A2 MATHEMATICS HOMEWORK C3...HW10 Trigonometry 3 HWX C3 June 2010 . 4 |Page HW1 Algebraic Fraction...

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A2 MATHEMATICS HOMEWORK C3

Mathematics Department September 2014 Version 1.1

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Contents

Contents ........................................................................................................... 2

Introduction ...................................................................................................... 3

HW1 Algebraic Fractions ..................................................................................... 4

HW2 Mappings and Functions .............................................................................. 6

HW3 The Modulus Function ................................................................................. 8

HW5 The Exponential Functions ......................................................................... 10

HW6 Differentiation: Product and Quotient Rules ................................................. 12

HW7 Numerical Methods ................................................................................... 13

HW8 Trigonometric Equations 1 ......................................................................... 15

HW9 Trigonometric Equations 2 ......................................................................... 17

HW10 Trigonometric Equations 3 ....................................................................... 19

HWX C3 June 2010 .......................................................................................... 21

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Introduction

Aim to complete all the questions. If you find the work difficult then get help [lunchtime

workshops in room 216, online, friends, teacher etc].

To learn effectively you should check your work carefully and mark answers � � ? If you have questions or comments, please write these on your homework. Your teacher will then

review and mark your mathematics.

If you spot an error in this pack please let your teacher know so we can make changes for

the next edition!

Homework Tasks – These cover the main topics in C3. Your teacher may set homework

from this or other tasks.

Week Topic Date

completed

Mark

HW0 Review summer work and revise for test

HW1 Algebraic fractions and long division

HW2 Mappings and functions

HW3 The modulus function and transformations

HW4 Exponential functions

HW5 Differentiation 1

HW6 Differentiation 2

HW7 Numerical methods

HW8 Trigonometry 1

HW9 Trigonometry 2

HW10 Trigonometry 3

HWX C3 June 2010

4 |Page

HW1 Algebraic Fraction

Complete on a separate sheet of paper. Show clear working. Mark your answers.

Key words Numerator, Denominator, Factorising, Cancelling like terms, improper

fractions, polynomial division

Exercise A

1. Simplify the following expressions as much as possible:

a) �� b)

�� c) ��

d) � � �� e)

�� f) ��

g) �2+4�−21�2−8�+15 h) �� i)

� �� 2. Combine and simplify the following algebraic fractions

a) �� + �� b) �� + � c)

�� − ��

d) ��+ �� e)

�� − �� f) ��+ ��

g) �� − �� h)

�� + �� i) ��+ ��

3. Combine and simplify the following algebraic fractions

a) ��− �� b) ��+ �� c)

�� − ��

d) �� − �� e) 2��+2 + �

�+3 + 4�2+5�+6 f)

� ��− ��

4. Express the following as impartial fractions (hint: polynomial division)

a) �!�� b)

�!�� c) ��

d) ��!�� e)

��"��!��

5 |Page

Exercise B - Exam Questions

1. [C3 June 2007]

Given that #�� = �� − �� ,� > ��

Show that #�� = �� (7 marks)

2. [C3 June 2012]

Express �� − �� as a single fraction in its simplest form.

(4 marks)

Exercise C – Extension tasks

Queen Mary University Essential Maths Questions

1. a) Add and simplify � �� + � ��

b) Solve 4� − �� = ��

c) Simplify ��+ ��− ��

2. Go to www.mathsnetalevel.com (username: cityisli, password: ask teacher).

www.mathsnetalevel.com/7244 or 5651

Answers

Exercise A

1a) �� b) �� c) �� d) � e) �� f) �� g) �� h) �� i) � � �

2a) �� b) � � c) − �� d) �� e) �� f)

�� g) �� h) ��i) ��

3a)� − 5 b) �� c) � − 1 d) − �� e) �� f) ��

4a) �� + 3� + 6 + �� b) �� + 4� + 12 + �� c) 2 + �� d) 4� − 13 + �� e) 2�� + � + 1 + ��

