Sec 3 G1 Maths Algebraic manipulation worksheet with answers

A free Sec 3 G1 Maths worksheet on algebraic manipulation: 7 questions, 30 marks, short questions first and then structured ones, in the school exam style. The answers and full worked solutions are on the last page.

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Sec 3 G1 Maths worksheet: Algebraic manipulation

Total: 30 marksSuggested time: 50 minutes

Name: ______________________________Class: ____________ Date: ____________

Answer all questions. Show all your working. Give non-exact answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless the question says otherwise.

Section A

(a)Expand and simplify

(x - 5)(x + 1)

Answer (a)[1]

(b)Expand and simplify

(3x + 7)(3x + 4)

Answer (b)[1]

(a)Expand and simplify

(x + 4)(x + 8)

Answer (a)[1]

(b)Expand and simplify

(x - 3)(x + 4)

Answer (b)[1]

Find an expression, in terms of

n

, for the

n

th term of the sequence

12, \ 16, \ 20, \ 24, \ \ldots

Answer[2]

(a)Find an expression, in terms of

n

, for the

n

th term of the sequence

16, \ 23, \ 30, \ 37, \ \ldots

Answer (a)[2]

(b)Find an expression, in terms of

n

, for the

n

th term of the sequence

-5, \ -3, \ -1, \ 1, \ \ldots

Answer (b)[2]

(a)Simplify

\dfrac{11b}{6a} \times \dfrac{12b}{5a}

, giving your answer in its simplest form.

Answer (a)[2]

(b)Simplify

\dfrac{11q^2}{2} \times \dfrac{8}{9pq}

, giving your answer in its simplest form.

Answer (b)[2]

Section B

(a)Find the 25th term of the sequence

-5, \ -3, \ -1, \ 1, \ \ldots

Answer (a)[2]

(b)(i)Expand and simplify

(x - 9)(x - 4)

Answer (b)(i)[1]

(b)(ii)Expand and simplify

(3x + 6)(x - 6)

Answer (b)(ii)[1]

(c)(i)Simplify

\dfrac{9xy^2}{4} \times \dfrac{5x}{9y^2}

, giving your answer in its simplest form.

Answer (c)(i)[2]

(c)(ii)Simplify

\dfrac{q^2}{10p^2} \div \dfrac{q^2}{12p}

, giving your answer in its simplest form.

Answer (c)(ii)[2]

(a)(i)Expand and simplify

(x + 3)(x + 5)

Answer (a)(i)[1]

(a)(ii)Expand and simplify

(3x - 6)(4x - 7)

Answer (a)(ii)[1]

(b)Find the 50th term of the sequence

7, \ 15, \ 23, \ 31, \ \ldots

Answer (b)[2]

(c)(i)Simplify

\dfrac{mn}{4} \times \dfrac{2}{3m}

, giving your answer in its simplest form.

Answer (c)(i)[2]

(c)(ii)Simplify

\dfrac{4y^2}{3x} \div \dfrac{2}{3x^2}

, giving your answer in its simplest form.

Answer (c)(ii)[2]

END OF PAPER

Answers and worked solutions

Question 1

(a)Answer:

x^2-4x-5

[1]

x^2-4x-5

(b)Answer:

9x^2+33x+28

[1]

9x^2+33x+28

Question 2

(a)Answer:

x^2+12x+32

[1]

x^2+12x+32

(b)Answer:

x^2+x-12

[1]

x^2+x-12

Question 3

Answer:

T_n = 4n + 8

[2]

\mathstrut 16 - 12 =

4

\mathstrut 12 - 4 =

8

T_n = 4n + 8

Question 4

(a)Answer:

T_n = 7n + 9

[2]

\mathstrut 23 - 16 =

7

\mathstrut 16 - 7 =

9

T_n = 7n + 9

(b)Answer:

T_n = 2n - 7

[2]

\mathstrut (-3) - (-5) =

2

\mathstrut (-5) - 2 =

-7

T_n = 2n - 7

Question 5

(a)Answer:

