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UNIT 2 MATHEMATICAL METHODS 2014
MASTER CLASS PROGRAM
WEEK 11 EXAMINATION 1 — SOLUTIONS
FOR ERRORS AND UPDATES, PLEASE VISIT
WWW.TSFX.COM.AU/MC-UPDATES
QUESTION 1
4 y
4
2
1
x
0
2.5π
−2π
−1.5π
−π
−0.5π
0
0.5π
π
1.5π
2π
2.5π
3π
-2
2
3
-4
1. Draw y = 2 sin( x) i.e. Stretch curve along Y axis (multiply each value of Y by 2).
2. Draw y = −2 sin( x) i.e. Reflect in the X axis.


3. Draw y = −2 sin  x +
π
π
units to left.
 i.e. Move curve
2
2
π

4. Draw y = 1 − 2 sin  x +  i.e. Move curve 1 unit up.
2

 The School For Excellence 2014
Unit 2 Master Classes – Maths Methods – Exam 1
Page 1
QUESTION 2
X intercepts – Let y = 0 :
Y intercepts – Let x = 0 :
2e 3− 2 x − 5 = 0
2e 3− 2 x = 5
5
e 3− 2 x =
2
y = 2e 3− 2 x − 5
y = 2e 3 − 5 i.e. 0, 2e 3 − 5
(
)
5
log e e 3− 2 x = log e  
2
5
3 − 2 x = log e  
2
x=
3 1
5
− log e  
2 2
2
3 1
5 
− log e  , 0 
2 
2 2
y = −5
i.e. 
QUESTION 3
 The School For Excellence 2014
Unit 2 Master Classes – Maths Methods – Exam 1
Page 2
QUESTION 4
a.
b.
c.
QUESTION 5
a.
(i)
3 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1× 2 = 6 × 8!
With 3 Mathematics books available, there are 3 ways in which the first position
can be filled leaving 2 ways of filling the last position at the end of the row.
The remaining 8 books are not restricted. Therefore, the number of ways in which
to fill the second position is 8, 7 for the third position etc.
(ii)
b.
Pr =
Number of ways of arranging as in part (i )
Number of arrangements without restriction
Pr =
6 × 8!
6
6
1
=
=
=
10! 10 × 9 90 15
Mathematics text books together = 1 unit.
These textbooks can be arranged in 3! ways.
In total, there are 8 units to be arranged (7 Chemistry and 1 Maths).
Number of arrangements in a circle if there are no restrictions = (n − 1)!
∴ (8 − 1)!× 3! = 7!3!
 The School For Excellence 2014
Unit 2 Master Classes – Maths Methods – Exam 1
Page 3
QUESTION 6
a.
 0.3 0.4
 , initial state S0 =
0.7 0.6
Transition matrix P = 
0 
1 . To find the probabilities for
 
three days later, find S3 = P3S0 i.e. n = 3 .
As Pr(eats hom e) = 0.636 then Pr(eats out ) = 1 − 0.636 = 0.364
3
 0.3 0.4 0 0.364
0.7 0.6 1 = 0.636

   

b.
0.63
QUESTION 7
Stationary Points exist at x = 3 and x = 0 .
(0, 0)
X intercept occurs at (4, 0)
Y intercept occurs at (0, 0)
(4, 0)
Gradient is negative when x < 0 and
between x = 0 and x = 3 .
Gradient is positive when x > 3 .
QUESTION 8
a.
b.
f ( x) = ( x − 1) 2 ( x − 2) + 1
Using the Quotient Rule: a = 4, b = −19, c = 2
 The School For Excellence 2014
Unit 2 Master Classes – Maths Methods – Exam 1
Page 4
QUESTION 9
π
3

sin 2 x +  =
3
2

 3 π

 2 = 3


1st Quadrant Angle = Sin −1 
Solutions are to lie in the quadrants where sine is positive i.e. the 1st and 2nd quadrants:


Let 2 x +
π
π


=
3 3
Let 2 x +
π
π
=π −
3
3
π  π 2π

2 x +  = ,
3 3 3

π  π 2π

x +  = ,
3 6 6

x=−
T=
π
6
,0
2π
6π
=π =
,
2
6
∴x=
5π
11π
,π,
, 2π
6
6
QUESTION 10
(a)
5 3
log b 

 2 
= log b (5 3) − log b (2)
= log b (5) + log b ( 3 ) − log b (2)
( )
= log b (5) + log b 31 / 2 − log b (2)
1
= log b (5) + log b (3) − log b (2)
2
1
=r+ q− p
2
 The School For Excellence 2014
Unit 2 Master Classes – Maths Methods – Exam 1
Page 5
(b)
QUESTION 11
 The School For Excellence 2014
Unit 2 Master Classes – Maths Methods – Exam 1
Page 6