Exercise B – Exam questions – check using www.examsolutions.net

2. ��

Exercise C – Extension tasks

1a) � �� b) � = ��±√�� c)

��

6 |Page

HW2 Mappings and Functions


Key words Function, Mapping, Domain, Range, Object, Input, Image, Output, One-

One, Many-One, One-Many, Composite Function, Inverse Function

Read the chapter on algebra and functions (p14- 18, 20-22, 25-29)

Exercise A

1. Evaluate the following mappings

a) #�� = �� − 4, � ∈ ℝ i) #�4� ii) #�−3� iii) #�0� b) ,: ./01., ⟼ /314567#,589 i) ,�./�� ii) ,��0856� iii) ,�:ℎ.06� c) For a quadratic equation the real solutions are given by the following function:

8�.�� + 4� + :� = �<±√<��=�� , 4� ≥ 4.:. Find: i) 8�� + 5� + 6� ii) 8�2�� + 7� + 6� iii) 8�� − 3� + 12�

2. For each mapping

i) Sketch the graph

ii) State the range of the mapping

iii) Say whether the mapping is one-one, many-one, one-many, or many-many

iv) State whether the mapping is a function

a) #�� = �� + 3 , � ∈ ℝ b) 8�� = �� , � ≠ 2

c) ℎ: � ⟼ 3� − 2, � > −1 d) B = ±√36 − �� , −6 ≤ � ≤ 6

3. Given #�� = �� + 1, 8�� = 1 − 3�, evaluate: a) #8�� b) 8#�� c) #�� d) 8��3�

7 |Page

4. a) What has to be true for a function to have an inverse?

b) Find the inverse of the following functions:

i) B = 2� + 7 , � ∈ ℝ ii) #�� = �� − 1 , � ≥ 0

iii) ℎ: � ⟼ ��, � ∈ ℝ iv) 8: � ⟼ �� , � ≠ 3 Exercise B - Exam Questions

1. [C3 Jun 08 Q3]

The function f is defined by #: � ⟼ ��− �� , � > 1. a) Show that#�� = �� , � > 1 (4)

b) Find #��. (3)

The function g is defined by 8: � ⟼�� + 5, � ∈ ℝ c) Solve #8�� = ��. (3)

2. [C3 Jan 06 Q8]

The functions f and g are defined by

#:�a 2� + ln2, �∈ℝ,

8:�a 52x, �∈ℝ.

a) Prove that the composite function 8# is 8#:�a 454x, x ∈ �. (4)

b) Sketch the curve with equation B = 8#��, and show the coordinates of the point where the curve cuts the B-axis. (1)

c) Write down the range of 8#. (1)

Extension

1. Go to www.mathsnetalevel.com (username: cityisli, password: ask teacher).

C3 Sequences. You can try some of the tests.

Answers Exercise A 1ai) 12, ii) 5, iii) -4 bi) 6, ii) 4, iii) undefined, ci) -2,-3, ii) −2,−3/2, ii) undefined. 2aii) #�� ≥ 3, iii) many – one, (iv) yes, bii) 8�� ∈ ℝ8�� ≠ 0, ii) one-one, iii) function, ci)ℎ�� ∈ ℝ, ii) one-one, iii) function d) −6 ≤ B ≤ 6, many-many, no.

3a) �1 − 3x�� + 1, b) 1 − 3�x� + 1�, c) �x� + 1�� + 1, d) 25.

4a) function must be one-one mapping b) i) B = �� ii) ℎ�� = �H! iii) #�� = √� + 1 iv) 8�� Exercise B

1b) #�� = �� ,c) � = ± 2 2b) 8#�� > 0, B > 0, c)

8 |Page

HW3 The Modulus Function


Key words Modulus Function, Absolute value Transformations

Exercise A

1. Calculate the following:

a) | − 3| b) |5| c) |4 − 7| d) |−3 + 4| e) |−4|– 7 f) K|−4|– 7K

2. True or false; which of the following statements are correct.

a) L|−2�|L = L−|2�|L b) �−|2|�� = |−2|� c) |2 + 3| = |2| + |3| d) |−2�| = −|2�| e) |3 − 2| = |3| − |2|

