Practice CT3 — Algebraic Fractions, Formulae & Quadratics
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A Graphic Display Calculator and/or calculator can be used for this paper.
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.
You must show all the relevant working to gain full marks and you will be given marks for correct methods, including sketches, even if your answer is incorrect.
The number of marks is given in brackets [ ] at the end of each question or part question.
Solving Algebraic Fraction Equations (No GDC)
[8 marks](a) Without using GDC, solve the equation.
Answer x =
[2]
(b) Without using GDC, solve the equation.
Answer x =
[3]
(c) Without using GDC, solve the equation.
Answer y =
[3]
Simplifying Algebraic Fractions
[7 marks](a) Simplify.
Answer =
[2]
(b) Simplify.
Answer =
[2]
(c) Simplify.
Answer =
[3]
Express as a Single Fraction in Simplest Form
[6 marks](a) Express each of the following as a single fraction in its simplest form.
Answer =
[3]
(b) Express each of the following as a single fraction in its simplest form.
Answer =
[3]
Making the Subject of a Formula
[6 marks](a) Given that y − xx + y = 3, make x the subject of the formula.
[3]
(b) Given that v = u + at, make t the subject of the formula.
[3]
Quadratic Graphs — Use of GDC
[8 marks](a) Sketch the graph of y = x2 − 6x + 5.
[2]
(b) Write down the coordinates of the turning point.
Answer (_______ , _______) [1]
(c) Write down the equation of the line of symmetry.
Answer x = _______ [1]
(d) Solve x2 − 6x + 5 = 0.
Answer x = _______ or _______ [2]
(e) P and Q are points on the graph y = x2 − 6x + 5. Line PQ is parallel to the x-axis. The coordinates of P are (1, k). Find the coordinates of P and Q.
Answer P = (_______ , _______) [1]
Q = (_______ , _______) [1]
(f) Find the number of solutions for the following simultaneous equations.
Answer ______________________ [1]
Pythagoras' Theorem
[4 marks]In the diagram, ∠BAC = 90°, AC = 8 cm, AB = 15 cm, BD = 17 cm and CD = 25 cm.
(a) Find the length of BC.
Answer ______________________ cm [2]
(b) Determine if triangle BCD is a right-angled triangle. Show your working.
[2]
Not to scale
Work Rate Problem — Constructing Equations
[10 marks]Sarah and Tom are helping the school to pack gift boxes for the charity event.
(a) Sarah takes x minutes to pack one gift box. Write down an expression in terms of x, for the number of gift boxes Sarah can pack in 2 hours.
Answer ______________________ [1]
(b) Tom takes 3 minutes longer than Sarah to pack one gift box. Write down an expression in terms of x, for the number of gift boxes Tom can pack in 2 hours.
Answer ______________________ [1]
(c) Sarah and Tom together can pack a total of 30 gift boxes in 2 hours. Form an equation in x and show that it reduces to
[3]
(d) Solve the equation x2 + 15x − 72 = 0.
Answer x = _______ or _______ [2]
(e) How many gift boxes can Sarah pack in 3 hours?
[3]
Practice CT3 — Mark Scheme
Q1(a): x = −47
Q1(b): x = 2
Q1(c): y = 4 or y = −52
Q2(a): 2x23y
Q2(b): m − 2n3
Q2(c): m + 3
Q3(a): x + 56
Q3(b): 4t + 4t − 2
Q4(a): x = −3y + x2
Q4(b): t = v − ua
Q5(a): Parabola opening upwards, vertex at (3, −4), x-intercepts at (1, 0) and (5, 0)
Q5(b): (3, −4)
Q5(c): x = 3
Q5(d): x = 1 or x = 5
Q5(e): P = (1, 0), Q = (5, 0)
Q5(f): 2 solutions
Q6(a): BC = √(8² + 15²) = √289 = 17 cm [2]
Q6(b): No — BC² + BD² = 17² + 17² = 578 ≠ 625 (= CD²), so ∠CBD ≠ 90° (Pythagoras' theorem converse) [2]
Q7(a): 120x
Q7(b): 120x + 3
Q7(c): 120x + 120x+3 = 30 → x2 + 15x − 72 = 0
Q7(d): x = 3.6 or x = −20 (reject negative)
Q7(e): Sarah packs 3603.6 = 100 gift boxes in 3 hours