Secondary 1 & 2 Mathematics
Practice Paper: Functions, Algebra & Simultaneous Equations
Prepared by Miss Clarissa Ng
www.clartutors.com
www.clartutors.com
Instructions: Answer all questions. Show your working clearly. You may use a GDC where appropriate. Total: 88 marks.
Topics Covered
- Sec 1: Functions and Linear Graphs
- Sec 2: Simultaneous Equations · Algebraic Expansion (Special Identities) · Algebraic Factorisation (Special Identities) · Algebraic Fractions
Section A
Functions & Linear Graphs
Sec 1 Ch 6 · [4 marks each · 8 marks]Question 1
(a) Find the gradient and y-intercept of the line 3y + 6x = 9.
(b) Hence draw the graph of the line.
(c) Find the equation of the line passing through (2, 5) and (4, 11), giving your answer in the form y = mx + c.
(b) Hence draw the graph of the line.
(c) Find the equation of the line passing through (2, 5) and (4, 11), giving your answer in the form y = mx + c.
(a) Gradient = , y-intercept =
(b) [Draw graph]
(c) y =
Question 2
Line L₁ has equation y = 4x + 1.
(a) Find the equation of the line parallel to L₁ that passes through (3, 2).
(b) Find the equation of the line perpendicular to L₁ that passes through (0, 5).
(a) Find the equation of the line parallel to L₁ that passes through (3, 2).
(b) Find the equation of the line perpendicular to L₁ that passes through (0, 5).
(a) y =
(b) y =
Section B
Simultaneous Equations
Sec 2 Ch 1 · [4 marks each · 28 marks]Question 3
Solve the following pairs of simultaneous equations:
(a) 3x + 2y = 12 and 5x − 2y = 4
(b) 4x − 3y = 10 and 2x + 5y = 20
(a) 3x + 2y = 12 and 5x − 2y = 4
(b) 4x − 3y = 10 and 2x + 5y = 20
(a) x = , y =
(b) x = , y =
Question 4
3 notebooks and 2 pens cost $14. 5 notebooks and 4 pens cost $38.
(a) Form two simultaneous equations and find the cost of one notebook and one pen.
(b) Hence find the cost of 7 notebooks and 6 pens.
(a) Form two simultaneous equations and find the cost of one notebook and one pen.
(b) Hence find the cost of 7 notebooks and 6 pens.
(a) Notebook = $ , Pen = $
(b) Total = $
Question 5
Solve the simultaneous equations:
y = 2x and x² + y² = 45
y = 2x and x² + y² = 45
(x, y) =
Question 6
Solve the simultaneous equations:
y = x² − 6 and y = −x
y = x² − 6 and y = −x
(x, y) =
Question 7
A rectangle has perimeter 34 cm and area 60 cm². Find the dimensions of the rectangle.
Length = cm, Width = cm
Question 8
Solve the simultaneous equations:
1x + 1y = 76 and 1x − 1y = 16
1x + 1y = 76 and 1x − 1y = 16
x = , y =
Question 9
Solve the simultaneous equations:
x + y = 5 and 1x + 1y = 56
x + y = 5 and 1x + 1y = 56
(x, y) =
Section C
Algebraic Expansion
Sec 2 Ch 3 · [4 marks each · 16 marks]Question 8
Expand and simplify:
(a) (2x + 3)(x − 4)
(b) (3x − 2)(2x + 5)
(a) (2x + 3)(x − 4)
(b) (3x − 2)(2x + 5)
(a)
(b)
Question 9
Expand:
(a) (x + 7)²
(b) (3x − 4)²
(c) (2x + 5)²
(a) (x + 7)²
(b) (3x − 4)²
(c) (2x + 5)²
(a)