3. Sketch the following graphs:

a) B = |� − 5| b) B = |�| − 5 c) B = | − ��| d) B = K��K e) B = |cos �| f) B = |5�| + 3 g) B = |�� − 1��|

4. Solving the following equations (sketching the graphs is very important)

a) |� + 5| = 3 b) |2� − 3| = 7 c) |� + 1| = −2� − 5

d) |� + 3| = |� − 6| e) |3� + 9| = |2� + 1|

Exercise B – Exam Questions

1. [C3 Jun 05] Figure 1

Figure 1 shows part of the graph of y = f(x), x ∈ �. The graph consists of two line

segments that meet at the point (1, a), a < 0. One line meets the x-axis at (3, 0). The

other line meets the x-axis at (–1, 0) and the y-axis at (0, b), b < 0.

In separate diagrams, sketch the graph with equation

a) y = f(x + 1), (2)

b) y = f(x). (3)

Indicate clearly on each sketch the coordinates of any points of intersection with the

axes. Given that f(x) = x – 1 – 2, find

c) the value of a and the value of b, (2)

b

O

y

x –1

(1, a)

3

9 |Page

d) the value of �for which #�� = 5�. (4)

2. [C3 Jan 06] Figure 1

Figure 1 shows the graph of B = #�x�, – 5≤�≤5.

The point M (2, 4) is the maximum turning point of the graph.

Sketch, on separate diagrams, the graphs of

a) B = #�� + 3, (2)

b) y = f(x), (2)

c) y = f(x). (3)

3. [C3 Jun 07]

Find the exact values of � for which K ��K = 3. (3)

Extension

Go to www.mathsnetalevel.com (username: cityisli, password: ask teacher) for further

questions.

Answers

Exercise A

1a) 3 b) 5 c) 3 d) 1 e) -3 f) 3

2a) T b) T c) T d) F e) T

3 Check using graph plotter or teacher will check.

4a) � = −2,−8 b) � = −2, 5 c) � = −4 only sketch the graph d) � = �� e) � = −8,−2 For d) & e) you can use the squaring method. See Modulus equations on examsolutions

Exercise B – Exam questions - see www.examsolutions.net

2. c) . = −2, 4 = −1 d) � = − ��

3. � = 3 �� , 2 ��

•

y

x 5 O

M (2, 4)

–5

10 |Page

HW5 The Exponential Functions


Key words: Exponentials, Natural logarithms, exponential growth/decay, rates,

modelling

Exercise A

1. Sketch the following graphs on different axis

a) B = 5� and B = ln � (on the same axis)

b) B = 5�� c) B = 55�� d) B = 12 + 75��

2. Find the exact solutions of

a) ln � = 3 b) ln � + ln 2 = 8 c) ln � − ln 3 = 5 d) ln � + 4 = ln�� + 2� e) ln�3� + 4�=2 3. Find the exact solutions for � for the following equations a) 45� = 12 b) 5�� = 5 c) 55�� + 2 = 17 d) 0.5 = 5�� − 1 e) 5�� − 95� + 14 = 0 f) 25� − 7 + 65�� = 0

4. [Created by an AS student] A maths class is learning Calculus for the first time. After Q minutes, their stress levels, R micrograms of cortisol per deciliter, is given by:

R = 65ST

a) What are their stress levels when they enter the class?

b) After 10 minutes their stress level is 12 micrograms per deciliter. Show that U = 0.0693 to three significant figures.

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Exercise B – Exam question

1. [C3 June 2009 Q3] Rabbits were introduced onto an island. The number of

rabbits, V, � years after they were introduced is modelled by the equation

V = 805HW�� ∈ ℝ, � ≥ 0 a) Write down the number of rabbits that were introduced to the island.

(1)

b) Find the number of years it would take for the number of rabbits to first

exceed 1000.

