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Mathematics questions, chapter 2
Nama / NameKelas / ClassTarikh / Date

Factorisation and Algebraic Fractions

41 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

12 marks
Choose the correct answer. Each question carries 1 mark.
1

Question 1

Expand 4(2x − 3).
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A
Every term in the bracket is multiplied by 4: 4 × 2x = 8x and 4 × (−3) = −12. 8x − 3 forgets to multiply the second term.
For the teacher: Remembering, low difficulty, standard 2.1.1
Textbook p. 21
2

Question 2

Expand −3p(p − 4).
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B
(−3p) × p = −3p2 and (−3p) × (−4) = +12p, because (−) × (−) = +. The first option keeps the minus sign on the second term.
For the teacher: Understanding, low difficulty, standard 2.1.1
Textbook p. 22
3

Question 3

Expand (x + 5)(x − 2).
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C
x2 − 2x + 5x − 10 = x2 + 3x − 10. The first option multiplies only the first and last terms.
For the teacher: Applying, low difficulty, standard 2.1.2
Textbook p. 23
4

Question 4

Expand (2k − 3)2.
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D
(2k − 3)(2k − 3) = 4k2 − 6k − 6k + 9 = 4k2 − 12k + 9. (2k)2 is 4k2, not 2k2, and the middle term is 2 × 2k × 3 = 12k.
For the teacher: Applying, medium difficulty, standard 2.1.2
Textbook p. 24
5

Question 5

Simplify (2w + 1)(w − 3) − 5w.
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A
Expand first: 2w2 − 6w + w − 3 = 2w2 − 5w − 3. Then subtract 5w: 2w2 − 10w − 3. The last option forgets the −5w.
For the teacher: Applying, high difficulty, standard 2.1.3
Textbook p. 25
6

Question 6

Diagram 1 shows a rectangular carpet. What is the area of the carpet?
Rectangle with length (2x + 3) metres and width (x − 1) metres
Diagram 1
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B
Area = length × width = (2x + 3)(x − 1) = 2x2 − 2x + 3x − 3 = 2x2 + x − 3.
For the teacher: Applying, medium difficulty, standard 2.1.4
Textbook p. 25
7

Question 7

What is the highest common factor (HCF) of 12ab and 18b2?
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C
HCF of 12 and 18 is 6. The only letter in both terms is b, and the smaller power is b. So the HCF is 6b. 6ab is not a factor of 18b2 because 18b2 has no a.
For the teacher: Remembering, low difficulty, standard 2.2.1
Textbook p. 28
8

Question 8

Factorise 3y2 − 48 completely.
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D
Take out the HCF 3 first: 3(y2 − 16). Then y2 − 16 = (y + 4)(y − 4). The second and third options are not fully factorised.
For the teacher: Applying, medium difficulty, standard 2.2.2
Textbook p. 29
9

Question 9

Which is the factorisation of 25p2 − 16?
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A
25p2 = (5p)2 and 16 = 42, so the difference of squares gives (5p + 4)(5p − 4). (5p − 4)2 expands to a three-term expression.
For the teacher: Understanding, low difficulty, standard 2.2.2
Textbook p. 29
10

Question 10

Factorise x2 − x − 12.
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B
Find two numbers with product −12 and sum −1: −4 and +3. So (x − 4)(x + 3). (x + 4)(x − 3) expands to x2 + x − 12, which has the wrong sign on x.
For the teacher: Applying, medium difficulty, standard 2.2.2
Textbook p. 30
11

Question 11

Factorise 3x2 + 10x − 8.
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C
Try (3x − 2)(x + 4): cross products 12x and −2x give 10x, and −2 × 4 = −8. Check the others: (3x + 2)(x − 4) gives −10x, (3x − 4)(x + 2) gives +2x and (3x + 8)(x − 1) gives +5x.
For the teacher: Analysing, high difficulty, standard 2.2.2
Textbook p. 31
12

Question 12

Simplify 32x + 13x.
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D
The LCM of 2x and 3x is 6x. 96x + 26x = 116x. Adding the denominators (5x) is the usual mistake.
For the teacher: Applying, low difficulty, standard 2.3.1
Textbook p. 35

