Matematik Tingkatan 4 Kertas 2Mathematics Form 4 Paper 2

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1

Question 1

Sebuah kawasan latihan bola jaring berbentuk segi empat tepat. Panjangnya (2x + 1) m dan lebarnya x m. Luas kawasan itu ialah 105 m².A netball practice area is a rectangle. Its length is (2x + 1) m and its width is x m. The area of the court is 105 m².

[1 mark] (a)Bentukkan persamaan kuadratik dalam bentuk am.Form a quadratic equation in general form.

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• 2x² + x − 105 = 01

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[3 marks] (b)Selesaikan persamaan itu dengan pemfaktoran.Solve the equation by factorisation.

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• (x − 7)(2x + 15) = 01
• x = 7 or x = −7.51
• Reject x = −7.5 (length cannot be negative); x = 71

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[1 mark] (c)Nyatakan panjang kawasan itu.State the length of the court.

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• 2(7) + 1 = 15 m1

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For the teacher: Applying, medium difficulty, Chapter 1, standard 1.1.4

Textbook p. 15

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Question 2

Kod loker bagi sebuah kelab sukan ialah 2314₅.The locker code of a sports club is 2314₅.

[2 marks] (a)Tukarkan 2314₅ kepada nombor dalam asas sepuluh.Convert 2314₅ to a number in base ten.

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• 2(5³) + 3(5²) + 1(5) + 4 [1 mark]1
• 334 [1 mark]1

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[1 mark] (b)Nyatakan nilai digit bagi digit 3 dalam 2314₅.State the digit value of the digit 3 in 2314₅.

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• 3 × 5² = 751

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For the teacher: Applying, low difficulty, Chapter 2, standard 2.1.2

Textbook p. 42

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Question 3

Pernyataan p: "Jika suatu nombor ialah gandaan 6, maka nombor itu ialah gandaan 3."Statement p: "If a number is a multiple of 6, then the number is a multiple of 3."

[1 mark] (a)Tulis akas bagi pernyataan p.Write the converse of statement p.

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• If a number is a multiple of 3, then the number is a multiple of 6.1

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[2 marks] (b)Nyatakan sama ada akas itu benar atau palsu. Beri satu contoh penyangkal jika palsu.State whether the converse is true or false. Give a counter-example if it is false.

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• False [1 mark]1
• Counter-example: 9 (a multiple of 3 but not a multiple of 6) [1 mark]1

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[1 mark] (c)Tulis kontrapositif bagi pernyataan p.Write the contrapositive of statement p.

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• If a number is not a multiple of 3, then the number is not a multiple of 6.1

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For the teacher: Understanding, low difficulty, Chapter 3, standard 3.1

Textbook p. 73

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Question 4

Diberi set semesta ξ = {x : 1 ≤ x ≤ 15, x ialah integer}, set P = {nombor perdana} dan set Q = {nombor ganjil}.Given the universal set ξ = {x : 1 ≤ x ≤ 15, x is an integer}, set P = {prime numbers} and set Q = {odd numbers}.

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

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• {3, 5, 7, 11, 13}1

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[2 marks] (b)Senaraikan semua unsur bagi (P ∪ Q)′.List all the elements of (P ∪ Q)′.

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• P ∪ Q = {1, 2, 3, 5, 7, 9, 11, 13, 15} [1 mark]1
• (P ∪ Q)′ = {4, 6, 8, 10, 12, 14} [1 mark]1

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[1 mark] (c)Cari n(P ∪ Q).Find n(P ∪ Q).

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• 91

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For the teacher: Applying, low difficulty, Chapter 4, standard 4.2

Textbook p. 108

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Question 5

Rajah 1 menunjukkan rangkaian lima buah tempat sukan, A hingga E, di sebuah kompleks sukan. Setiap garis ialah laluan yang menyambung dua tempat.Diagram 1 shows a network of five sports venues, A to E, in a sports complex. Each line is a path joining two venues.

[1 mark] (a)Nyatakan darjah bagi bucu D.State the degree of vertex D.

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• 31

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[1 mark] (b)Nyatakan bilangan tepi dalam rangkaian itu.State the number of edges in the network.

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• 61

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[1 mark] (c)Sebuah pokok dibentuk daripada rangkaian ini dengan menggunakan semua bucu. Berapakah bilangan tepi yang perlu dibuang?A tree is to be formed from this network using all the vertices. How many edges must be removed?

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• A tree with 5 vertices has 4 edges, so 6 − 4 = 2 edges1

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[1 mark] (d)Senaraikan semua bucu yang berdarjah ganjil.List all the vertices of odd degree.

