Practice paper, not an official KPM paper

Percubaan Sijil Pelajaran MalaysiaTrial Sijil Pelajaran Malaysia

Matematik TambahanAdditional Mathematics

Tingkatan 5, 2 jam / 2 hours, 100 marks

Nama / Name

Kelas / Class

Tarikh / Date

Markah / Marks: / 100

Instructions

Kertas soalan ini mengandungi tiga bahagian: Bahagian A, Bahagian B dan Bahagian C.This question paper consists of three sections: Section A, Section B and Section C.
Jawab semua soalan.Answer all questions.
Tunjukkan langkah-langkah penting dalam kerja mengira anda. Markah diberikan untuk kaedah.Show your working. Marks are given for method.
Rajah yang mengiringi soalan tidak dilukis mengikut skala kecuali dinyatakan.The diagrams in this paper are not drawn to scale unless stated.
Anda dibenarkan menggunakan kalkulator saintifik yang tidak boleh diprogram.You may use a non-programmable scientific calculator.

Answer schemes are in English (DLP).

Practice shows answers as you go. Exam times you and hides answers until you submit.

Bahagian ASection A

40 marks

Jawab semua soalan.Answer all questions.

Question 1

Sebiji piza berbentuk bulat dipotong kepada beberapa keping. Satu keping berbentuk sektor yang mencangkum sudut 150° di pusat piza itu. Jejari piza ialah 12 cm.A round pizza is cut into slices. One slice is a sector that subtends an angle of 150° at the centre of the pizza. The radius of the pizza is 12 cm.

[1 mark] (a)Ungkapkan 150° dalam radian, dalam sebutan π.Express 150° in radians, in terms of π.

Show answerHide answer

150° = 5π/6 rad [1]1

Tick each point you got.

[2 marks] (b)Cari panjang lengkok keping piza itu, dalam sebutan π.Find the arc length of the slice of pizza, in terms of π.

Show answerHide answer

s = 12 × 5π/6 [1]1
s = 10π cm [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, Chapter 1, standard 1.2.1

Textbook p. 6

Question 2

Diberi lengkung y = 5x³ − 2x⁻² + 7.Given the curve y = 5x³ − 2x⁻² + 7.

[2 marks] (a)Cari dy/dx.Find dy/dx.

Show answerHide answer

Power rule on each term [1]1
dy/dx = 15x² + 4x⁻³ (= 15x² + 4/x³) [1]1

Tick each point you got.

[1 mark] (b)Cari kecerunan lengkung itu pada x = −1.Find the gradient of the curve at x = −1.

Show answerHide answer

15(1) + 4/(−1)³ = 11 [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, Chapter 2, standard 2.2.2

Textbook p. 41

Question 3

Lengkung y = x²/(2x + 1) melalui titik P yang x = 1. Cari persamaan tangen kepada lengkung itu di P.The curve y = x²/(2x + 1) passes through the point P where x = 1. Find the equation of the tangent to the curve at P.

[4 marks] (a)Tulis jawapan anda dalam bentuk ax + by + c = 0, dengan a, b dan c ialah integer.Write your answer in the form ax + by + c = 0, where a, b and c are integers.

Show answerHide answer

Quotient rule: dy/dx = [2x(2x + 1) − x²(2)] / (2x + 1)² [1]1
At x = 1: y = 1/3 and gradient = 4/9 [1]1
y − 1/3 = (4/9)(x − 1) [1]1
4x − 9y − 1 = 0 [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, Chapter 2, standard 2.4.2

Textbook p. 53

Question 4

Diberi ∫₁^k (4x − 3) dx = 6, dengan k > 1. Cari nilai k.Given ∫₁^k (4x − 3) dx = 6, where k > 1. Find the value of k.

[4 marks] (a)Cari nilai k.Find the value of k.

Show answerHide answer

[2x² − 3x] from 1 to k [1]1
(2k² − 3k) − (2 − 3) = 6, so 2k² − 3k − 5 = 0 [1]1
(2k − 5)(k + 1) = 0 [1]1
k = 5/2 (k = −1 rejected because k > 1) [1]1

Tick each point you got.

For the teacher: Analysing, high difficulty, Chapter 3, standard 3.3.1

Textbook p. 94

Question 5

Pada hari sukan, lapan orang pelari menyertai larian 100 m.On sports day, eight runners take part in the 100 m race.

