Practice CT3 — Constructing Equations & Pythagoras Theorem

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Class

Date

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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

Constructing equations from word problems
Making the subject of a formula
Graphing quadratic functions using GDC
Pythagoras' theorem in right-angled triangles

Section A

Making the Subject of a Formula & Constructing Equations

[10 marks]

Question 1

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 2

The sum of three consecutive even numbers is 78. Form an equation in x, where x is the smallest number, and solve it to find the three numbers.

Question 3

A rectangle has length (2x + 3) cm and width (x − 1) cm. The perimeter of the rectangle is 34 cm.

(a) Form an equation in x and solve it.
(b) Find the area of the rectangle.

(a)

(b)

Section B

Solving Quadratic Equations

[10 marks]

Question 4

Solve each of the following equations.

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

(a)

(b)

Question 5

Solve the equation

3x+2x−1

= 4. Give your answer as a fraction in its simplest form.

Section C

Quadratic Graphs

[10 marks]

Question 6

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 D

Pythagoras' Theorem

[10 marks]

Question 7

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 E

Constructing Equations from Word Problems

[5 marks]

Question 8

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)

Answer Key

Q1(a): r =

√(V/πh)

Q1(b): x =

p+33−p

Q2: 24, 26, 28

Q3(a): x = 5; (b) Area = 42 cm²

Q4(a): x = 1, −7

Q4(b): x = 3, −

Q5: x =

Q6(a): y = 3 (at x=2)

Q6(c): x ≈ 0.38, 3.62

Q7(i): (x+4)² + (x+12)² = (x+16)²

Q7(ii): x ≈ 13.79, −3.79

Q7(iii): Area = 240.5 cm²

Q8(a):

45z

Q8(b):

45z

−

45z+1

= 30

Q8(c): z ≈ 6.27, −7.27