 201 m [1]1

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For the teacher: Applying, medium difficulty, Chapter 7, standard 7.2.3

Textbook p. 200

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Question 8

Bilangan buku yang dipinjam oleh lapan orang murid dari perpustakaan sekolah dalam satu bulan ialah 3, 6, 6, 8, 9, 11, 14 dan 15.The numbers of books borrowed by eight pupils from the school library in a month are 3, 6, 6, 8, 9, 11, 14 and 15.

[2 marks] (a)Cari julat dan julat antara kuartil.Find the range and the interquartile range.

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• Range = 15 − 3 = 12 [1]1
• Q1 = 6, Q3 = 12.5, IQR = 6.5 [1]1

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[2 marks] (b)Hitung varians dan sisihan piawai, betul kepada dua tempat perpuluhan.Calculate the variance and the standard deviation, correct to two decimal places.

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• Mean = 72 ÷ 8 = 9; variance = 768 ÷ 8 − 9² = 15 [1]1
• √15 = 3.87 [1]1

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For the teacher: Applying, medium difficulty, Chapter 8, standard 8.2.2

Textbook p. 224

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Question 9

Sebiji dadu adil dilambung dan sekeping kad dipilih secara rawak daripada empat keping kad berlabel A, I, M dan R.A fair die is rolled and a card is chosen at random from four cards labelled A, I, M and R.

[1 mark] (a)Berapakah bilangan kesudahan dalam ruang sampel?How many outcomes are there in the sample space?

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• 6 × 4 = 24 [1]1

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[2 marks] (b)Cari kebarangkalian mendapat nombor genap dan kad vokal.Find the probability of getting an even number and a vowel card.

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• 3/6 × 2/4 [1]1
• 1/4 [1]1

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[2 marks] (c)Cari kebarangkalian mendapat nombor perdana atau kad M.Find the probability of getting a prime number or the card M.

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• P(prime) = 3/6, P(M) = 1/4, P(prime and M) = 3/6 × 1/4 = 1/8 [1]1
• 1/2 + 1/4 − 1/8 = 5/8 [1]1

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For the teacher: Applying, low difficulty, Chapter 9, standard 9.2.3

Textbook p. 250

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Question 10

Encik Adli menerima gaji RM4 200 dan sewa rumah RM300 sebulan. Perbelanjaan tetapnya RM1 650 dan perbelanjaan tidak tetapnya RM1 480 sebulan. Dia ingin menabung RM12 000 dalam 8 bulan untuk wang pendahuluan sebuah motosikal.Encik Adli receives a salary of RM4 200 and rent of RM300 a month. His fixed expenses are RM1 650 and his variable expenses are RM1 480 a month. He wants to save RM12 000 in 8 months for the down payment of a motorcycle.

[2 marks] (a)Hitung aliran tunai bulanan Encik Adli.Calculate Encik Adli's monthly cash flow.

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• Income 4 200 + 300 = RM4 500; expenses 1 650 + 1 480 = RM3 130 [1]1
• Cash flow = RM1 370 [1]1

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[1 mark] (b)Adakah matlamat kewangannya boleh dicapai dalam 8 bulan? Berikan sebab.Can his financial goal be achieved in 8 months? Give a reason.

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• No; he needs RM12 000 ÷ 8 = RM1 500 a month but his cash flow is only RM1 370 [1]1

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For the teacher: Applying, low difficulty, Chapter 10, standard 10.1.2

Textbook p. 284

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Question 11

Jawab soalan-soalan berikut tentang pernyataan dan hujah.Answer the following questions about statements and arguments.

[2 marks] (a)Nyatakan sama ada setiap ayat berikut ialah pernyataan. Jika ya, nyatakan sama ada ia benar atau palsu.
(i) 3⁴ = 12
(ii) Sila tutup pintu itu.State whether each sentence is a statement. If it is, state whether it is true or false.
(i) 3⁴ = 12
(ii) Please close the door.

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• (i) A statement; false (3⁴ = 81) [1]1
• (ii) Not a statement (a request) [1]1

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[2 marks] (b)p: 2.5 × 4 = 10. q: √36 = 18.
Nyatakan nilai kebenaran bagi "p dan q" dan "p atau q".p: 2.5 × 4 = 10. q: √36 = 18.
State the truth value of "p and q" and "p or q".

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• p is true, q is false1
• "p and q" is false [1]; "p or q" is true [1]1

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[2 marks] (c)Tulis dua implikasi daripada pernyataan "Suatu nombor boleh dibahagi tepat dengan 5 jika dan hanya jika digit terakhirnya ialah 0 atau 5."Write two implications from the statement "A number is divisible by 5 if and only if its last digit is 0 or 5."

