IB MYP Grade 10 Extended Mathematics
Algebra — Expansion, Factorisation & Algebraic Fractions
Time
90 minutes
Calculator
Not required
Total
60 marks
Instructions
• Answer all questions.
• Show all necessary working.
• Give exact answers unless otherwise stated.
• Where appropriate, state any restrictions on the values of variables.
• Unsupported answers may not receive full marks.
• The questions are arranged approximately from Level 1 to Level 8 in increasing difficulty.
Question 1 — Level 1
6 marks
Basic algebraic manipulation
a. Expand and simplify: 3(x + 4)
[1]
b. Expand and simplify: (x + 5)(x + 2)
c. Factorise completely: 8x + 20
[1]
d. Factorise completely: x² + 7x
[2]
Question 2 — Level 2
[2]
7 marks
Expansion and factorisation
a. Expand and simplify: (2x − 3)(x + 5)
b. Expand and simplify: (3x + 2)²
c. Factorise completely: x² − 9
[2]
[2]
[1]
d. Factorise completely: x² + 9x + 20
Question 3 — Level 3
[2]
7 marks
Recognising factorisation structures
a. Factorise completely: 6x² − 15x
[2]
b. Factorise completely: 4x² − 25
[2]
c. Factorise completely: x² − 10x + 25
[2]
d. Hence, or otherwise, write 9x² − 30x + 25 in the form (ax + b)².
Question 4 — Level 4
8 marks
Factorisation using different methods
a. Factorise completely: 2x² + 7x + 3
[3]
b. Factorise completely: 3x² − 12x + 12
[2]
[1]
c. Factorise completely: x³ + 3x² + 2x
[2]
d. A student says: “The expression x² − 16 cannot be factorised because there is no common factor.”
Explain why the student is incorrect and give the complete factorisation. [1]
Question 5 — Level 5
8 marks
Algebraic fractions and restrictions
a. Simplify: 6x² / 9x. State the restriction on x.
[2]
b. Simplify completely: (x² − 9)/(x² + 6x + 9). State all restrictions on x.
c. Simplify completely: (x² − 4x)/(x² − 16). State all restrictions on x.
Question 6 — Level 6
[3]
[3]
8 marks
Operations with algebraic fractions
a. Simplify: 2/x + 3/(x + 1). Give your answer as a single fraction.
[3]
b. Simplify completely: x/(x − 2) − 2/(x − 2). State the restriction on x.
[2]
c. Simplify completely: 2/(x + 1) × (x² − 1)/4. State all restrictions on x.
[3]
Question 7 — Level 7
8 marks
Equations involving algebraic fractions
Solve the equation: 3/(x − 2) = 2/(x + 1)
a. State the values of x that are not permitted.
b. Solve the equation.
[2]
[4]
c. Check that your solution satisfies the original equation.
Question 8 — Level 8
[2]
8 marks
Problem solving and algebraic reasoning
The area of a rectangular garden is x² − 9 square metres. The length of the garden is x + 3 metres.
a. Factorise the expression for the area.
[2]
b. Hence determine an expression for the width of the garden.
[2]
A path of constant width is added along one side of the garden. The new width becomes x − 1 metres.
The new area is therefore (x + 3)(x − 1).
c. Find the difference between the original area and the new area, and simplify your answer.
[2]
d. The ratio of the original area to the new area is (x² − 9)/((x + 3)(x − 1)). Simplify this ratio completely
and state the values of x for which the expression is defined. [2]
End of Test
Total: 60 marks