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Units 3 and 4 Mathematical

Methods Notes Solutions

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Units 3 and 4 Mathematical Methods The Engage Education Foundation

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Question 1Question 1Question 1Question 1

The correct answer is A.




The correct answer is D.

Question Question Question Question 4444

The correct answer is C.








� = −23 � + 143


The correct answer is E.





Question 12Question 12Question 12Question 12 14 �� − � �� − �� + 2� Question 13aQuestion 13aQuestion 13aQuestion 13a

k = -4

Question 13bQuestion 13bQuestion 13bQuestion 13b � ∈ � \ {−3, −4}

The Engage Education Foundation Units 3 and 4 Mathematical Methods

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Question 14aQuestion 14aQuestion 14aQuestion 14a

� = 25 �� − 5� − 10

Question 14bQuestion 14bQuestion 14bQuestion 14b � = −6�� − 1�� − 6�� + 4�

Question 14cQuestion 14cQuestion 14cQuestion 14c

� = −32�� + 3� �� − 3�� − 254 �




Question 15bQuestion 15bQuestion 15bQuestion 15b

QuestionQuestionQuestionQuestion 15c15c15c15c

Question 16aQuestion 16aQuestion 16aQuestion 16a � ∈ � Question 16bQuestion 16bQuestion 16bQuestion 16b � ∈ � \ {−2} Question 16cQuestion 16cQuestion 16cQuestion 16c

� ∈ �−∞, 112 � \ {−4, 72} ⊈ Question 17Question 17Question 17Question 17



Question 19aQuestion 19aQuestion 19aQuestion 19a � !��" = �log&�� − 4 � ∈ �0, ∞�



Question 19bQuestion 19bQuestion 19bQuestion 19b ! ��" does not exist since /0!1 �� ⊈ 234 !�� [−4, ∞� ⊈ �0, ∞� Question 20aQuestion 20aQuestion 20aQuestion 20a

�6 = 78�9: � + 3 � ∈ �−∞, 2] �6 = [3, ∞� Question 20bQuestion 20bQuestion 20bQuestion 20b

� = <−�� + 2�2 + 3 Question 21aQuestion 21aQuestion 21aQuestion 21a







The correct answer is B.



Question 25Question 25Question 25Question 25 37 , 8, 16, 3, 329

Question 26Question 26Question 26Question 26aaaa

Question Question Question Question 26262626bbbb


60g

Question 27bQuestion 27bQuestion 27bQuestion 27b 601 ! Question 27cQuestion 27cQuestion 27cQuestion 27c 4 log&�2�






a = 2

Question 29bQuestion 29bQuestion 29bQuestion 29b ?86 = √8 − � +2, x∈ �−∞, 16] Question 30Question 30Question 30Question 30




QuestionQuestionQuestionQuestion 32a32a32a32a

(2, ∞)

QuestionQuestionQuestionQuestion 32b32b32b32b 46�� = √� + 3 + 2 � ∈ [−3, ∞� Question 32cQuestion 32cQuestion 32cQuestion 32c




12,2,16,12


� = 7A12 , 11A12 , 19A12 , 27A12


= 4.5 0 = 43 Question 37Question 37Question 37Question 37

Max = 29m

Min =1m

Period = 90 sec



Question 38Question 38Question 38Question 38 13 Question 39Question 39Question 39Question 39 CDC9 = −12� sin �E CC9 = 86 √689

�F�� = 8�GHI 9: JKG 9�&L9 !F�� = 68MKNO� 9� 9L

CDC9 = 24�P9 cos�3�� − 181P9 sin�3�� CCC9 = 4�E cos�� − �P sin�� Question 40Question 40Question 40Question 40




Question 42aQuestion 42aQuestion 42aQuestion 42a 12 Q Question 42bQuestion 42bQuestion 42bQuestion 42b A4 Q/ℎ Question 43Question 43Question 43Question 43