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• If a number is divisible by 5, then its last digit is 0 or 5 [1]1
• If the last digit of a number is 0 or 5, then it is divisible by 5 [1]1

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[3 marks] (d)Premis 1: Jika hujan esok, bazar akan ditangguhkan.
Premis 2: Bazar tidak ditangguhkan.
(i) Tulis kesimpulan bagi hujah ini.
(ii) "Semua nombor perdana ialah nombor ganjil. 2 ialah nombor perdana. Maka, 2 ialah nombor ganjil." Nyatakan sama ada hujah ini sah dan sama ada ia munasabah. Berikan sebab anda.Premise 1: If it rains tomorrow, the bazaar will be postponed.
Premise 2: The bazaar is not postponed.
(i) Write the conclusion of this argument.
(ii) "All prime numbers are odd numbers. 2 is a prime number. So, 2 is an odd number." State whether this argument is valid and whether it is sound. Give your reason.

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• (i) It does not rain tomorrow [1]1
• (ii) Valid: the conclusion follows from the premises [1]1
• Not sound: premise 1 is false (2 is a prime number but it is even) [1]1

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For the teacher: Understanding, low difficulty, Chapter 3, standard 3.1.4

Textbook p. 64

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Question 12

Jadual 1 menunjukkan bilangan barang yang dibeli oleh 20 orang pelanggan di Kedai A, sebuah kedai runcit.

Bilangan barang	1	2	3	4	5
Bilangan pelanggan	4	7	5	3	1

Jadual 1Table 1 shows the number of items bought by 20 customers at Shop A, a grocery shop.

Number of items	1	2	3	4	5
Number of customers	4	7	5	3	1

Table 1

[2 marks] (a)Hitung min bilangan barang yang dibeli.Calculate the mean number of items bought.

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• Σfx = 4 + 14 + 15 + 12 + 5 = 50 [1]1
• 50 ÷ 20 = 2.5 [1]1

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[2 marks] (b)Hitung sisihan piawai, betul kepada dua tempat perpuluhan.Calculate the standard deviation, correct to two decimal places.

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• Σfx² = 150; variance = 150 ÷ 20 − 2.5² = 1.25 [1]1
• √1.25 = 1.12 [1]1

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[2 marks] (c)Cari julat antara kuartil.Find the interquartile range.

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• Q1 = 2 (average of the 5th and 6th values), Q3 = 3 (average of the 15th and 16th values) [1]1
• IQR = 3 − 2 = 1 [1]1

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[3 marks] (d)Kedai B mempunyai min 2.7 dan sisihan piawai 1.6.
(i) Kedai manakah yang bilangan barang dibelinya lebih konsisten? Berikan sebab.
(ii) Dalam satu promosi, setiap pelanggan di Kedai A membeli 2 barang tambahan. Nyatakan min dan sisihan piawai yang baharu.Shop B has a mean of 2.7 and a standard deviation of 1.6.
(i) Which shop has more consistent numbers of items bought? Give a reason.
(ii) During a promotion, every customer at Shop A buys 2 more items. State the new mean and standard deviation.

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• (i) Shop A: its standard deviation (1.12) is smaller than 1.6 [1]1
• (ii) New mean = 4.5 [1]1
• Standard deviation stays 1.12 (adding a constant does not change it) [1]1

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For the teacher: Applying, low difficulty, Chapter 8, standard 8.2.4

Textbook p. 228

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Question 13

Sebuah kotak mengandungi 4 keping biskut coklat dan 6 keping biskut vanila. Dua keping biskut diambil satu demi satu tanpa pemulangan.A box contains 4 chocolate biscuits and 6 vanilla biscuits. Two biscuits are taken one after another without replacement.

[2 marks] (a)Cari kebarangkalian kedua-dua biskut ialah biskut coklat.Find the probability that both biscuits are chocolate.

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• 4/10 × 3/9 [1]1
• 2/15 [1]1

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[2 marks] (b)Cari kebarangkalian satu biskut coklat dan satu biskut vanila diperoleh, dalam sebarang tertib.Find the probability that one chocolate biscuit and one vanilla biscuit are obtained, in any order.

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• (4/10 × 6/9) + (6/10 × 4/9) [1]1
• 8/15 [1]1

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[2 marks] (c)Cari kebarangkalian sekurang-kurangnya satu daripada biskut itu ialah biskut vanila.Find the probability that at least one of the biscuits is vanilla.