Section B: structured questions

29 marks
Answer all questions. Show your working.
13

Question 13

Expand and simplify each expression.
[1 mark] (a)(m + 4)(m − 7)
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Tick each point you got.
[1 mark] (b)(2a − 3b)(a + 5b)
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Tick each point you got.
[1 mark] (c)(3x − 4)2
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(a − b)2 = a2 − 2ab + b2 with a = 3x and b = 4.
Tick each point you got.
For the teacher: Applying, low difficulty, standard 2.1.2
Textbook p. 23
14

Question 14

Simplify each expression.
[2 marks] (a)(2n + 1)(n + 3) − 4n − 2
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Tick each point you got.
[2 marks] (b)(x + 4)2 − (x − 4)2
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The bracket after the minus sign changes the sign of all three terms.
Tick each point you got.
For the teacher: Applying, medium difficulty, standard 2.1.3
Textbook p. 25
15

Question 15

Diagram 2 shows a rectangular floor of a hall. A rug measuring 3 m by (x + 1) m is placed on the floor.
Rectangle (2x + 5) metres by (x + 3) metres with a pink rug 3 metres by (x + 1) metres in the bottom left corner
Diagram 2
[2 marks] (a)Write the area of the floor in expanded form.
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Tick each point you got.
[2 marks] (b)Find the area of the floor that is not covered by the rug.
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Tick each point you got.
For the teacher: Applying, medium difficulty, standard 2.1.4
Textbook p. 26
16

Question 16

Answer these questions about factors.
[1 mark] (a)List all the factors of 5y.
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Tick each point you got.
[1 mark] (b)Find the HCF of 12x2y and 18xy2.
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6 is the HCF of 12 and 18; the lowest power of x is x and of y is y.
Tick each point you got.
[1 mark] (c)Factorisation is the reverse process of
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expansion
Expansion removes brackets and factorisation puts them back.
For the teacher: Understanding, low difficulty, standard 2.2.1
Textbook p. 27
17

Question 17

Factorise each expression completely.
[1 mark] (a)6ab − 9a2
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Tick each point you got.
[1 mark] (b)4m2 − 81
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Tick each point you got.
[1 mark] (c)6k2 + k − 2
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Cross products 4k and −3k add to k.
Tick each point you got.
For the teacher: Applying, medium difficulty, standard 2.2.2
Textbook p. 28
18

Question 18

Faiz factorised 3x2 − 12 as (3x + 6)(x − 2).
[1 mark] (a)Expand (3x + 6)(x − 2) to check whether his answer is correct.
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Tick each point you got.
[1 mark] (b)Explain why his answer is not completely factorised.
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Tick each point you got.
[1 mark] (c)Write the complete factorisation.
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Take out HCF 3 first, then use the difference of squares.
Tick each point you got.
For the teacher: Evaluating, high difficulty, standard 2.2.2
Textbook p. 29
19

Question 19

The floor area of a rectangular hall is (6x2 + 15x) m2. Its width is 3x m.
[2 marks] (a)Find the length of the hall in terms of x.
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Tick each point you got.
[1 mark] (b)Find the area of the hall when x = 4, if the length is (2x + 5) m.
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Width 3x = 12 m and length 2x + 5 = 13 m. Check: 6 × 16 + 15 × 4 = 96 + 60 = 156.
Tick each point you got.
For the teacher: Analysing, medium difficulty, standard 2.2.3
Textbook p. 32
20

Question 20

Simplify the expression.
[2 marks] (a)x + 34 − x − 16
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3x + 9 − 2x + 2 = x + 11. The minus sign changes both terms of (x − 1).
Tick each point you got.
For the teacher: Applying, medium difficulty, standard 2.3.1
Textbook p. 35
21

Question 21

Simplify each expression.
[2 marks] (a)p2 − 43p × 6pp + 2
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Tick each point you got.
[2 marks] (b)2x + 1 + x − 1x2 − 1
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Factorise and simplify the second fraction before adding.
Tick each point you got.
For the teacher: Applying, high difficulty, standard 2.3.2
Textbook p. 36