Mathematics questions, chapter 1

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Variation

45 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

The wage of a part-time promoter varies directly as the number of hours worked. What happens to the wage if the hours decrease by 40%?

It increases by 40%
It decreases by 40%
It is halved
It decreases by 60%

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In a direct variation both quantities change at the same rate, so a 40% fall in hours is a 40% fall in wage.

For the teacher: Remembering, low difficulty, standard 1.1.1

Textbook p. 3

Question 2

The values of x and y are shown below.
x: 1, 2, 3, 4
y: 3, 6, 9, 12
Which relation is correct?

y ∝ x
y ∝ x²
y ∝ 1/x
y ∝ √x

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y ÷ x = 3 every time, so y/x is constant and y ∝ x. y/x² is 3, 1.5, 1, 0.75, which is not constant.

For the teacher: Understanding, low difficulty, standard 1.1.2

Textbook p. 5

Question 3

y varies directly as the square of x. y = 18 when x = 3. Find y when x = 5.

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y = kx². 18 = 9k, so k = 2. When x = 5, y = 2 × 25 = 50. 30 comes from treating it as y ∝ x.

For the teacher: Applying, low difficulty, standard 1.1.2

Textbook p. 8

Question 4

m varies directly as the square root of n. m = 6 when n = 9. Find n when m = 10.

25
15
√50
100

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m = k√n. 6 = 3k, so k = 2. 10 = 2√n gives √n = 5 and n = 25.

For the teacher: Applying, low difficulty, standard 1.1.2

Textbook p. 8

Question 5

p varies directly as the square of q and inversely as r. Which equation shows this?

p = kq²r
p = kq²/r²
p = kq²/r
p = kr/q²

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Direct variation goes on the top and inverse variation on the bottom: p = kq²/r.

For the teacher: Understanding, low difficulty, standard 1.3.1

Textbook p. 27

Question 6

y varies inversely as x. y = 6 when x = 4. Find y when x = 3.

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xy = k = 24. When x = 3, y = 24 ÷ 3 = 8. y gets bigger when x gets smaller. 4.5 comes from using direct variation.

For the teacher: Applying, low difficulty, standard 1.2.2

Textbook p. 23

Question 7

y varies inversely as the square of x. y = 2 when x = 6. Find x when y = 8.

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x²y = k = 36 × 2 = 72. When y = 8, x² = 72 ÷ 8 = 9, so x = 3. 9 is x², not x.

For the teacher: Applying, medium difficulty, standard 1.2.2

Textbook p. 23

Question 8

y varies directly as x and inversely as the square of z. y = 6 when x = 3 and z = 2. Find y when x = 5 and z = 4.

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y = kx/z². 6 = 3k/4, so k = 8. y = 8(5)/16 = 2.5.

For the teacher: Applying, low difficulty, standard 1.3.2

Textbook p. 28

Question 9

x varies jointly as y and z. x = 24 when y = 3 and z = 4. Find x when y = 5 and z = 6.

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x = kyz. 24 = 12k, so k = 2. x = 2 × 5 × 6 = 60.

For the teacher: Applying, low difficulty, standard 1.1.3

Textbook p. 12

Question 10

The volume V of a cylinder varies jointly as the square of its radius r and its height h. The radius increases by 10% and the height stays the same. What is the change in V?

Increases by 10%
Increases by 11%
Increases by 20%
Increases by 21%

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V is multiplied by 1.1² = 1.21, so V increases by 21%. 20% comes from adding 10% twice.

For the teacher: Analysing, high difficulty, standard 1.1.3

Textbook p. 12

Question 11

Which table shows that y varies inversely as x?

x: 1, 2, 3, 4 and y: 2, 4, 6, 8
x: 1, 2, 3, 6 and y: 12, 6, 4, 2
x: 1, 2, 3, 4 and y: 12, 10, 8, 6
x: 2, 4, 6, 8 and y: 1, 4, 9, 16

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xy = 12 in every column of the second table. The first is direct variation; the third has y falling by 2 each time (not a constant product); the fourth has neither a constant ratio nor a constant product.

For the teacher: Analysing, low difficulty, standard 1.2.2

Textbook p. 21

Section B: structured questions

34 marks

Answer all questions. Show your working.

Question 12

The table shows the values of x and y.
x: 1, 2, 3, 4, 5
y: 3, 12, 27, 48, 75

[2 marks] (a)Calculate y ÷ x² for each pair of values.

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3 ÷ 1, 12 ÷ 4, 27 ÷ 9 [1]1
all equal 3 (48 ÷ 16 = 3, 75 ÷ 25 = 3) [1]1

Tick each point you got.

[1 mark] (b)Hence write the relation between y and x using the symbol ∝.

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y ∝ x² [1]1

y/x² is constant, so y varies directly as the square of x.

Tick each point you got.

For the teacher: Understanding, low difficulty, standard 1.1.2

Textbook p. 7

Question 13

Diagram 1 shows the graph of y against x². The straight line passes through the origin and the point (16, 40).

Graph of y against x squared: a straight line from the origin through the point (16, 40) — Diagram 1

[1 mark] (a)State the relation between y and x.