= \dfrac{22b^2}{5a^2}

[2]

\dfrac{11b}{6a} \times \dfrac{12b}{5a} = \dfrac{132 \times b \times b}{30 \times a \times a}

The HCF of

132

and

30

6

\dfrac{132}{30} = \dfrac{22}{5}

= \dfrac{22b^2}{5a^2}

(b)Answer:

= \dfrac{44q}{9p}

[2]

\dfrac{11q^2}{2} \times \dfrac{8}{9pq} = \dfrac{88 \times q \times q}{18 \times p \times q}

The HCF of

88

and

18

2

\dfrac{88}{18} = \dfrac{44}{9}

= \dfrac{44q}{9p}

Question 6

(a)Answer:

2 \times 25 - 7 = 43

[2]

\mathstrut (-3) - (-5) =

2

\mathstrut (-5) - 2 =

-7

T_n = 2n - 7

2 \times 25 - 7 = 43

(b)(i)Answer:

x^2-13x+36

[1]

x^2-13x+36

(b)(ii)Answer:

3x^2-12x-36

[1]

3x^2-12x-36

(c)(i)Answer:

= \dfrac{5x^2}{4}

[2]

\dfrac{9xy^2}{4} \times \dfrac{5x}{9y^2} = \dfrac{45 \times x \times x \times y \times y}{36 \times y \times y}

The HCF of

45

and

36

9

\dfrac{45}{36} = \dfrac{5}{4}

= \dfrac{5x^2}{4}

(c)(ii)Answer:

= \dfrac{6}{5p}

[2]

\dfrac{q^2}{10p^2} \div \dfrac{q^2}{12p} = \dfrac{q^2}{10p^2} \times \dfrac{12p}{q^2}

= \dfrac{12 \times p \times q \times q}{10 \times p \times p \times q \times q}

The HCF of

12

and

10

2

\dfrac{12}{10} = \dfrac{6}{5}

= \dfrac{6}{5p}

Question 7

(a)(i)Answer:

x^2+8x+15

[1]

x^2+8x+15

(a)(ii)Answer:

12x^2-45x+42

[1]

12x^2-45x+42

(b)Answer:

8 \times 50 - 1 = 399

[2]

\mathstrut 15 - 7 =

8

\mathstrut 7 - 8 =

-1

T_n = 8n - 1

8 \times 50 - 1 = 399

(c)(i)Answer:

= \dfrac{n}{6}

[2]

\dfrac{mn}{4} \times \dfrac{2}{3m} = \dfrac{2 \times m \times n}{12 \times m}

The HCF of

2

and

12

2

\dfrac{2}{12} = \dfrac{1}{6}

= \dfrac{n}{6}

(c)(ii)Answer:

= 2xy^2

[2]

\dfrac{4y^2}{3x} \div \dfrac{2}{3x^2} = \dfrac{4y^2}{3x} \times \dfrac{3x^2}{2}

= \dfrac{12 \times x \times x \times y \times y}{6 \times x}

The HCF of

12

and

6

6

\dfrac{12}{6} = 2

= 2xy^2

Sec 3 G1 Maths Algebraic manipulation worksheet with answers

In the print window, choose Save as PDF to download it.

Or practise it online with step-by-step help when you get one wrong.

Section B

(a)Find the 25th term of the sequence

-5, \ -3, \ -1, \ 1, \ \ldots

Answer (a)[2]

(b)(i)Expand and simplify

(x - 9)(x - 4)

Answer (b)(i)[1]

(b)(ii)Expand and simplify

(3x + 6)(x - 6)

Answer (b)(ii)[1]

(c)(i)Simplify

\dfrac{9xy^2}{4} \times \dfrac{5x}{9y^2}

, giving your answer in its simplest form.

Answer (c)(i)[2]

(c)(ii)Simplify

\dfrac{q^2}{10p^2} \div \dfrac{q^2}{12p}

, giving your answer in its simplest form.