(2)

c) Find XYX�

(2)

d) Find V when XYX� = 50 (3)

(Total 8 marks)

Extension

1. Create a model for exponential growth/decay for the following examples:

(a) A cup of tea cooling down from 80C to room temperature.

(b) Population of lizards on an island growing from 10 to 500 over 3 years.

Do your models seem accurate for all situations? If not how could you alter it?

2. Try C3 Jan 2010 Q9

3. What is the maximum value of #�� = ��Z[\ ? try to explain why.

Answers

2.a) � = 5� b) � = Z]� c) � = 35� d) � = �Z"�� e) � = Z��

3.a) � = ln3 b) � = ^_�� c) � = ^_�� d) � = ^_`�!a� e) � = ln 7 or ln 2 f) � = ln `��a or ln 2 4.a) Q = 6

Exam

1.a) 80 b) � = 12.6 c) XbX� = 165 cW d) 250

12 |Page

HW6 Differentiation: Product and Quotient Rules


Key words: derivative, function, gradient function, natural log, exponential, chain

rule, product rule, quotient, substitution

Read pages 113-117; 129-134 and make sure you understand. Try to help this consolidate

what you’ve already learnt about differentiation

Exercise A

1. Differentiate the following using the product rule:

a) B = ��5� b) B = cos � sin �

c) B = ln � 5� d) #�� = ��1 + 3�� e) B = 3��3� − 1�� f) #�� = �� − 2��2� + 3� 2. Differentiate the following using the quotient rule:

a) B = tan � b) #�� = sec �

c) B = hij��Z\ d) #�� = ��

e) #�� = �� f) B = ��H�

Exercise B – Exam questions

1. [C3 Jun 05]

a) Differentiate with respect to � i) 3 sin� � + sec 2� (3)

ii) k� + ,/�2��l� (3)

b) Given that B = �� , � ≠ 1, show that X X� = − ��! (6)

2. [C3 Jan 06]

a) Differentiate with respect to �

i) ��5��, (4)

ii) hij��!�� (4)

b) Given that � = 4 sin�2B + 6� find X X� in terms of �. (5)

Answers

1a) 3��5� + ��5�, b) cos� � − sin� � = 1 − 2 cos� � c) Z\� + ln��5� d) �1 + 3��1 + 18�� e) 3��5� − 2��5� − 1�� 2a)

=mn��nop!�=mn�� =95:�� b) jq_��=mn�� = sin��:795:�� c) Z\ jq_��Z\�hij��Z�\ d) �� e) ��! f) ��!�

Exam Questions

1a) #r�� = 35� − �� 2a) 6 sin � cos � + 2 sec 2� tan 2� b) 3�� + ln 2��^2�1 + �� 2a)

X X� = �� leading to B = −9� + 27 4ai) X X� = 3��5�� + 2�5�� aii) X X� = ��! jq_��!�� hij��!�� b) X X� = �� hij�tuhjq_�\"�� = �±� ��√��

13 |Page

HW7 Numerical Methods


Key words: roots, interval, algebraic, accuracy, continuous function, domain,

iteration, converge, diverge

Exercise A

1. Sketch the following curves and say for which ones the change on sign method will

work. If it doesn’t work can you give a reason?

a) B = k�– 2l�, � ∈ v−1, 3w b) B = ��, � ∈ v−2, 3w c) B = sin � , � ∈ v0, 2xw 2. Figure 1 shows a sketch of #�� = 5� − 5�

a) How many roots does the equation 5� − 5� = 0 have?

b) Find an interval of unit length (e.g v−8,−7w) containing each root. c) Show that one root is 2.54 to 2 d.p. and find the other root.