[2 marks] (a)Hadiah pertama, kedua dan ketiga diberikan kepada tiga pelari yang berlainan. Cari bilangan cara hadiah itu boleh diberikan.The first, second and third prizes are given to three different runners. Find the number of ways the prizes can be given.

Show answerHide answer

⁸P₃ [1]1
= 336 [1]1

Tick each point you got.

[3 marks] (b)Sebuah pasukan lari berganti-ganti yang terdiri daripada 2 murid lelaki dan 2 murid perempuan dipilih daripada 6 murid lelaki dan 5 murid perempuan. Cari bilangan cara pasukan itu boleh dipilih.A relay team of 2 boys and 2 girls is chosen from 6 boys and 5 girls. Find the number of ways the team can be chosen.

Show answerHide answer

⁶C₂ × ⁵C₂ [1]1
15 × 10 [1]1
= 150 [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, Chapter 4, standard 4.2.3

Textbook p. 135

Question 6

Jisim beg sekolah murid-murid tertabur secara normal dengan min μ kg dan varians 9 kg². Kebarangkalian seorang murid membawa beg yang berjisim lebih daripada 40 kg ialah 0.0668. Cari nilai μ.The masses of pupils' school bags are normally distributed with mean μ kg and variance 9 kg². The probability that a pupil carries a bag of mass more than 40 kg is 0.0668. Find the value of μ.

[4 marks] (a)Cari nilai μ. [Gunakan jadual taburan normal piawai atau kalkulator.]Find the value of μ. [Use the standard normal table or a calculator.]

Show answerHide answer

P(Z > k) = 0.0668 [1]1
k = 1.5 [1]1
(40 − μ)/3 = 1.5 [1]1
μ = 35.5 [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, Chapter 5, standard 5.3.4

Textbook p. 174

Question 7

Selesaikan persamaan cos 2x + 3 cos x = 1 untuk 0° ≤ x ≤ 360°.Solve the equation cos 2x + 3 cos x = 1 for 0° ≤ x ≤ 360°.

[4 marks] (a)Selesaikan persamaan itu.Solve the equation.

Show answerHide answer

cos 2x = 2cos²x − 1 [1]1
2cos²x + 3cos x − 2 = 0 [1]1
(2cos x − 1)(cos x + 2) = 0, cos x = 1/2 (cos x = −2 rejected) [1]1
x = 60°, 300° [1]1

Tick each point you got.

For the teacher: Applying, high difficulty, Chapter 6, standard 6.6.1

Textbook p. 223

Question 8

Sebuah kedai kokurikulum menjual x buah penanda buku dan y buah buku nota. Jumlah barang yang dijual tidak melebihi 150. Bilangan buku nota sekurang-kurangnya separuh daripada bilangan penanda buku. Kedai itu menjual sekurang-kurangnya 20 buah buku nota.A co-curriculum shop sells x bookmarks and y notebooks. The total number of items sold does not exceed 150. The number of notebooks is at least half the number of bookmarks. The shop sells at least 20 notebooks.

[3 marks] (a)Tulis tiga ketaksamaan linear yang memenuhi semua syarat itu.Write three linear inequalities that satisfy all the conditions.

Show answerHide answer

x + y ≤ 150 [1]1
y ≥ x/2 (x ≤ 2y) [1]1
y ≥ 20 [1]1

Tick each point you got.

[1 mark] (b)Adakah (100, 40) memenuhi semua syarat itu? Beri sebab.Does (100, 40) satisfy all the conditions? Give a reason.

Show answerHide answer

No, because x ≤ 2y is not satisfied: 100 > 2(40) = 80 [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, Chapter 7, standard 7.1.1

Textbook p. 237

Question 9

Satu zarah bergerak di sepanjang suatu garis lurus. Sesarannya, s m, dari titik tetap O diberi oleh s = 2t³ − 9t² + 12t, dengan t ialah masa, dalam saat, selepas bermula dari O.A particle moves along a straight line. Its displacement, s m, from a fixed point O is given by s = 2t³ − 9t² + 12t, where t is the time, in seconds, after leaving O.