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• 1 − P(both chocolate) [1]1
• 1 − 2/15 = 13/15 [1]1

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[3 marks] (d)Hana berkata, "Jika biskut pertama dikembalikan ke dalam kotak sebelum biskut kedua diambil, kebarangkalian mendapat dua biskut coklat menjadi lebih besar." Adakah Hana betul? Tunjukkan pengiraan anda.Hana says, "If the first biscuit is put back into the box before the second is taken, the probability of getting two chocolate biscuits becomes greater." Is Hana correct? Show your working.

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• With replacement: 4/10 × 4/10 = 0.16 [1]1
• Without replacement: 2/15 = 0.133 [1]1
• 0.16 > 0.133, so Hana is correct [1]1

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For the teacher: Applying, low difficulty, Chapter 9, standard 9.3.2

Textbook p. 256

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Question 14

Rajah 3 menunjukkan lima buah kampung, A, B, C, D dan E, yang akan disambungkan dengan kabel gentian optik. Nilai pada setiap tepi ialah jarak, dalam km, yang boleh dipasang kabel.Diagram 3 shows five villages, A, B, C, D and E, that will be joined with optical fibre cable. The value on each edge is the distance, in km, where a cable can be laid.

[2 marks] (a)Nyatakan darjah bucu C dan hitung jumlah darjah semua bucu.State the degree of vertex C and calculate the sum of the degrees of all the vertices.

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• Degree of C = 4 [1]1
• Sum = 2 × 7 = 14 [1]1

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[2 marks] (b)Subgraf dengan tepi AB, BC dan AC bukan sebuah pokok. Berikan satu sebab dan nyatakan bilangan tepi dalam pokok yang merangkumi kelima-lima kampung.The subgraph with edges AB, BC and AC is not a tree. Give one reason and state the number of edges in a tree that includes all five villages.

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• It forms a cycle (A, B, C) [1]1
• 4 edges [1]1

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[3 marks] (c)Senaraikan tepi bagi pokok yang mempunyai jumlah jarak minimum dan nyatakan jumlah jarak itu.List the edges of the tree with the minimum total distance and state that total distance.

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• DE (2), BC (3) [1]1
• AB (4), CD (5) [1]1
• Total = 14 km [1]1

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[2 marks] (d)Kos memasang kabel ialah RM1 800 bagi setiap km. Peruntukan ialah RM25 000. Adakah peruntukan itu mencukupi? Tunjukkan pengiraan anda.The cost of laying the cable is RM1 800 per km. The budget is RM25 000. Is the budget enough? Show your working.

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• 14 × 1 800 = RM25 200 [1]1
• RM25 200 > RM25 000, so it is not enough [1]1

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For the teacher: Analysing, medium difficulty, Chapter 5, standard 5.1.5

Textbook p. 146

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Question 15

Seramai 70 orang penduduk sebuah taman ditemu bual. Seramai 40 orang memiliki kereta (C), 30 orang memiliki motosikal (M) dan 12 orang memiliki kedua-duanya. Rajah 4 ialah gambar rajah Venn bagi maklumat ini.70 residents of a housing estate are interviewed. 40 residents own a car (C), 30 own a motorcycle (M) and 12 own both. Diagram 4 is a Venn diagram of this information.

[2 marks] (a)Cari nilai x dan y.Find the values of x and y.

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• x = 40 − 12 = 28 [1]1
• y = 30 − 12 = 18 [1]1

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[2 marks] (b)Cari nilai z.Find the value of z.

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• n(C ∪ M) = 28 + 12 + 18 = 58 [1]1
• z = 70 − 58 = 12 [1]1

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[2 marks] (c)Cari n(C′ ∪ M).Find n(C′ ∪ M).

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• C′ = 18 + 12 = 30 residents [1]1
• Add those in C ∩ M: 30 + 12 = 42 [1]1

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[3 marks] (d)Seorang ahli majlis berkata, "Lebih separuh daripada penduduk memiliki sama ada kereta atau motosikal sahaja, tetapi bukan kedua-duanya." Adakah kenyataan itu betul? Tunjukkan pengiraan anda.A councillor says, "More than half of the residents own either a car or a motorcycle, but not both." Is the statement correct? Show your working.

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• 28 + 18 = 46 residents [1]1
• Half of 70 = 35 [1]1
• 46 > 35, so the statement is correct [1]1

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For the teacher: Applying, high difficulty, Chapter 4, standard 4.3.2

Textbook p. 118

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