Question doesn’t work

Question 44Question 44Question 44Question 44 � = 66T

xxxx 1111 UUV 3333

g’x -ve 0 +ve

∴ \ _ / local minimum Question 45Question 45Question 45Question 45

� = 5�3 − 163


2 − 12√AY

Z = 108 √AY





Question 48Question 48Question 48Question 48 9YE − E9L

+ 2� + [ 66\ �2� − 1�] + [ − cos�5� + 2�5 + [ −2√−3� + 4 + [ 2A sin ^A�2 _ − 6 log&|6� + 5| + [ 1 92 − 18 9

2 − 2� + [ Question 49aQuestion 49aQuestion 49aQuestion 49a

981


6


0 and 7


a −bc + db + Uefbgh

Question 50cQuestion 50cQuestion 50cQuestion 50c 6376 ij k0lmi Question 51Question 51Question 51Question 51

-4, 12, 8, 14, 4


Question does not work


(1,1) and (25,25)





128 sq units

Question 54Question 54Question 54Question 54aaaa �F�� = 2� cos � − cos � + 2 sin � Question 54bQuestion 54bQuestion 54bQuestion 54b � sin � + cos � + [ Question 54cQuestion 54cQuestion 54cQuestion 54c

− A2 − 1 Question 55aQuestion 55aQuestion 55aQuestion 55a

1000 sq units


1000 sq units


4000 sq units

Question 56Question 56Question 56Question 56aaaa 320 Question 56bQuestion 56bQuestion 56bQuestion 56b 310 Question 57Question 57Question 57Question 57





Not in course anymore.




2.5

Question 61bQuestion 61bQuestion 61bQuestion 61b 58 Question 61cQuestion 61cQuestion 61cQuestion 61c

1.8


0.1881




0.2922


0.1505






10


8






= A2 Question 68bQuestion 68bQuestion 68bQuestion 68b � = 0.5903 Question 68cQuestion 68cQuestion 68cQuestion 68c [ = 0.3634 Question 69Question 69Question 69Question 69 12 log&�3� Question 70Question 70Question 70Question 70





= 12 Question 72b iQuestion 72b iQuestion 72b iQuestion 72b i

0.7

Question 72b iiQuestion 72b iiQuestion 72b iiQuestion 72b ii 35








0.585


0.585


0.0156

Question 75dQuestion 75dQuestion 75dQuestion 75d

(0.5545,0.6155)

Question 75eQuestion 75eQuestion 75eQuestion 75e

(0.5719,0.5981)

Question 100Question 100Question 100Question 100 n1m k = � o = sin � kF = 2� oF = cos � ∴ 2�2� = kFo + oFk = 2� sin � + � cos � Question 101Question 101Question 101Question 101

�� = ��E + 3�6 ∴ �� = 6 × 2� ×(�E + 3�qrL

= �√� + 3 ∴ �F�1� = 1√4 =

6 Question 102Question 102Question 102Question 102

cos�2�� = √32

2� = A6 , 11A6

� = A12 , 11A12

2� ∈ [0,20]



Question 103Question 103Question 103Question 103 log&�� + 2� − log&�� = log&�2� + 1� log& �� + 2�

� = log&�2� + 1� �� + 2�

� = 2� + 1 �� + 2� = 2� + � � + 2� + 4 = 2� + � 0 = � − 3� − 4 0 = �� − 4�� + 1� � = −1, 4 � = 4, il0[1 � ≠ 1 Question 104Question 104Question 104Question 104 2E98E = 8 89 2E98E = 2E� 89� 3� − 3 = 6 − 3� 6� − 3 = 6 6� = 9 � = 32




a 2 cos�� − sin�2��2� = 2 sin�� + 12 cos 2� + [ ∴ �� = 2 sin�� + 12 cos 2� + [ ��t � =2sin t + 6 cos A + [ = 6

∴ 2 − 12 + [ = 12 [ = −1 ∴ �� = 2 sin � + 12 cos 2� − 1 Question 106Question 106Question 106Question 106