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• B, C, D, E1

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

Textbook p. 133

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Question 6

Syafiq mempunyai RM30. Dia membeli x batang pen berharga RM2 setiap satu dan y buah buku nota berharga RM3 setiap satu. Jumlah wang yang dibelanjakan tidak melebihi RM30.Syafiq has RM30. He buys x pens at RM2 each and y notebooks at RM3 each. The total amount he spends does not exceed RM30.

[1 mark] (a)Tulis satu ketaksamaan linear dalam x dan y untuk situasi ini.Write a linear inequality in x and y for this situation.

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• 2x + 3y ≤ 301

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[2 marks] (b)Adakah titik (6, 6) memuaskan ketaksamaan itu? Tunjukkan pengiraan anda.Does the point (6, 6) satisfy the inequality? Show your calculation.

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• 2(6) + 3(6) = 30 [1 mark]1
• 30 ≤ 30, so yes, it satisfies the inequality [1 mark]1

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[1 mark] (c)Garis sempadan bagi rantau itu dilukis sebagai garis padu atau garis sempang? Beri alasan.Is the boundary line of the region drawn as a solid line or a dashed line? Give a reason.

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• Solid line, because the inequality contains "≤" so points on the line are included1

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For the teacher: Applying, medium difficulty, Chapter 6, standard 6.1

Textbook p. 160

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Question 7

Rajah 2 ialah graf laju-masa bagi seorang atlet berbasikal dalam tempoh 14 saat.Diagram 2 is the speed-time graph of a cyclist over a period of 14 seconds.

[2 marks] (a)Hitung pecutan, dalam m s⁻², dalam 4 saat pertama.Calculate the acceleration, in m s⁻², in the first 4 seconds.

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• (8 − 0) ÷ 4 [1 mark]1
• 2 m s⁻² [1 mark]1

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[1 mark] (b)Hitung jarak yang dilalui semasa dia bergerak dengan laju seragam.Calculate the distance travelled while the cyclist moves at uniform speed.

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• 8 × (10 − 4) = 48 m1

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[2 marks] (c)Hitung jumlah jarak yang dilalui dalam 14 saat itu.Calculate the total distance travelled in the 14 seconds.

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• ½(6 + 14) × 8 [1 mark]1
• 80 m [1 mark]1

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

Textbook p. 198

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Question 8

Skor yang diperoleh tujuh orang pemain dalam suatu pertandingan ialah 6, 9, 4, 8, 12, 7 dan 10.The scores obtained by seven players in a competition are 6, 9, 4, 8, 12, 7 and 10.

[1 mark] (a)Cari julat skor itu.Find the range of the scores.

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• 12 − 4 = 81

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

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• Ordered: 4, 6, 7, 8, 9, 10, 12; Q1 = 6, Q3 = 10 [1 mark]1
• IQR = 10 − 6 = 4 [1 mark]1

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For the teacher: Applying, low difficulty, Chapter 8, standard 8.2.1

Textbook p. 222

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Question 9

Hana dan Dina masing-masing membuat satu balingan percuma. Kebarangkalian Hana menjaringkan gol ialah ³⁄₅ dan kebarangkalian Dina menjaringkan gol ialah ²⁄₃. Kedua-dua kejadian adalah tidak bersandar.Hana and Dina each take one free throw. The probability that Hana scores is ³⁄₅ and the probability that Dina scores is ²⁄₃. The two events are independent.

[1 mark] (a)Cari kebarangkalian kedua-duanya menjaringkan gol.Find the probability that both of them score.

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• 3/5 × 2/3 = 2/51

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[1 mark] (b)Cari kebarangkalian Hana menjaringkan gol tetapi Dina tidak.Find the probability that Hana scores but Dina does not.

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• 3/5 × 1/3 = 1/51

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[2 marks] (c)Cari kebarangkalian sekurang-kurangnya seorang daripada mereka menjaringkan gol.Find the probability that at least one of them scores.

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• P(neither) = 2/5 × 1/3 = 2/15 [1 mark]1
• 1 − 2/15 = 13/15 [1 mark]1

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

Textbook p. 252

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Question 10

Farah menerima gaji RM1 800 dan pendapatan sampingan mengajar RM200 sebulan. Perbelanjaan tetapnya ialah RM950 dan perbelanjaan tidak tetapnya ialah RM780 sebulan.Farah receives a salary of RM1 800 and a side income of RM200 from tutoring each month. Her fixed expenses are RM950 and her variable expenses are RM780 each month.

[2 marks] (a)Hitung aliran tunai bersih bulanan Farah dan nyatakan sama ada ia lebihan atau defisit.Calculate Farah’s net monthly cash flow and state whether it is a surplus or a deficit.

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• 2 000 − (950 + 780) = RM270 [1 mark]1
• Surplus [1 mark]1

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[1 mark] (b)Dia ingin membeli sebuah komputer riba berharga RM1 620 dalam masa 6 bulan. Berapakah simpanan bulanan yang diperlukan?She wants to buy a laptop costing RM1 620 within 6 months. How much must she save each month?