(b)
(c)
Question 10
Expand:
(a) (x + 9)(x − 9)
(b) (2x + 3)(2x − 3)
(c) (5 + 3y)(5 − 3y)
(a) (x + 9)(x − 9)
(b) (2x + 3)(2x − 3)
(c) (5 + 3y)(5 − 3y)
(a)
(b)
(c)
Question 12
Expand and simplify:
(a) (x + 4)² − (x − 4)²
(b) (3x + 2)(3x − 2)(9x² + 4)
(c) (2x + 3)² − (2x − 3)²
(a) (x + 4)² − (x − 4)²
(b) (3x + 2)(3x − 2)(9x² + 4)
(c) (2x + 3)² − (2x − 3)²
(a) Answer:
(b) Answer:
(c) Answer:
Section D
Algebraic Factorisation
Sec 2 Ch 4 · [4 marks each · 16 marks]Question 13
Factorise completely:
(a) 6x² + 9x
(b) x² + 5x + 6
(c) 2x² − 7x − 15
(a) 6x² + 9x
(b) x² + 5x + 6
(c) 2x² − 7x − 15
(a)
(b)
(c)
Question 14
Factorise:
(a) x² − 49
(b) 4x² − 25
(c) 9 − 16y²
(d) x² − 100
(a) x² − 49
(b) 4x² − 25
(c) 9 − 16y²
(d) x² − 100
(a)
(b)
(c)
(d)
Question 15
Factorise:
(a) x² + 10x + 25
(b) 4x² − 12x + 9
(c) 25y² + 30y + 9
(a) x² + 10x + 25
(b) 4x² − 12x + 9
(c) 25y² + 30y + 9
(a)
(b)
(c)
Question 17
Factorise completely:
(a) x⁴ − 16
(b) x² + 6x + 9 − y²
(c) 3x² − 12
(a) x⁴ − 16
(b) x² + 6x + 9 − y²
(c) 3x² − 12
(a) Answer:
(b) Answer:
(c) Answer:
Section E
Algebraic Fractions
Sec 2 Ch 6 · [4 marks each · 20 marks]Question 18
Simplify:
(a) 12x²y8xy²
(b) x² − 9x² + 4x + 3
(c) 2x² − 8x² + x − 6
(a) 12x²y8xy²
(b) x² − 9x² + 4x + 3
(c) 2x² − 8x² + x − 6
(a)
(b)
(c)
Question 19
(a) Simplify x² − 4x + 1 × x + 1x − 2
(b) Simplify x² − 1x² + 3x + 2 ÷ x − 1x + 2
(b) Simplify x² − 1x² + 3x + 2 ÷ x − 1x + 2
(a)
(b)
Question 20
Simplify:
(a) 3x − 2 + 1x + 3
(b) 52x − 23x
(c) xx + 1 − 1x − 1
(a) 3x − 2 + 1x + 3
(b) 52x − 23x
(c) xx + 1 − 1x − 1
(a)
(b)
(c)
Question 21
Simplify:
(a) 1 + 2x − 1
(b) x + 2xx + 4x
(a) 1 + 2x − 1
(b) x + 2xx + 4x
(a)
(b)
Question 22
Solve:
(a) 2x = 1x − 3
(b) x + 1x − 2 = 3
(a) 2x = 1x − 3
(b) x + 1x − 2 = 3
(a) x =
(b) x =
Answer Key
Section A — Functions & Linear Graphs
Q1(a) m = −2, c = 3
Q1(c) y = 3x − 1
Q2(a) y = 4x − 10
Q2(b) y = −¼x + 5
Section B — Simultaneous Equations
Q3(a) x = 2, y = 3
Q3(b) x = 5, y = 2
Q4(a) Notebook = $2, Pen = $4
Q4(b) $38
Q5 (3, 6) and (−3, −6)
Q6 (−3, 3) and (2, −2)
Q7 12 cm × 5 cm
Q8 x = 3/2, y = 2
Q9 (2, 3) and (3, 2)
Section C — Algebraic Expansion
Q8(a) 2x² − 5x − 12
Q8(b) 6x² + 11x − 10
Q9(a) x² + 14x + 49
Q9(b) 9x² − 24x + 16
Q9(c) 4x² + 20x + 25
Q10(a) x² − 81
Q10(b) 4x² − 9
Q10(c) 25 − 9y²
Q12(a) 16x
Q12(b) 81x⁴ − 16
Q12(c) 24x
Section D — Algebraic Factorisation
Q13(a) 3x(2x + 3)
Q13(b) (x + 2)(x + 3)
Q13(c) (2x + 3)(x − 5)
Q14(a) (x + 7)(x − 7)
Q14(b) (2x + 5)(2x − 5)
Q14(c) (3 + 4y)(3 − 4y)
Q14(d) (x + 10)(x − 10)
Q15(a) (x + 5)²
Q15(b) (2x − 3)²
Q15(c) (5y + 3)²
Q17(a) (x + 2)(x − 2)(x² + 4)
Q17(b) (x + 3 + y)(x + 3 − y)
Q17(c) 3(x + 2)(x − 2)
Section E — Algebraic Fractions
Q18(a) 3x/(2y)
Q18(b) (x−3)/(x+1)
Q18(c) 2(x−2)/(x−3)
Q19(a) x + 2
Q19(b) 1
Q20(a) (4x+7)/((x−2)(x+3))
Q20(b) 11/(6x)
Q20(c) (x²−x−1)/(x²−1)
Q21(a) (x+1)/(x−1)
Q21(b) (x²+2)/(x²+4)
Q22(a) x = 6
Q22(b) x = 3.5