� = � + 3 − ��−3

= 1 − 4�� + 7

−�3 = 4� + 7 −� − 7� = 12 � + 7� + 12 = 0 �� + 4�� + 3� = 0 � = −3, −4 Check k = -3

−3� − 3� = 0 4� + 4� = 1 ∴ � = −3 li 03 i3nkml30 Check k = -4

−4� − 3� = −1 4� − 3� = 1 ∴ � = −4 Question 107Question 107Question 107Question 107

sin ^2� + tE_ = 6 2� + tE ∈ [tE , u\E ]



2� + A3 = 5A6 , 13A6

2� = 3A6 , 11A6

� = A4 , 11A12

QuQuQuQuestionestionestionestion 108108108108 � = −4� + 9 ∴ 4m0!10m = 14 2�2� = 12 �86

= 12√� ∴ 12√� = 14 2√� = 4 √� = 2 � = 4 ∴ � = 2 2 = −4 × 4 + = 18 Question 109Question 109Question 109Question 109

a �√�v\ 2� = 27

a ��6 2� = 27v\

[2�3 �E ] = 27\v

2�3 × 9E = 27



2�3 = 1 � = 32 Question 10Question 10Question 10Question 10aaaa

Find h

� − �9 �=ℎ

� − �4 = ℎ

3� 4 = ℎ

ℎ = √3�2

∴ w3nk41 = 12 × � × √3�2 × � = 1000 √3�4 � = 1000 � = 4000√3�

Question Question Question Question 10b10b10b10b

x = 3�� + 12 × 2 × √3�2 × � = 3 × 4000�√3� + √3�

2

= P\\\√E9L +√E× as req.


x = 4000√3�86 + √32 � 2x2� = −4000√3�8 + √3�



= −4000√3� + √3� For min let

CyC9 = 0 ∴ 0 = −4000√3� + √3� 0 = −4000√3�E 4000√3 = √3�E �E = 4000 � = √4000Y

Question Question Question Question 111a111a111a111a �2,4� ∴ 4 = 4 + 2m − 1jkml30 1 2�2� = 2� + � !/2l10m 3� m/!1m = 0 − 46 − 2 = -1

∴ −1 = 4 + � − 1jkml30 2 Solving equation 1 and equation 2

5 = � ∴ −1 = 4 + 5 = −32

= −32 , � = 5 Question 111bQuestion 111bQuestion 111bQuestion 111b iiii o − 40 − 2 = 0 − 4k − 2 4 − o2 = −kk − 2 4 − o = −8k − 2 o = 4 + 8k − 2 Question 111b iiQuestion 111b iiQuestion 111b iiQuestion 111b ii

x = 12 × k × o − 8 = k2 × �4 + 8k − 2z − 8



x = 2k + 4kk − 2 − 8 ∴ 2x2k = 2 − 8�k − 2�

For max/min let CyC{ = 0

∴ 0 = 2 − 8�k − 2� 8�k − 2� = 2

4 = �2 − k� 0 = 4 − 4k + k − 4 0 = k − 4k 0 = o�k − 4� o = 0,4 ∴ o = 4 il0[1 o ≠ 0 ∴ x = 8 |1im 102 }3l0mi k = 52 x = 17 k = 6 x = 10 ∴ k = 4, x~�� = 8 ij k0lmi Question 111cQuestion 111cQuestion 111cQuestion 111c

k = 52 , x = 17 ij k0lmi Question 112Question 112Question 112Question 112


































Past Exam QuestionsPast Exam QuestionsPast Exam QuestionsPast Exam Questions

2013 Exam 2 Q1

Question 1aQuestion 1aQuestion 1aQuestion 1a �� = 27°� m = 0, 16ℎ Question 1bQuestion 1bQuestion 1bQuestion 1b