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• 1 620 ÷ 6 = RM2701

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[1 mark] (c)Adakah rancangan itu boleh dilaksanakan? Beri alasan.Is the plan feasible? Give a reason.

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• Yes, because RM270 needed is not more than the surplus of RM2701

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For the teacher: Applying, low difficulty, Chapter 10, standard 10.1.1

Textbook p. 276

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Question 11

Sebuah pasukan robotik sekolah menggunakan nombor dalam pelbagai asas untuk mengekod gerakan robot.A school robotics team uses numbers in different bases to code the movements of its robot.

[2 marks] (a)Tukarkan 1101011₂ kepada nombor dalam asas lapan.Convert 1101011₂ to a number in base eight.

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• Group in threes: 001 101 011 [1 mark]1
• 153₈ [1 mark]1

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[2 marks] (b)Tukarkan 1101011₂ kepada nombor dalam asas sepuluh.Convert 1101011₂ to a number in base ten.

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• 1(2⁶) + 1(2⁵) + 1(2³) + 1(2¹) + 1 [1 mark]1
• 107 [1 mark]1

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[3 marks] (c)Hitung 3412₅ + 1234₅. Beri jawapan dalam asas lima.Calculate 3412₅ + 1234₅. Give the answer in base five.

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• Add by columns with carrying in base five [1 mark]1
• Correct carries: 2 + 4 = 11₅, 1 + 3 + 1 = 10₅, 4 + 2 + 1 = 12₅, 3 + 1 + 1 = 10₅ [1 mark]1
• 10201₅ [1 mark]1

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[2 marks] (d)Robot A diprogramkan dengan 213₅ langkah dan robot B dengan 111011₂ langkah. Robot yang manakah bergerak lebih banyak langkah? Tunjukkan pengiraan anda.Robot A is programmed with 213₅ steps and robot B with 111011₂ steps. Which robot makes more steps? Show your calculation.

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• 213₅ = 58 and 111011₂ = 59 [1 mark]1
• Robot B [1 mark]1

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For the teacher: Applying, low difficulty, Chapter 2, standard 2.1.3

Textbook p. 49

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Question 12

Sebuah kelab sukan memilih x orang pelajar lelaki dan y orang pelajar perempuan untuk pasukan larian berganti-ganti. Rajah 3 menunjukkan rantau berlorek yang memenuhi semua syarat pemilihan. Semua garis sempadan ialah garis padu.A sports club selects x boys and y girls for a relay team. Diagram 3 shows the shaded region that satisfies all the conditions for selection. All the boundary lines are solid lines.

[3 marks] (a)Tulis tiga ketaksamaan linear yang mentakrifkan rantau berlorek itu.Write the three linear inequalities that define the shaded region.

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• y ≥ 2 [1 mark]1
• y ≤ x + 2 [1 mark]1
• x + y ≤ 8 [1 mark]1

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[2 marks] (b)Adakah memilih 4 orang pelajar lelaki dan 3 orang pelajar perempuan memenuhi semua syarat itu? Tunjukkan bagaimana anda tahu.Does selecting 4 boys and 3 girls satisfy all the conditions? Show how you know.

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• 3 ≥ 2, 3 ≤ 4 + 2, 4 + 3 ≤ 8 [1 mark]1
• Yes, the point (4, 3) is inside the shaded region [1 mark]1

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[2 marks] (c)Berapakah bilangan maksimum pelajar perempuan yang boleh dipilih?What is the maximum number of girls that can be selected?

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• Highest point of the region is the vertex (3, 5) [1 mark]1
• 5 girls [1 mark]1

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[2 marks] (d)Jurulatih berkata 6 orang pelajar lelaki dan 3 orang pelajar perempuan boleh dipilih. Adakah jurulatih itu betul? Berikan alasan anda.The coach says that 6 boys and 3 girls can be selected. Is the coach correct? Give your reason.

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• x + y = 6 + 3 = 9, which is more than 8 [1 mark]1
• The coach is not correct; (6, 3) lies outside the region [1 mark]1

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For the teacher: Analysing, high difficulty, Chapter 6, standard 6.2

Textbook p. 170

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Question 13

Rajah 4 ialah graf jarak-masa bagi perjalanan Haziq berbasikal dari rumahnya ke sebuah stadium.Diagram 4 is the distance-time graph of Haziq’s cycling journey from his house to a stadium.

[1 mark] (a)Berapa lama Haziq berhenti rehat?For how long did Haziq stop to rest?

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• 35 − 20 = 15 minutes1

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[2 marks] (b)Hitung laju Haziq, dalam km/j, dalam 20 minit pertama.Calculate Haziq’s speed, in km/h, during the first 20 minutes.

