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Variation · Form 5

Variation: Practice Questions

Original SPM-style practice questions for Variation, each with full worked solutions, Paper 1 multiple-choice and Paper 2 structured.

Written by the spmmath.com.my editorial team.

Original practice questions for Variation, in the style of Mathematics Paper 1 (Objective) and Mathematics Paper 2 (Subjective). Each answer is worked so you can check your method, not just the result.

Multiple-choice (Paper 1 style)

Question 1

y varies directly as the square of x. Which of the following relationships is correct?

(k is a constant.)

  1. A. y = k/x²
  2. B. y = kx²
  3. C. y = k√x
  4. D. y = kx

Question 2

p varies directly as q. Given that p = 12 when q = 3, find the value of the constant k.

  1. A. 3
  2. B. 9
  3. C. 4
  4. D. 36

Question 3

y varies inversely as x. When x = 2, y = 15.

Find the value of y when x = 6.

  1. A. 20
  2. B. 10
  3. C. 5
  4. D. 45

Question 4

w varies directly as x and inversely as y. Which relationship is correct?

(k is a constant.)

  1. A. w = kxy
  2. B. w = kx/y
  3. C. w = ky/x
  4. D. w = k/(xy)

Question 5

y varies directly as √x. Given y = 6 when x = 4, find y when x = 9.

  1. A. 13.5
  2. B. 12
  3. C. 9
  4. D. 4.5

Question 6

F varies inversely as the square of d. When d = 3, F = 8.

Find the value of the constant k.

  1. A. 24
  2. B. 72
  3. C. 2.67
  4. D. 8

Question 7

z varies jointly as x and y. Given z = 24 when x = 2 and y = 3, find z when x = 5 and y = 4.

  1. A. 40
  2. B. 60
  3. C. 100
  4. D. 80

Question 8

y varies directly as x. When x = 8, y = 20.

Using the proportion method, find y when x = 14.

  1. A. 11.43
  2. B. 35
  3. C. 32
  4. D. 26

Question 9

y varies inversely as x. If x is multiplied by 4, what happens to y?

  1. A. multiplied by 4
  2. B. divided by 4
  3. C. unchanged
  4. D. multiplied by 16

Question 10

y varies directly as x² and inversely as z. Which expression is correct?

(k is a constant.)

  1. A. y = kx²z
  2. B. y = kx²/z
  3. C. y = kz/x²
  4. D. y = k/(x²z)

Structured (Paper 2 style)

Question 1 (6 marks)

The mass, M grams, of a copper wire varies directly as its length, L cm. A wire of length 25 cm has a mass of 40 g.

(a) Express M in terms of L. (b) Find the mass of a wire of length 60 cm.

(c) Find the length of a wire that has a mass of 72 g.

  1. Write the direct variation as M = kL, where k is a constant.
  2. Substitute M = 40 and L = 25: 40 = k × 25, so k = 40 ÷ 25 = 1.6.
  3. (a) Therefore M = 1.6L.
  4. (b) When L = 60: M = 1.6 × 60 = 96 g.
  5. (c) When M = 72: 72 = 1.6L, so L = 72 ÷ 1.6 = 45 cm.

Question 2 (6 marks)

The time, t hours, taken to fill a water tank varies inversely as the number of pipes, n, that are used. When 4 pipes are used, the tank is filled in 3 hours.

(a) Form an equation connecting t and n. (b) Find the time taken to fill the tank when 6 pipes are used.

(c) Find the number of pipes needed to fill the tank in 1.5 hours.

  1. Write the inverse variation as t = k/n, where k is a constant.
  2. Substitute t = 3 and n = 4: 3 = k ÷ 4, so k = 3 × 4 = 12.
  3. (a) Therefore t = 12/n.
  4. (b) When n = 6: t = 12 ÷ 6 = 2 hours.
  5. (c) When t = 1.5: 1.5 = 12/n, so n = 12 ÷ 1.5 = 8 pipes.

Question 3 (7 marks)

The quantity w varies directly as u and inversely as the square of v. When u = 8 and v = 2, w = 10.

(a) Express w in terms of u and v. (b) Find the value of w when u = 12 and v = 5.

(c) Find the value of v when w = 25 and u = 45.

  1. Write the combined variation as w = ku/v², where k is a constant.
  2. Substitute w = 10, u = 8, v = 2: 10 = k × 8 ÷ 2² = 8k ÷ 4 = 2k, so k = 5.
  3. (a) Therefore w = 5u/v².
  4. (b) When u = 12 and v = 5: w = 5 × 12 ÷ 5² = 60 ÷ 25 = 2.4.
  5. (c) When w = 25 and u = 45: 25 = 5 × 45 ÷ v² = 225/v², so v² = 225 ÷ 25 = 9 and v = √9 = 3.

Question 4 (6 marks)

The electrical power, P watts, dissipated in a resistor varies directly as the square of the current, I amperes. When I = 3 A, P = 45 W.

(a) Express P in terms of I. (b) Find the power when I = 5 A.

(c) Find the current when the power is 80 W.

  1. Write the direct variation as P = kI², where k is a constant.
  2. Substitute P = 45 and I = 3: 45 = k × 3² = 9k, so k = 45 ÷ 9 = 5.
  3. (a) Therefore P = 5I².
  4. (b) When I = 5: P = 5 × 5² = 5 × 25 = 125 W.
  5. (c) When P = 80: 80 = 5I², so I² = 80 ÷ 5 = 16 and I = √16 = 4 A.

Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)

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Frequently asked questions

What is the difference between the Paper 1 and Paper 2 questions in this variation practice set?

Paper 1 questions are multiple-choice, checking whether you can quickly identify direct, inverse, or joint variation from a given relationship. Paper 2 questions are structured, usually asking you to find the constant of variation, form an equation, then use it to calculate an unknown value.

Why solve variation questions myself before looking at the worked solution?

Setting up the correct relationship, direct, inverse, or joint, is the part most likely to go wrong, and only attempting it yourself shows whether you chose the right form before finding the constant. Reading the solution first hides that decision, so the same mistake can repeat in the exam.

What mistakes commonly appear in variation questions?

A common error is writing an inverse variation as y = kx instead of y = k/x, or forgetting to square or cube a variable when the question states 'varies as the square of' or 'the cube of'. Students also sometimes substitute values in the wrong order when finding the constant, k.

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