Additional Mathematics questions, chapter 7

Nama / NameKelas / ClassTarikh / Date

Linear Programming

44 marks. Use Practice to check each answer as you go, or Exam to answer everything first. Show your working for Section B.

Section A: objective questions

11 marks

Choose the correct answer. Each question carries 1 mark.

Question 1

In a linear programming problem, what is the objective function?

A linear inequality that limits the resources
A linear expression, k = ax + by, that is maximised or minimised
The region that satisfies all the constraints
A variable, such as x or y, that stands for a decision

Show answerHide answer

The objective function is the quantity to optimise, such as profit or cost, written as k = ax + by. The inequalities are the constraints; the region is the feasible region.

For the teacher: Remembering, low difficulty, standard 7.1.1

Textbook p. 234

Question 2

Which inequality is graphed with a dashed line?

y ≥ 3
x ≤ 5
2x + y > 6
x + y ≤ 9

Show answerHide answer

A strict symbol (< or >) gives a dashed line because points on the line are not included. ≤ and ≥ give a solid line.

For the teacher: Remembering, low difficulty, standard 7.1.1

Textbook p. 235

Question 3

Which describes the graph of 3x − 2y ≥ 6?

Region above the line, solid line
Region above the line, dashed line
Region below the line, dashed line
Region below the line, solid line

Show answerHide answer

3x − 2y ≥ 6 gives −2y ≥ 6 − 3x, so y ≤ 1.5x − 3 (dividing by −2 reverses the symbol). That is below the line, and ≥ in the original means a solid line. Test (0, 0): 0 ≥ 6 is false, and (0, 0) is above the line.

For the teacher: Understanding, low difficulty, standard 7.1.1

Textbook p. 236

Question 4

A company uses x lorries and y vans. The number of vans is at least three times the number of lorries. Which inequality represents this?

y ≥ 3x
x ≥ 3y
y ≤ 3x
y ≥ x + 3

Show answerHide answer

'At least' means ≥, and three times the lorries is 3x, so y ≥ 3x. The second option swaps the variables.

For the teacher: Applying, low difficulty, standard 7.1.1

Textbook p. 235

Question 5

A factory makes x cars of type M and y cars of type N. The total made is at most 80 and at least 10 are type N. Which pair of inequalities represents this?

x + y ≥ 80 and y ≤ 10
x + y ≤ 80 and y ≤ 10
x + y ≤ 80 and y ≥ 10
x + y < 80 and y > 10

Show answerHide answer

'At most 80' is x + y ≤ 80 and 'at least 10' is y ≥ 10. The last option uses strict symbols, which would exclude exactly 80 and exactly 10.

For the teacher: Applying, low difficulty, standard 7.1.1

Textbook p. 238

Question 6

A table P costs RM200 and a table Q costs RM100. The money spent on x tables P and y tables Q must not exceed RM6 000. Which inequality, in its simplest form, represents this?

x + 2y ≤ 60
2x + y ≤ 60
2x + y ≥ 60
x + y ≤ 60

Show answerHide answer

200x + 100y ≤ 6 000. Divide by 100: 2x + y ≤ 60. 'Must not exceed' is ≤.

For the teacher: Applying, low difficulty, standard 7.2.1

Textbook p. 242

Question 7

Which point lies in the region defined by x + 2y ≥ 30, 2x + y ≤ 60 and y ≤ 2x?

(12, 20)
(5, 10)
(20, 30)
(10, 25)

Show answerHide answer

(12, 20): 12 + 40 = 52 ≥ 30, 24 + 20 = 44 ≤ 60 and 20 ≤ 24. (5, 10) gives x + 2y = 25. (20, 30) gives 2x + y = 70. (10, 25) has y > 2x.

For the teacher: Understanding, low difficulty, standard 7.2.1

Textbook p. 243

Question 8

The vertices of a feasible region are (2, 1), (2, 6) and (8, 4). Find the maximum value of 3x + 2y.

Show answerHide answer

3x + 2y at (2, 1) is 8, at (2, 6) is 18 and at (8, 4) is 24 + 8 = 32. The maximum is 32.