16 hours


25 + 2 cos �Am8 z = 26 ∴ m = 83 ℎ Question 1dQuestion 1dQuestion 1dQuestion 1d

| = 26 m = 83 , 403 , 563

∴ 83 + 563 − 403 = 8 ℎ3k/i



Question 1e iQuestion 1e iQuestion 1e iQuestion 1e i

cos �2A3 z = −12

Question 1e iiQuestion 1e iiQuestion 1e iiQuestion 1e ii

‘Show that’ question

−12 = 0 − √32[ − 2A3

−12 = −√323[ − 2A3

− 12 = −√32 × 33[ − 2A 12 = 3√36[ − 4A 6[ − 4A = 6√3 6[ = 6√3 + 4A [ = √3 + 2A3 i /1j. Question 1f iQuestion 1f iQuestion 1f iQuestion 1f i √32 � = 10 ∴ � = 20√33

√34 = 30 ∴ 4 = 10√3 Question 1f iiQuestion 1f iiQuestion 1f iiQuestion 1f ii

�20√3A3 , 10� 2008 Exam 2 Q1

Question 1a iQuestion 1a iQuestion 1a iQuestion 1a i 0. 8T = 0.1678 Question 1a iiQuestion 1a iiQuestion 1a iiQuestion 1a ii � = !3n � − �� = �8,0.8� Pr�� − 6� = 0.2936 Question 1b iQuestion 1b iQuestion 1b iQuestion 1b i 0.84u = 0.2951 Question 1b iiQuestion 1b iiQuestion 1b iiQuestion 1b ii �� 0.84 × 0.84 × 0.16 = 0.1129



�� 0.84 × 0.16 × 0.64 = 0.0860 �� 0.16 × 0.64 × 0.84 = 0.08600.2849 Question 1b iiiQuestion 1b iiiQuestion 1b iiiQuestion 1b iii


Question 1c iQuestion 1c iQuestion 1c iQuestion 1c i

Question 1c iiQuestion 1c iiQuestion 1c iiQuestion 1c ii

a ��2� = 0.1211E

Question 1c iiiQuestion 1c iiiQuestion 1c iiiQuestion 1c iii

a ��2� = 4.1333E

2012 Exam 2 Q3

Question 3aQuestion 3aQuestion 3aQuestion 3a iiii 164 Question 3a iiQuestion 3a iiQuestion 3a iiQuestion 3a ii

� ∼ ��20, 14� Pr�� ≥ 10� = 0.0139 Question 3a iiiQuestion 3a iiiQuestion 3a iiiQuestion 3a iii

0 × 14 × 34 = 7516 ∴ 0 = 25 Question 3b iQuestion 3b iQuestion 3b iQuestion 3b i

�� 13 × 34 × 34 × 34 = 964 �� 23 × 13 × 34 × 34 = 18 964 + 18 = 1764 Question 3b iiQuestion 3b iiQuestion 3b iiQuestion 3b ii


Question 3cQuestion 3cQuestion 3cQuestion 3c Pr�� > 23� = 6 Pr�� = 25�



Pr�� = 25� + Pr�� = 24� = 6 Pr�� = 25� � ]}�1 − }�\ + � P} P�1 − }� = 6 ]� ]} ]�1 − }� ] ]

∴ } ] + 25} P�1 − 9� = 6} ] −5} ] + 25} P = 0 } P�−30} + 25� = 0 } = 0 3/ 56 ∴ } = 56 il0[1 } ≠ 0 Question 3dQuestion 3dQuestion 3dQuestion 3d

�~�� 25, 56z Pr�� ≥ 18� = 0.955 Pr� � ≥ 22� = 0.381 *Pr�� > �� = 0.955 � = −1.698 Pr�� > 7 � = 0.381 � = 0.301 kil0! � = � − 03

∴ −1.698 = 20 − � , 0.301 = 25 − �

Solving gives

= 24.246, � = 2.500 2014 Exam 2 Q4

See Examiners Report for detailed solutions.

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