3. What do you need to be true for the iteration method to work?

4. Find all the roots of �� − 5� + 2 = 0 to 1 d.p. [hint; it might be useful to make a

sketch using http://www.wolframalpha.com/examples/PlottingAndGraphics.html and

note all the roots are in the range [-3,3]]


1. [C3 Jan 07] #�� = �� − 4� − 8

a) Show that there is a root of #�� = 0 in the interval v−2,−1w (3)

b) Sketch the graph of B = #�� (3)

2. [C3 Jun 08 Q7]

#�� = 3�� − 2� − 6. a) Show that #�� = 0 has a root, y, between � = 1.4 and � = 1.45. (2)

b) Show that the equation #�� =0 can be written as � = z�� + ��, � ≠ 0. (3)

c) Starting with � = 1.43, use the iteration �p� = {| 2�p + 23}

to calculate the values of ��, �� and ��, giving your answers to 4 decimal places. (3)

d) By choosing a suitable interval, show that y = 1.435 is correct to 3 d.p. (3)

Figure 1

~�

14 |Page

3. [C3Jan 06 Q5]

#�� = 2�� − � − 4 a) Show that the equation #�� = 0 can be written as � = z�� + �� (3)

The equation 2�� − � − 4 = 0 has a root between 1.35 and 1.4. b) Use the iteration formula

�p� = { 2�p + 12 with � = 1.35, to find, to 2 decimal places, the value of ��, �� and ��. (3)

The only real root of #�� = 0is α. c) By choosing a suitable interval, prove that α = 1.392, to 3 decimal places. (3)

4. [C3 Jun 05 Q4]

Consider #�� = 35� − �� ,/� − 2, � > 0. The iterative formula �p� = �� 5�� , � = 1, is used to find an approximate value for α.

a) Calculate the values of ��, ��, �� and ��, giving your answers to 4 decimal

places. (2)

b) By considering the change of sign of #′�� in a suitable interval, prove that α = 0.1443correct to 4 decimal places. (2)

Extension

Find the solutions to the simultaneous equations B = 2� and B = ��.

ANSWERS

Exercise A

2.b) 2, c) v0,1w, v2,3w, d) #�2.535� < 0, #�2.545� > 0 and #�� continuous 3. successive iteration must converge, and must converge to the root you are looking for.

4. −2.4, 0.4, 2. Exercise B

1. a) #�−2� = 16 + 8 − 8�= 16� > 0 #�−1� = 1 + 4 − 8�= −3� < 0. Change of sign (and continuity) ⇒root in interval �−2,−1�

2. a) #�1.4� = −0.568 < 0, #�1.45� = 0.245 > 0 ⇒ y ∈ �1.4,1.45� … < 0 … > 0 Change of sign (and continuity)

b) �� = 1.4371, �� = 1.4347, �� = 1.4347 , c) Choosing the interval �1.4345, 1.4355�. d) #�1.4345� = −0.01, #�1.4355� = 0.003 ⇒ y = 1.435 Due to change of sign (and continuity) 3. b) �� = 1.41, �� = 1.39, �� = 1.39. c) Choosing (1.3915, 1.3925), #�1.3915� ≈ −3 × 10��, #�1.3925� ≈ 7 × 10�� , Change of sign (and

continuity) ⇒ y ∈ �1.3915, 1.3925�

4. c) �� = 0.0613, �� = 0.1568, �� = 0.1425, �� = 0.1445 d) Using �� = �5� − �� #r�0.14425� = −0.0007, #r�0.14435� = +0.002�1�

15 |Page

HW8 Trigonometric Equations 1


Key words principal value, secondary value, repeat period, radians, interval

Exercise A

1. Read pages 51 -61 and make sure you understand. Try to link this with what you

are learning about mappings and functions. Learn the following

:795:� = 190/� sin� � +cos�� ≡ 1 95:� = 1:79� tan� � + 1 ≡ sec� � :7�� = 1�./� 1 + cot� � ≡ :795:��

