Practice CT3 — Algebraic Fractions & Quadratic Equations

Instructions: Answer all the questions. All answers should be given in their simplest form. Unless instructed otherwise, give your answers exactly or correct to three significant figures as appropriate. Answers in degrees should be given to one decimal place. For π, use your calculator value. A Graphic Display Calculator and/or calculator can be used for this paper. Total: 45 marks.

Syllabus Topics Covered

Algebraic fractions and simplification
Making the subject of a formula
Constructing equations from word problems
Solving quadratic equations (completing the square)
Graphing quadratic functions using GDC
Pythagoras' theorem in right-angled triangles

Section A

Algebraic Fractions & Formulae

[12 marks]

Question 1

Simplify each of the following. Give your answers in their simplest form.

(a)
6x²y9xy²
(b)
3b+1
+
2b−1
(c)
(x² − 4)(x² + 4x + 4)
÷
1(x+2)

(a)

(b)

(c)

Question 2

Make the stated letter the subject of the formula.

(a) Given that V = πr²h, make r the subject.
(b) Given that p = 3
(x−2)(x+1)
, make x the subject.

(a)

(b)

Question 3

Solve the equation

2x+1x−3

= 5. Give your answer as a fraction in its simplest form.

Question 4

Simplify each of the following. Give your answers in their simplest form.

(a) (3x + 2)(x − 4)
(b)
5x−1
−
3x+2

(a)

(b)

Section B

Quadratic Equations & Graphs

[13 marks]

Question 5

Solve each of the following equations.

(a) x² + 6x − 7 = 0
(b) 2x² − 5x − 3 = 0

(a)

(b)

Question 6

Solve x² + 8x − 3 = 0 by completing the square. Give your answers correct to two decimal places.

Question 7

Complete the table of values for y = x² − 4x + 3 for x from −1 to 5.

x	−1	0	1	2	3	4	5
y	6	3	2		0	3	8

(a) Write down the missing value in the table.
(b) On the grid below, sketch the graph of y = x² − 4x + 3 for −1 ≤ x ≤ 5. Indicate the axes intercepts clearly.

(a)

(c)

Section C

Pythagoras' Theorem

[8 marks]

Question 8

The diagram shows a right-angled triangle with sides (x + 4) cm, (x + 12) cm and (x + 16) cm. The hypotenuse is the longest side.

(i) By forming an equation in x, show that it reduces to x² − 8x − 48 = 0.
(ii) Solve the equation x² − 8x − 48 = 0. Give your answers correct to 2 decimal places.
(iii) Calculate the area of the triangle.

(i)

(ii)

(iii)

Section D

Constructing Equations from Word Problems

[12 marks]

Question 9

Some students are planning to buy some pastries for a party. The budget for pastries is $45.

(a) The price of a muffin is $z. Write down, in terms of z, an expression for the number of muffins the students can buy if they decide to buy muffins only.
(b) The price of a brownie is $(z + 1). If they spend $45 on muffins only, they can buy 30 more muffins than when they spend $45 on brownies only. Write down an equation in z and show that it simplifies to z² + z − 45 = 0.
(c) Solve z² + z − 45 = 0. Give your answers correct to 2 decimal places.

(a)

(b)

(c)

Question 10

A rectangular garden has a length that is 3 m longer than its width. The area of the garden is 28 m².

(a) Let w be the width of the garden in metres. Write down, in terms of w, an expression for the length of the garden.
(b) Form an equation in w and show that it simplifies to w² + 3w − 28 = 0.
(c) Solve w² + 3w − 28 = 0. Hence find the dimensions of the garden.

(a)

(b)

(c)

Answer Key

Q1(a):

2x3y

Q1(b):

5b+1(b+1)(b−1)

Q1(c):

x−2x+2

Q2(a): r =

√(V/πh)

Q2(b): x =

p+33−p

Q3: x =

167

Q4(a): 3x² − 10x − 8

Q4(b):

2x+7(x−1)(x+2)

Q5(a): x = 1, −7

Q5(b): x = 3, −

Q6: x = −4.57, 0.57

Q7(a): y = 3 (at x=2)

Q7(c): x ≈ 0.38, 3.62

Q8(i): (x+4)² + (x+12)² = (x+16)²

Q8(ii): x ≈ 13.79, −3.79

Q8(iii): Area = 240.5 cm²

Q9(a):

45z

Q9(b):

45z

−

45z+1

= 30

Q9(c): z ≈ 6.27, −7.27

Q10(a): Length = w + 3

Q10(c): Width = 4 m, Length = 7 m