For the teacher: Applying, low difficulty, standard 7.1.1

Textbook p. 237

Question 9

The vertices of the feasible region R are (1, 4), (1, 9), (6, 9) and (6, 2). Find the minimum value of the cost 2x + 3y.

Show answerHide answer

2x + 3y at the vertices is 14, 29, 39 and 18. The least is 14, at (1, 4).

For the teacher: Applying, low difficulty, standard 7.2.1

Textbook p. 241

Question 10

Diagram 1 shows the shaded region R. Which set of inequalities defines R?

Graph from 0 to 8 on x and 0 to 14 on y. Solid lines y = 2x, x + y = 6 and x = 6 enclose a shaded triangle R with vertices at about (2, 4), (6, 0) and (6, 12) — Diagram 1

y ≥ 2x, x + y ≥ 6, x ≤ 6
y ≤ 2x, x + y ≤ 6, x ≤ 6
y ≤ 2x, x + y ≥ 6, x ≤ 6
y ≤ 2x, x + y ≥ 6, x ≥ 6

Show answerHide answer

R is below y = 2x, above x + y = 6 and to the left of x = 6. Test (5, 5): 5 ≤ 10, 10 ≥ 6, 5 ≤ 6, all true.

For the teacher: Analysing, high difficulty, standard 7.1.1

Textbook p. 236

Question 11

In a feasible region, the maximum or minimum value of the objective function occurs at

the origin
a vertex of the region
the midpoint of the region
any point inside the region

Show answerHide answer

Moving the objective line parallel to itself, the last (or first) point it touches is a vertex of the feasible region.

For the teacher: Remembering, low difficulty, standard 7.1.1

Textbook p. 241

Section B: structured questions

33 marks

Answer all questions. Show your working.

Question 12

Diagram 2 shows two shaded regions. In (a) the line passes through (0, 4) and (6, 0). In (b) the line passes through (0, −2) and (4, 0).

(a) A solid line through (0, 4) and (6, 0) with the region below it, including the origin, shaded. (b) A dashed line through (0, −2) and (4, 0) with the region above it, including the origin, shaded — Diagram 2

[2 marks] (a)Write the inequality for the shaded region in (a).

Show answerHide answer

Line: 2x + 3y = 12 [1]1
2x + 3y ≤ 12 (solid line, origin inside) [1]1

Tick each point you got.

[2 marks] (b)Write the inequality for the shaded region in (b).

Show answerHide answer

Line: x − 2y = 4 [1]1
x − 2y < 4 (dashed line, origin inside: 0 < 4) [1]1

Tick each point you got.

For the teacher: Understanding, low difficulty, standard 7.1.1

Textbook p. 236

Question 13

Xin Tian plants x hectares of banana trees and y hectares of papaya trees on at most 80 hectares of land. Each hectare of banana trees needs 3 workers and each hectare of papaya trees needs 6 workers; he can hire at most 360 workers. Maintenance costs RM800 per hectare for banana and RM300 per hectare for papaya, and his capital is at least RM24 000. Other than x ≥ 0 and y ≥ 0, write three inequalities.

[1 mark] (a)The land used.

Show answerHide answer

x + y ≤ 80 [1]1

Tick each point you got.

[2 marks] (b)The number of workers, in its simplest form.

Show answerHide answer

3x + 6y ≤ 360 [1]1
x + 2y ≤ 120 [1]1

Tick each point you got.

[2 marks] (c)The capital.

Show answerHide answer

800x + 300y ≥ 24 000 [1]1
8x + 3y ≥ 240 [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 7.1.1

Textbook p. 238

Question 14

Diagram 3 shows the feasible region of a situation. The objective function is k = 2x + y.

Graph from 0 to 40 on x and 0 to 50 on y. A triangle with vertices (10, 40), (30, 20) and (10, 5), bounded by x = 10, x + y = 50 and 3x − 4y = 10 — Diagram 3

[2 marks] (a)Find the value of k at each vertex.

Show answerHide answer

(10, 40): 60; (30, 20): 80 [1]1
(10, 5): 25 [1]1

Tick each point you got.

[1 mark] (b)State the maximum value of k.