2. Starting with sin�� +cos�� ≡ 1 , prove the following identities

a) 1 + cot� � ≡ cosec2� b) tan� � + 1 ≡ sec� �

3. Sketch the graphs of

a) B = :795:� b) B = 95:� c) B = :7��

4. Solve each of the following equations giving your answers in the specified interval.

a) sec� = 2 0 ≤ � ≤ 3x

b) cosec� = 5 − x ≤ � < x

c) cot � = 0.50 ≤ � < 2x

5. Solve each of the following equations giving your answers in the range specified.

a) sec� � = 3 + tan �0 ≤ � < 360

b) cot `2� + ��a = √3− x ≤ � < 2x

Exercise B - Exam questions

1. [C3 Jun 06 Q6]

a) Using sin� � + cos� � ≡ 1 show that cosec2� − cot� � ≡ 1. (2)

b) Hence, or otherwise, prove that cosec4� − cot� � ≡ cosec2� + cot� � (2)

c) Solve, for 90° < � < 180°, cosec� � − cot� � = 2 − cot � (6)

2. [C3 Jan 07 Q8]

Prove that sec�� − cosec2� ≡ tan�� −cot�� Exercise C - Extension

1. Solve cot � = 0 � ∈ v0, 2xw

16 |Page

2. [C3 Jan 08 Q7]

Given that B = arccos�, −1 ≤ � ≤ 1 and 0 ≤ B ≤ x a) Express arcsin�� in terms of B

b) Hence evaluate arccos�� + arcsin��. give your answer in terms of x

Answers Exercise A

3) Sec x cot x

4. a) (-��), �� , �� , �� , ��

b) 0.201, 2.94 c) 1.11, 4.25, 7.39, 10.53

5. a) 63.4, 135, 243.4, 315

b) �� , �� , �� , �� , �� , ��

Exercise B 1 � = 135° Extension

1 � = �� , ��

2 a) B = .6:79� ⇒ � = cosB, � = sin `�� − Ba ⇒ arcsin � = �� − B,

b) arccos � + arcsin � = B + �� − B = ��

17 |Page



Key words compound angle formula, double angle formula

sin�� + �� = 90/�:79� + :79�90/�sin�� − �� = 90/�:79� − :79�90/� cos�� + �� = :79�:79� − 90/�90/�cos�� − �� = :79�:79� + 90/�90/� tan�� + �� = �./� + �./�1 − �./��./� tan�� − �� = �./� − �./�1 + �./��./�

90/2� = 290/�:79� :792� = cos� � − sin� �= 2 cos� � − 1= 1 − 2 sin� ��./2� = 2�./�1 − tan� �

Exercise A

1. Read pages 63- 70 and make sure you understand [key points on p79,80] Learn the

identities.

Sketch a graph of cos�� and show that cos�−�� = cos��

Sketch a graph of sin�� and show that sin�−�� = −sin��

2. Use the compound angle formula to expand simplifying your answers without a

calculator:

a) cos�60 − ��

b) sin�2� + 30�

c) tan�45 − ��

3. Solve the following;

a) sin� = cos�� + 120� in the range 0 ≤ � ≤ 180

b) 2 cos `� − ��a = cos�� + ��

4. Start with the compound angle formulae and prove the following:

a) cos2� ≡ 1 − 2 sin� � [eg start with cos�� + �� = :79�:79� − 90/�90/�]

b) cos2� ≡ 2 cos� � − 1

c) tan 2� ≡ ��p��t_� �

d) sin2� cos� − cos 2� sin � ≡ sin�

18 |Page

Exercise B - Exam questions

1. [C3 Jun 10 Q1].

a) Show that jq_ ��hij �� = tan�

(2)

b) Hence find, for – 180° ≤ � < 180°,all the solutions of � jq_ ��hij �� = 1 Give your answers to 1 decimal place.

(3)

2. [C3 Jan 2011 Q3].