[1 mark] (a)Ungkapkan halaju zarah itu, v, dalam sebutan t.Express the velocity of the particle, v, in terms of t.

Show answerHide answer

v = ds/dt = 6t² − 18t + 12 [1]1

Tick each point you got.

[2 marks] (b)Cari masa apabila zarah itu berhenti seketika.Find the times when the particle is instantaneously at rest.

Show answerHide answer

v = 0: 6(t − 1)(t − 2) = 0 [1]1
t = 1 s and t = 2 s [1]1

Tick each point you got.

[2 marks] (c)Cari pecutan zarah itu pada t = 3.Find the acceleration of the particle when t = 3.

Show answerHide answer

a = dv/dt = 12t − 18 [1]1
a = 12(3) − 18 = 18 m s⁻² [1]1

Tick each point you got.

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

Textbook p. 260

Question 10

Tanpa menggunakan kalkulator, nyatakan nilai tepat bagi setiap yang berikut.Without using a calculator, state the exact value of each of the following.

[1 mark] (a)sin 150°

Show answerHide answer

1/2 [1]1

Tick each point you got.

[1 mark] (b)tan 5π/4

Show answerHide answer

1 [1]1

Tick each point you got.

[1 mark] (c)kos (−60°)cos (−60°)

Show answerHide answer

1/2 [1]1

Tick each point you got.

[1 mark] (d)sek 300°sec 300°

Show answerHide answer

2 [1]1

Tick each point you got.

For the teacher: Remembering, low difficulty, Chapter 6, standard 6.2.2

Textbook p. 197

Bahagian BSection B

40 marks

Jawab semua soalan.Answer all questions.

Question 11

Lengkung y = x³ − 3x² − 9x + 4 mempunyai dua titik pusingan.The curve y = x³ − 3x² − 9x + 4 has two turning points.

[2 marks] (a)Cari dy/dx dan d²y/dx².Find dy/dx and d²y/dx².

Show answerHide answer

dy/dx = 3x² − 6x − 9 [1]1
d²y/dx² = 6x − 6 [1]1

Tick each point you got.

[2 marks] (b)Cari nilai-nilai x pada titik pusingan itu.Find the values of x at the turning points.

Show answerHide answer

3x² − 6x − 9 = 0, (x + 1)(x − 3) = 0 [1]1
x = −1 and x = 3 [1]1

Tick each point you got.

[3 marks] (c)Tentukan koordinat dan jenis setiap titik pusingan itu.Determine the coordinates and the nature of each turning point.

Show answerHide answer

x = −1: y = 9; d²y/dx² = −12 < 0, so (−1, 9) is a maximum point [1]1
x = 3: y = −23; d²y/dx² = 12 > 0, so (3, −23) is a minimum point [1]1
Both coordinates correct [1]1

Tick each point you got.

[3 marks] (d)Sebuah reban ayam berbentuk segi empat tepat dibahagi kepada dua bahagian yang sama oleh sebatang pagar selari dengan lebarnya. Panjang reban ialah x m dan lebarnya y m. Jumlah panjang pagar yang digunakan ialah 120 m. Cari luas maksimum reban itu dan buktikan ia maksimum.A rectangular chicken coop is divided into two equal parts by a fence parallel to its width. The length of the coop is x m and its width is y m. The total length of fencing used is 120 m. Find the maximum area of the coop and prove that it is a maximum.

Show answerHide answer

2x + 3y = 120, A = xy = x(120 − 2x)/3 = 40x − (2/3)x² [1]1
dA/dx = 40 − (4/3)x = 0, x = 30 [1]1
d²A/dx² = −4/3 < 0, so A is a maximum; A = 40(30) − (2/3)(900) = 600 m² [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, Chapter 2, standard 2.4.4

Textbook p. 59

Question 12

Huruf-huruf bagi perkataan FOOTBALL disusun dalam satu baris.The letters of the word FOOTBALL are arranged in a row.

[2 marks] (a)Cari bilangan susunan yang berlainan bagi semua huruf itu.Find the number of different arrangements of all the letters.

Show answerHide answer

8!/(2! 2!) [1]1
= 10 080 [1]1

Tick each point you got.

[2 marks] (b)Cari bilangan susunan yang bermula dengan huruf F dan berakhir dengan huruf B.Find the number of arrangements that begin with the letter F and end with the letter B.