Show answerHide answer

80 [1]1

Tick each point you got.

[1 mark] (c)State the minimum value of k.

Show answerHide answer

25 [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 7.1.1

Textbook p. 237

Question 15

A canteen prepares x packets of nasi lemak and y packets of mee goreng. The conditions are: I at most 60 packets in all; II the packets of mee goreng are at most twice the packets of nasi lemak; III at least 10 packets of mee goreng. Diagram 4 shows the region R that satisfies all the conditions.

Graph from 0 to 60 on x and 0 to 70 on y. Lines y = 10, y = 2x and x + y = 60 enclose a shaded triangle R with vertices at about (5, 10), (20, 40) and (50, 10) — Diagram 4

[3 marks] (a)Write the three inequalities for conditions I, II and III.

Show answerHide answer

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

Tick each point you got.

[2 marks] (b)Use Diagram 4 to find the least and greatest number of packets of nasi lemak when 20 packets of mee goreng are made.

Show answerHide answer

y = 20 meets y = 2x at x = 10, so least 10 [1]1
y = 20 meets x + y = 60 at x = 40, so greatest 40 [1]1

Tick each point you got.

[3 marks] (c)The profit is RM2 per packet of nasi lemak and RM3 per packet of mee goreng. Find the maximum profit.

Show answerHide answer

k = 2x + 3y [1]1
Vertices (5, 10), (20, 40), (50, 10) give 40, 160, 130 [1]1
Maximum profit RM160 [1]1

(20, 40) is where y = 2x meets x + y = 60: 3x = 60.

Tick each point you got.

For the teacher: Analysing, medium difficulty, standard 7.2.1

Textbook p. 241

Question 16

A post office hires x lorries and y vans to carry at least 600 parcels. A lorry carries 100 parcels and a van carries 50 parcels. The number of vans is at most twice the number of lorries, and at least 2 vans are used. Diagram 5 shows region R, which continues beyond the grid. A lorry costs RM150 and a van costs RM100.

Graph from 0 to 10 on x and 0 to 14 on y. Lines 2x + y = 12, y = 2x and y = 2 bound a shaded region R that lies above 2x + y = 12 and y = 2, below y = 2x, and extends upwards and to the right — Diagram 5

[3 marks] (a)Write three inequalities, other than x ≥ 0 and y ≥ 0, in simplest form.

Show answerHide answer

100x + 50y ≥ 600, so 2x + y ≥ 12 [1]1
y ≤ 2x [1]1
y ≥ 2 [1]1

Tick each point you got.

[1 mark] (b)Find the least number of lorries needed if only 2 vans are used.

Show answerHide answer

2x + 2 ≥ 12, x ≥ 5, so 5 lorries [1]1

Tick each point you got.

[3 marks] (c)Find the minimum cost.

Show answerHide answer

k = 150x + 100y [1]1
Vertices (5, 2) and (3, 6) (from 2x + y = 12 with y = 2 and with y = 2x) [1]1
k = 950 and 1 050, so minimum RM950 [1]1

In a region that is unbounded, check the vertices for a minimum: cost only grows when moving away.

Tick each point you got.

For the teacher: Analysing, medium difficulty, standard 7.2.1

Textbook p. 247

Question 17

Aina says: 'The region for 2x − 3y > 6 is above the line 2x − 3y = 6 because the symbol is >.'

[1 mark] (a)Is Aina correct?

AYesBNo

Show answerHide answer

The coefficient of y is negative, so the sides swap.

[2 marks] (b)Write the inequality as y < ... or y > ... and state which side of the line is the region.

Show answerHide answer

−3y > 6 − 2x, so y < (2/3)x − 2 (dividing by −3 reverses the symbol) [1]1
The region is below the line [1]1

Tick each point you got.

[2 marks] (c)Use the point (0, 0) to check your answer, and state the type of line.

Show answerHide answer

0 > 6 is false, so (0, 0) is not in the region (it is above the line) [1]1
Dashed line, because the symbol is strict [1]1

Tick each point you got.

For the teacher: Evaluating, high difficulty, standard 7.1.1

Textbook p. 236

Back to notes Chapter 8 questions Model papers