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• 8 ÷ (20/60) [1 mark]1
• 24 km/h [1 mark]1

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[2 marks] (c)Hitung purata laju bagi seluruh perjalanan, dalam km/j.Calculate the average speed for the whole journey, in km/h.

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• 28 ÷ (75/60) [1 mark]1
• 22.4 km/h [1 mark]1

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[4 marks] (d)Daniel bertolak dari rumah yang sama pada masa yang sama dan berbasikal pada laju seragam 20 km/j mengikut laluan yang sama. Siapakah yang sampai ke stadium dahulu, dan berapa minit lebih awal? Stadium itu berada 28 km dari rumah.Daniel leaves the same house at the same time and cycles at a uniform speed of 20 km/h along the same route. Who reaches the stadium first, and by how many minutes? The stadium is 28 km from the house.

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• Daniel’s time = 28 ÷ 20 = 1.4 hours [1 mark]1
• = 84 minutes [1 mark]1
• Haziq takes 75 minutes, so Haziq arrives first [1 mark]1
• 84 − 75 = 9 minutes earlier [1 mark]1

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For the teacher: Evaluating, low difficulty, Chapter 7, standard 7.1.2

Textbook p. 190

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Question 14

Seramai 40 orang murid menyertai kelab sukan. Daripada mereka, 22 orang bermain badminton (B), 16 orang bermain tenis meja (T) dan 8 orang bermain kedua-dua sukan itu. Seorang murid dipilih secara rawak.There are 40 students in a sports club. Of them, 22 play badminton (B), 16 play table tennis (T) and 8 play both sports. A student is chosen at random.

[1 mark] (a)Cari kebarangkalian murid itu bermain kedua-dua sukan.Find the probability that the student plays both sports.

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• 8/40 = 1/51

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[2 marks] (b)Cari kebarangkalian murid itu bermain badminton atau tenis meja.Find the probability that the student plays badminton or table tennis.

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• 22 + 16 − 8 = 30 [1 mark]1
• 30/40 = 3/4 [1 mark]1

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[2 marks] (c)Dua orang murid dipilih satu demi satu dengan pemulangan. Cari kebarangkalian kedua-duanya tidak bermain sebarang sukan itu.Two students are chosen one after the other with replacement. Find the probability that neither of them plays either sport.

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• Neither = 40 − 30 = 10, P = 10/40 = 1/4 [1 mark]1
• 1/4 × 1/4 = 1/16 [1 mark]1

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[4 marks] (d)Dua orang murid dipilih satu demi satu tanpa pemulangan. Maria berkata kebarangkalian kedua-duanya bermain kedua-dua sukan adalah sama seperti jika dengan pemulangan. Adakah Maria betul? Tunjukkan pengiraan anda.Two students are chosen one after the other without replacement. Maria says the probability that both play both sports is the same as it would be with replacement. Is Maria correct? Show your calculation.

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• With replacement: 8/40 × 8/40 = 1/25 [1 mark]1
• Without replacement: 8/40 × 7/39 = 7/1952
• Not equal (0.04 and about 0.036), so Maria is not correct [1 mark]1

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For the teacher: Evaluating, low difficulty, Chapter 9, standard 9.2.2

Textbook p. 255

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Question 15

Tinggi, h meter, sebiji bola sepak yang ditendang selepas t saat diberi oleh h = −t² + 6t.The height, h metres, of a football kicked after t seconds is given by h = −t² + 6t.

[2 marks] (a)Cari nilai t apabila bola berada di atas tanah (h = 0).Find the values of t when the ball is at ground level (h = 0).

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• −t(t − 6) = 0 [1 mark]1
• t = 0 and t = 6 [1 mark]1

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[2 marks] (b)Cari masa bola mencapai tinggi maksimum dan tinggi maksimum itu.Find the time at which the ball reaches its maximum height, and the maximum height.

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• t = (0 + 6) ÷ 2 = 3 s [1 mark]1
• h = −9 + 18 = 9 m [1 mark]1

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[3 marks] (c)Pada bila-bila masa bola berada pada ketinggian 5 m?At what times is the ball at a height of 5 m?

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• −t² + 6t = 5, so t² − 6t + 5 = 0 [1 mark]1
• (t − 1)(t − 5) = 0 [1 mark]1
• t = 1 s and t = 5 s [1 mark]1

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[2 marks] (d)Sebuah dinding setinggi 10 m berada di laluan bola. Bolehkah bola melepasi dinding itu? Beri alasan.A 10 m high wall stands in the path of the ball. Can the ball clear the wall? Give a reason.

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• Maximum height is 9 m [1 mark]1
• 9 m < 10 m, so the ball cannot clear the wall [1 mark]1

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

Textbook p. 26

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