Show answerHide answer

Remaining letters O, O, T, A, L, L: 6!/(2! 2!) [1]1
= 180 [1]1

Tick each point you got.

[3 marks] (c)Cari bilangan susunan yang dua huruf L tidak bersebelahan.Find the number of arrangements in which the two letters L are not next to each other.

Show answerHide answer

L's together: 7!/2! = 2 520 [1]1
Total − together: 10 080 − 2 520 [1]1
= 7 560 [1]1

Tick each point you got.

[3 marks] (d)Satu susunan dipilih secara rawak. Cari kebarangkalian bahawa semua huruf vokal (O, O, A) berada bersebelahan.One arrangement is chosen at random. Find the probability that all the vowels (O, O, A) are together.

Show answerHide answer

Vowels as one block: 3!/2! = 3 ways inside the block [1]1
6 items (F, T, B, L, L and the block): 6!/2! = 360, so 3 × 360 = 1 080 [1]1
P = 1 080/10 080 = 3/28 [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, Chapter 4, standard 4.1.3

Textbook p. 129

Question 13

Rajah 1 menunjukkan graf bagi fungsi y = a kos bx untuk 0 ≤ x ≤ 2π, dengan a dan b ialah pemalar dan b > 0.Diagram 1 shows the graph of the function y = a cos bx for 0 ≤ x ≤ 2π, where a and b are constants and b > 0.

Cosine curve starting at minus 3 when x is 0, maximum 3 at pi over 2, minimum at pi, maximum at 3 pi over 2 and minimum at 2 pi — Rajah 1Diagram 1

[2 marks] (a)Nyatakan nilai a dan nilai b.State the value of a and the value of b.

Show answerHide answer

a = −3 (the graph starts at its minimum value −3) [1]1
b = 2 (two complete cycles in 0 to 2π) [1]1

Tick each point you got.

[2 marks] (b)Nyatakan nilai maksimum y dan nilai-nilai x apabila nilai maksimum itu berlaku.State the maximum value of y and the values of x at which it occurs.

Show answerHide answer

Maximum value = 3 [1]1
x = π/2 and x = 3π/2 [1]1

Tick each point you got.

[3 marks] (c)Dengan melukis satu garis lurus yang sesuai pada rajah itu, cari bilangan penyelesaian bagi persamaan 3π kos 2x = π − x untuk 0 ≤ x ≤ 2π. Nyatakan persamaan garis lurus itu.By drawing a suitable straight line on the diagram, find the number of solutions of the equation 3π cos 2x = π − x for 0 ≤ x ≤ 2π. State the equation of the straight line.

Show answerHide answer

Divide by −π: −3 cos 2x = x/π − 1 [1]1
Straight line y = x/π − 1, through (0, −1), (π, 0) and (2π, 1) [1]1
Number of solutions = 4 (the line meets the curve 4 times) [1]1

Tick each point you got.

[3 marks] (d)Selesaikan 6 kos 2x + 3 = 0 untuk 0 ≤ x ≤ 2π. Beri jawapan dalam sebutan π.Solve 6 cos 2x + 3 = 0 for 0 ≤ x ≤ 2π. Give the answers in terms of π.

Show answerHide answer

cos 2x = −1/2 [1]1
2x = 2π/3, 4π/3, 8π/3, 10π/3 [1]1
x = π/3, 2π/3, 4π/3, 5π/3 [1]1

Tick each point you got.

For the teacher: Analysing, medium difficulty, Chapter 6, standard 6.3.2

Textbook p. 208

Question 14

Kelab Sains membuat x buah layang-layang kecil dan y buah layang-layang besar untuk Karnival Sekolah. Pembuatan itu tertakluk kepada syarat-syarat berikut:
I: Jumlah layang-layang yang dibuat tidak melebihi 60.
II: Sekurang-kurangnya 10 buah layang-layang kecil dan sekurang-kurangnya 10 buah layang-layang besar dibuat.
III: Sebuah layang-layang kecil menggunakan 2 helai kertas dan sebuah layang-layang besar menggunakan 5 helai kertas. Kelab itu mempunyai 240 helai kertas.The Science Club makes x small kites and y large kites for the School Carnival. The making is subject to the following conditions:
I: The total number of kites made does not exceed 60.
II: At least 10 small kites and at least 10 large kites are made.
III: A small kite uses 2 sheets of paper and a large kite uses 5 sheets of paper. The club has 240 sheets of paper.