Answer (c)(ii)[2]

Answers and worked solutions

Question 1

(a)Answer:

x^2-4x-5

[1]

x^2-4x-5

(b)Answer:

9x^2+33x+28

[1]

9x^2+33x+28

Question 2

(a)Answer:

x^2+12x+32

[1]

x^2+12x+32

(b)Answer:

x^2+x-12

[1]

x^2+x-12

Question 3

Answer:

T_n = 4n + 8

[2]

\mathstrut 16 - 12 =

4

\mathstrut 12 - 4 =

8

T_n = 4n + 8

Question 4

(a)Answer:

T_n = 7n + 9

[2]

\mathstrut 23 - 16 =

7

\mathstrut 16 - 7 =

9

T_n = 7n + 9

(b)Answer:

T_n = 2n - 7

[2]

\mathstrut (-3) - (-5) =

2

\mathstrut (-5) - 2 =

-7

T_n = 2n - 7

Question 5

(a)Answer:

= \dfrac{22b^2}{5a^2}

[2]

\dfrac{11b}{6a} \times \dfrac{12b}{5a} = \dfrac{132 \times b \times b}{30 \times a \times a}

The HCF of

132

and

30

6

\dfrac{132}{30} = \dfrac{22}{5}

= \dfrac{22b^2}{5a^2}

(b)Answer:

= \dfrac{44q}{9p}

[2]

\dfrac{11q^2}{2} \times \dfrac{8}{9pq} = \dfrac{88 \times q \times q}{18 \times p \times q}

The HCF of

88

and

18

2

\dfrac{88}{18} = \dfrac{44}{9}

= \dfrac{44q}{9p}

Question 6

(a)Answer:

2 \times 25 - 7 = 43

[2]

\mathstrut (-3) - (-5) =

2

\mathstrut (-5) - 2 =

-7

T_n = 2n - 7

2 \times 25 - 7 = 43

(b)(i)Answer:

x^2-13x+36

[1]

x^2-13x+36

(b)(ii)Answer:

3x^2-12x-36

[1]

3x^2-12x-36

(c)(i)Answer:

= \dfrac{5x^2}{4}

[2]

\dfrac{9xy^2}{4} \times \dfrac{5x}{9y^2} = \dfrac{45 \times x \times x \times y \times y}{36 \times y \times y}

The HCF of

45

and

36

9

\dfrac{45}{36} = \dfrac{5}{4}

= \dfrac{5x^2}{4}

(c)(ii)Answer:

= \dfrac{6}{5p}

[2]

\dfrac{q^2}{10p^2} \div \dfrac{q^2}{12p} = \dfrac{q^2}{10p^2} \times \dfrac{12p}{q^2}

= \dfrac{12 \times p \times q \times q}{10 \times p \times p \times q \times q}

The HCF of

12

and

10

2

\dfrac{12}{10} = \dfrac{6}{5}

= \dfrac{6}{5p}

Question 7

(a)(i)Answer:

x^2+8x+15

[1]

x^2+8x+15

(a)(ii)Answer:

12x^2-45x+42

[1]

12x^2-45x+42

(b)Answer:

8 \times 50 - 1 = 399

[2]

\mathstrut 15 - 7 =

8

\mathstrut 7 - 8 =

-1

T_n = 8n - 1

8 \times 50 - 1 = 399

(c)(i)Answer:

= \dfrac{n}{6}

[2]

\dfrac{mn}{4} \times \dfrac{2}{3m} = \dfrac{2 \times m \times n}{12 \times m}

The HCF of

2

and

12

2

\dfrac{2}{12} = \dfrac{1}{6}

= \dfrac{n}{6}

(c)(ii)Answer:

= 2xy^2

[2]

\dfrac{4y^2}{3x} \div \dfrac{2}{3x^2} = \dfrac{4y^2}{3x} \times \dfrac{3x^2}{2}

= \dfrac{12 \times x \times x \times y \times y}{6 \times x}

The HCF of

12

and

6

6

\dfrac{12}{6} = 2

= 2xy^2