Find all the solutions of 2 cos 2� = 1 − 2 sin �in the interval 0 ≤ � < 360° (6)

3. [C3 Jan 2012 Q8]

a) Starting from the formulae for sin�� + ��and cos�� + ��, prove that tan�� + �� = ��p��p��p��p� (4)

b) Deduce that tan `� + ��a = �√�t_� √��t_� (3)

c) Hence, or otherwise, solve, for 0 ≤ � ≤ x, 1 + √3 tan � = �√3 − tan�� tan�x − �� Give your answers as multiples of x. (6)

Exercise C - Extension

1. Prove sin 3� ≡ 3 sin � −490/��

Answers: Exercise A

2. a) �� cos�� + √3 sin��

b) �� cos�2�� + √3 sin�2��

c) ��t_��t_��

3. a) 165° b) 2.79 radians (This is a hard question – expand both sides then use �./� = 90/�/:79� Exercise B – Exam questions

1. � = 26.6°,−154.4°, 2. � = 54°, 126°, 198°, 342° 3. � = ��x � = �� x

19 |Page



Key words compound angle formula, double angle formula

Read pages 71-74 and have a look at the examples.

�90/�� + y� = �90/�:79y + �:79�90/y �:79�� + y� = �:79�:79y − �90/�90/y �90/�� − y� = �90/�:79y − �:79�90/y �:79�� − y� = �:79�:79y + �90/�90/y

Exercise A

1. Express each of the following in the form shown, where� > 0 and 0 < y < ��

a) sin� + 3 cos� = � sin�� + y� b) 3 sin� − 4 cos � = � sin�� − y�

c) 2 cos � + 7 sin� = � cos�� − y� d) cos 2� − 2 sin2� = � cos�2� + y� 2. Sketch the graph of B = 5 sin� � + 30� and mark on this the maximum and minimum

points on the graph.

Use this graph to help solve the equation 5 sin� � + �� = 4, � ∈ v−x, xw

3. For the following functions find the max value of the function and the value of � ‘theta’ for which it occurs

a) #�� = 13 sin� � + 67.4°� b) #�� = 3cos� � − 35.5°�

c) B = −3 cos� � + 41.8°�


1. [C3 Jun 07 Q6]

a) Express 3 sin � + 2 cos� in the form �sin�� + α� where � > 0 and 0 < α <��. (4)

b) Hence find the greatest value �3sinx + 2cosx��. (2)

c) Solve, for 0 < � < 2x, the equation 3 sin � + 2 cos � = 1, giving your answers to 3 decimal places. (5)

20 |Page

2. [C3 Jun 2008 Q2]

#�� = 5 cos � + 12 sin � Given that #�� = � cos��– y� where � > 0 and 0 < y < ��

a) find the value of R and the value of α to 3 decimal places. (4)

b) Hence solve the equation

5 cos � + 12 sin� = 6 for 0 ≤ � ≤ 2x (5)

c) i) Write down the maximum value of 5 cos � + 12 sin�. (1)

ii) Find the smallest positive value of � for which this maximum value

occurs. (2)

3. [C3 Jan 09 Q8]

a) Express 3:79� + 490/� in the form �:79�� − y�, where � and y are constants, � > 0 and 0 < y < 90°. (4)

b) Hence find the maximum value of 3:79� + 490/� and the smallest positive

value of θ for which this maximum occurs. (3)

The temperature, #��, of a warehouse is modelled using the equation

#�� = 10 + 3:79�15�� + 490/�15��where t is the time in hours from midday and 0≤� < 24.

c) Calculate the minimum temperature of the warehouse as given by this model.

(2)

d) Find the value of t when this minimum temperature occurs. (3)

Extension

For #�� = ��hij��jq_� find the max and minimum value of this function

Also see page 77 in text book for more questions.

Answers: 1a √10sin�� + 1.25�. b 5sin�� −53.1�. c √53cos�� − 74.1�d √5cos( 2� + 63.4� 2 � = 0.404,−1.45 3a f(x)max = 13, � = 22.6 b f(x)max = 3, � = 35.5 c ymax = 3, � = 228.2

Jun 07, 6a � = √��, α = 0.588 (Allow 33.7°), b 169, c x = 2.273 or x = 5.976 (awrt)Both (radians only) Jun 08, 2a R =13, y = 1.176.b 2.3, awrt 0.084 or 0.085. c Rmax=13 at max y = 1.176 Jun 09, 8a R=5, y = 53°. b max value = 5 and this occurs at y = 53°. c min temp is 5°. D t = 15.5 Extension 5 max = 2, min = 1/3

21 |Page

HWX C3 June 2010

1. a) Show that

θ

θ

2cos1

2sin

+ = tan θ. (2)

b) Hence find, for –180° ≤ θ < 180°, all the solutions of

θ

θ

2cos1

2sin2

+ = 1.