Quadrilateral region R bounded by x = 10, y = 10, x + y = 60 and 2x + 5y = 240 — Rajah 2Diagram 2

[4 marks] (a)Tulis empat ketaksamaan linear, selain x ≥ 0 dan y ≥ 0, yang memenuhi semua syarat itu.Write four linear inequalities, other than x ≥ 0 and y ≥ 0, that satisfy all the conditions.

Show answerHide answer

x + y ≤ 60 [1]1
x ≥ 10 [1]1
y ≥ 10 [1]1
2x + 5y ≤ 240 [1]1

Tick each point you got.

[3 marks] (b)Rajah 2 menunjukkan rantau berlorek R yang memenuhi semua ketaksamaan itu. (i) Cari koordinat bucu R yang merupakan persilangan x + y = 60 dan 2x + 5y = 240. (ii) Adakah (30, 30) berada di dalam R?Diagram 2 shows the shaded region R that satisfies all the inequalities. (i) Find the coordinates of the vertex of R that is the intersection of x + y = 60 and 2x + 5y = 240. (ii) Is (30, 30) inside R?

Show answerHide answer

Substitute x = 60 − y: 2(60 − y) + 5y = 240, 3y = 120 [1]1
y = 40, x = 20, so the vertex is (20, 40) [1]1
Yes: 30 + 30 = 60 ≤ 60 and 2(30) + 5(30) = 210 ≤ 240 [1]1

Tick each point you got.

[3 marks] (c)Untung bagi sebuah layang-layang kecil ialah RM4 dan sebuah layang-layang besar ialah RM9. Cari untung maksimum yang diperoleh. Bucu-bucu R ialah (10, 10), (50, 10), (20, 40) dan (10, 44).The profit for a small kite is RM4 and for a large kite is RM9. Find the maximum profit. The vertices of R are (10, 10), (50, 10), (20, 40) and (10, 44).

Show answerHide answer

Profit k = 4x + 9y [1]1
Values at the vertices: 130, 290, 440 and 436 [1]1
Maximum profit = RM440, at (20, 40) [1]1

Tick each point you got.

For the teacher: Applying, high difficulty, Chapter 7, standard 7.2.1

Textbook p. 241

Bahagian CSection C

20 marks

Jawab semua soalan.Answer all questions.

Question 15

Masa yang diambil oleh murid Tingkatan 5 sebuah sekolah untuk berlari sejauh 1.6 km tertabur secara normal dengan min 8 minit dan sisihan piawai 1.5 minit. Guru Pendidikan Jasmani memilih 20% murid yang paling laju untuk mewakili sekolah ke pertandingan daerah. [Gunakan jadual taburan normal piawai.]The time taken by the Form 5 pupils of a school to run 1.6 km is normally distributed with a mean of 8 minutes and a standard deviation of 1.5 minutes. The Physical Education teacher selects the fastest 20% of the pupils to represent the school at the district competition. [Use the standard normal table.]

[3 marks] (a)Seorang murid mengambil masa 9.5 minit. Cari skor-z bagi masa itu dan kebarangkalian bahawa seorang murid dipilih secara rawak mengambil masa lebih daripada 9.5 minit.A pupil takes 9.5 minutes. Find the z-score for that time and the probability that a pupil chosen at random takes more than 9.5 minutes.

Show answerHide answer

z = (9.5 − 8)/1.5 = 1 [1]1
P(X > 9.5) = P(Z > 1) [1]1
= 0.1587 [1]1

Tick each point you got.

[3 marks] (b)Cari kebarangkalian bahawa masa seorang murid berada di antara 5.75 minit dengan 9.5 minit.Find the probability that a pupil's time is between 5.75 minutes and 9.5 minutes.

Show answerHide answer

z = (5.75 − 8)/1.5 = −1.5 [1]1
P(−1.5 < Z < 1) = 1 − P(Z > 1.5) − P(Z > 1) [1]1
= 1 − 0.0668 − 0.1587 = 0.7745 [1]1

Tick each point you got.