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y ∝ x² [1]1

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[2 marks] (b)Find the equation that links y and x.

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k = 40 ÷ 16 = 2.5 [1]1
y = 2.5x² [1]1

k is the gradient of the line.

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[1 mark] (c)Calculate y when x = 6.

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y = 2.5 × 36 = 90 [1]1

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For the teacher: Applying, low difficulty, standard 1.1.2

Textbook p. 7

Question 14

The kinetic energy E joules of an object varies jointly as its mass m kg and the square of its speed v m s⁻¹. E = 100 when m = 2 and v = 10.

[2 marks] (a)Write an equation that links E, m and v.

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E = kmv², 100 = k(2)(100), k = 0.5 [1]1
E = 0.5mv² [1]1

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[2 marks] (b)Calculate the speed of an object of mass 5 kg and kinetic energy 250 J.

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250 = 0.5(5)v², v² = 100 [1]1
v = 10 m s⁻¹ [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 1.1.4

Textbook p. 15

Question 15

The interest I received varies jointly as the principal p and the period t in years. Hana receives RM90 when she saves RM3 000 for 2 years.

[2 marks] (a)Write an equation that links I, p and t.

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I = kpt, 90 = k(3 000)(2), k = 0.015 [1]1
I = 0.015pt [1]1

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[2 marks] (b)Calculate the period needed to receive RM270 from a principal of RM4 000.

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270 = 0.015(4 000)t [1]1
t = 4.5 years [1]1

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[1 mark] (c)Hana wants the same interest in a shorter period. Should she increase or decrease the principal? Explain.

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Increase. I is fixed, so pt is fixed: when t decreases, p must increase [1]1

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For the teacher: Applying, medium difficulty, standard 1.1.4

Textbook p. 15

Question 16

The time t minutes to set up chairs varies with the number of workers x. The table shows the values.
x: 2, 3, 4, 6
t: 36, 24, 18, 12

[2 marks] (a)Calculate xt and x²t for each pair of values.

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xt = 72 for every pair [1]1
x²t = 144, 216, 288, 432 [1]1

Tick each point you got.

[1 mark] (b)Hence state the relation between t and x, using ∝.

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t ∝ 1/x [1]1

xt is constant, x²t is not.

Tick each point you got.

[1 mark] (c)Find the time needed by 9 workers.

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t = 72 ÷ 9 = 8 minutes [1]1

Tick each point you got.

For the teacher: Analysing, low difficulty, standard 1.2.2

Textbook p. 21

Question 17

At a fixed temperature the pressure p kPa of a gas varies inversely as its volume V cm³. The pressure is 150 kPa when the volume is 60 cm³.

[2 marks] (a)Express p in terms of V.

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k = 150 × 60 = 9 000 [1]1
p = 9 000/V [1]1

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[1 mark] (b)Calculate the volume when the pressure is 120 kPa.

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V = 9 000 ÷ 120 = 75 cm³ [1]1

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[1 mark] (c)The volume is halved to 30 cm³. Calculate the new pressure and describe the change.

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p = 300 kPa, the pressure is doubled [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 1.2.3

Textbook p. 25

Question 18

T varies directly as the square of x and inversely as y. T = 12 when x = 3 and y = 6.

[2 marks] (a)Express T in terms of x and y.

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T = kx²/y, 12 = k(9)/6, k = 8 [1]1
T = 8x²/y [1]1

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[1 mark] (b)Calculate T when x = 5 and y = 4.

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T = 8(25) ÷ 4 = 50 [1]1

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[1 mark] (c)Calculate y when T = 16 and x = 4.

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16 = 8(16)/y, y = 8 [1]1

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For the teacher: Applying, medium difficulty, standard 1.3.2

Textbook p. 28

Question 19

A shop sells pizzas of three sizes: diameter 20 cm for RM10, 30 cm for RM21 and 40 cm for RM32. The area of a pizza is directly proportional to the square of its diameter d cm.

[2 marks] (a)Calculate price ÷ d² for each pizza, correct to 4 decimal places.

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10 ÷ 400 = 0.0250 [1]1
21 ÷ 900 = 0.0233 and 32 ÷ 1 600 = 0.0200 [1]1

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[1 mark] (b)Does the price vary directly as the area? Give a reason.

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No, price ÷ d² is not constant [1]1

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[1 mark] (c)Which size gives the best value? Explain.

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40 cm, it has the lowest price for each cm² of pizza [1]1

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For the teacher: Evaluating, high difficulty, standard 1.1.4

Textbook p. 16

Question 20

Create your own example of an inverse variation.

[2 marks] (a)Describe a real situation in which one quantity falls when another rises, and write an equation y = k/x for it with k = 24 (say what x and y stand for).

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x workers take y days to finish a job. y = 24/x, so 4 workers take 6 days.

A sensible real situation with both quantities named [1]1
Equation of the form y = 24/x that matches it, e.g. x workers take y days to do a job of 24 worker-days [1]1

Tick each point you got.

For the teacher: Creating, high difficulty, standard 1.2.1

Textbook p. 17

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