Give your answers to 1 decimal place. (3)

2. A curve C has equation

y = 235

3

)( x−, x ≠

3

5.

The point P on C has x-coordinate 2. Find an equation of the normal to C at P in the form

ax + by + c = 0, where a, b and c are integers. (7)

3. f(x) = 4 cosec x − 4x +1, where x is in radians.

a) Show that there is a root α of f(x) = 0 in the interval [1.2, 1.3].

(2) b) Show that the equation f(x) = 0 can be written in the form

x = xsin

1 +

4

1 (2)

c) Use the iterative formula

xn+ 1 = nxsin

1 +

4

1, x0 = 1.25,

to calculate the values of x1, x2 and x3, giving your answers to 4 decimal places. (3)

d) By considering the change of sign of f(x) in a suitable interval, verify that α = 1.291

correct to 3 decimal places.

(2)

4. The function f is defined by

f : x |→ |2x − 5|, x ∈ �.

a) Sketch the graph with equation y = f(x), showing the coordinates of the points where the

graph cuts or meets the axes.

(2)

b) Solve f(x) =15 + x.

(3)

The function g is defined by

g : x |→ x2 – 4x + 1, x ∈ �, 0 ≤ x ≤ 5.

c) Find fg(2).

(2)

d) Find the range of g.

(3)

22 |Page

5. Figure 1

Figure 1 shows a sketch of the curve C with the equation y = (2x

2 − 5x + 2)e

−x.

a) Find the coordinates of the point where C crosses the y-axis.

(1)

b) Show that C crosses the x-axis at x = 2 and find the x-coordinate of the other point where

C crosses the x-axis.

(3)

c) Find x

y

d

d.

(3) d) Hence find the exact coordinates of the turning points of C.

(5)

7. a) Express 2 sin θ – 1.5 cos θ in the form R sin (θ – α), where R > 0 and 0 < α < 2

π.

Give the value of α to 4 decimal places. (3)

b) (i) Find the maximum value of 2 sin θ – 1.5 cos θ.

(ii) Find the value of θ, for 0 ≤ θ < π, at which this maximum occurs.

(3) Tom models the height of sea water, H metres, on a particular day by the equation

H = 6 + 2 sin

25

4 tπ – 1.5 cos

25

4 tπ, 0 ≤ t <12,

where t hours is the number of hours after midday.

c) Calculate the maximum value of H predicted by this model and the value of t, to 2

decimal places, when this maximum occurs.

(3)

d) Calculate, to the nearest minute, the times when the height of sea water is predicted, by

this model, to be 7 metres. (6)

23 |Page

6. Figure 2

Figure 2 shows a sketch of the curve with the equation y = f(x), x ∈ �.

The curve has a turning point at A(3, − 4) and also passes through the point (0, 5).

a) Write down the coordinates of the point to which A is transformed on the curve with equation

i) y = |f(x)|,

ii) y = 2f(21 x).

(4) b) Sketch the curve with equation y = f(|x|).

On your sketch show the coordinates of all turning points and the coordinates of the

point at which the curve cuts the y-axis.

(3) The curve with equation y = f(x) is a translation of the curve with equation y = x

2.

c) Find f(x).

(2)

d) Explain why the function f does not have an inverse.

(1)

8. a) Simplify fully

152

5922

2

−+

−+

xx

xx.

(3)

Given that ln (2x

2 + 9x− 5) = 1 + ln (x

2 + 2x −15), x ≠ −5,

b) find x in terms of e.

(4)

TOTAL FOR PAPER: 75 MARKS

END

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