[3 marks] (c)Cari masa maksimum, betul kepada dua tempat perpuluhan, yang melayakkan seorang murid dipilih.Find the maximum time, correct to two decimal places, that qualifies a pupil to be selected.

Show answerHide answer

P(Z < −k) = 0.2, so P(Z > k) = 0.2 and k = 0.84 (0.842 on a calculator) [1]1
(c − 8)/1.5 = −0.84 [1]1
c = 8 − 1.26 = 6.74 minutes [1]1

Tick each point you got.

[1 mark] (d)Seorang murid mengambil masa 6.5 minit. Adakah dia layak dipilih? Beri sebab.A pupil takes 6.5 minutes. Does she qualify to be selected? Give a reason.

Show answerHide answer

Yes, because 6.5 < 6.74 minutes (her time is less than the cut-off) [1]1

Tick each point you got.

For the teacher: Evaluating, medium difficulty, Chapter 5, standard 5.3.5

Textbook p. 181

Question 16

Rajah 3 menunjukkan logo Kelab Seni sekolah. Logo itu ialah sebuah bulatan berpusat O dengan jejari 10 cm. Tali lengkok AB mencangkum sudut 2π/3 rad di O. Tembereng AB yang berlorek akan dicat dengan cat emas. [Guna π = 3.142]Diagram 3 shows the logo of the school Art Club. The logo is a circle with centre O and radius 10 cm. The chord AB subtends an angle of 2π/3 rad at O. The shaded segment AB is to be painted with gold paint. [Use π = 3.142]

Circle with centre O, radii OA and OB of 10 cm with angle 2 pi over 3 between them, and the segment below chord AB shaded — Rajah 3Diagram 3

[3 marks] (a)Cari panjang lengkok minor AB, betul kepada dua tempat perpuluhan.Find the length of the minor arc AB, correct to two decimal places.

Show answerHide answer

s = rθ [1]1
s = 10 × 2(3.142)/3 [1]1
s = 20.95 cm [1]1

Tick each point you got.

[3 marks] (b)Cari perimeter kawasan berlorek itu, betul kepada dua tempat perpuluhan.Find the perimeter of the shaded region, correct to two decimal places.

Show answerHide answer

Chord AB = 2 × 10 × sin(π/3) [1]1
AB = 17.32 cm [1]1
Perimeter = 20.95 + 17.32 = 38.27 cm [1]1

Tick each point you got.

[3 marks] (c)Cari luas kawasan berlorek itu, betul kepada dua tempat perpuluhan.Find the area of the shaded region, correct to two decimal places.

Show answerHide answer

Area of sector = ½(10)²(2.0947) = 104.73 cm² [1]1
Area of triangle OAB = ½(10)²(sin 120°) = 43.30 cm² [1]1
Segment = 104.73 − 43.30 = 61.43 cm² [1]1

Tick each point you got.

[1 mark] (d)Satu tin cat emas boleh menutup 50 cm². Kelab itu akan mengecat 12 logo. Cari bilangan tin cat emas yang perlu dibeli. Tunjukkan pengiraan anda.One tin of gold paint covers 50 cm². The club will paint 12 logos. Find the number of tins of gold paint that must be bought. Show your calculation.

Show answerHide answer

12 × 61.43 = 737.16 cm²; 737.16 ÷ 50 = 14.7, so 15 tins [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, Chapter 1, standard 1.4.1

Textbook p. 20

Test specification (JSU)

Chapter	Remembering	Understanding	Applying	Analysing	Evaluating	Creating	Items	marks
1. Circular Measure	–	–	2 (13)	–	–	–	2	13
2. Differentiation	–	–	3 (17)	–	–	–	3	17
3. Integration	–	–	–	1 (4)	–	–	1	4
4. Permutation and Combination	–	–	2 (15)	–	–	–	2	15
5. Probability Distribution	–	–	1 (4)	–	1 (10)	–	2	14
6. Trigonometric Functions	1 (4)	–	1 (4)	1 (10)	–	–	3	18
7. Linear Programming	–	–	2 (14)	–	–	–	2	14
8. Kinematics of Linear Motion	–	–	1 (5)	–	–	–	1	5
Total	1 (4)	0 (0)	12 (72)	2 (14)	1 (10)	0 (0)	16	100

Each cell shows items (marks).