Victorian Certificate of Education SUPERVISOR TO ATTACH PROCESSING LABEL HERE

2019

Letter
STUDENT NUMBER

SPECIALIST MATHEMATICS
Written examination 1

Friday 8 November 2019

Reading time: 9.00 am to 9.15 am (15 minutes)
Writing time: 9.15 am to 10.15 am (1 hour)

QUESTION AND ANSWER BOOK

Structure of book
Number of Number of questions Number of
questions to be answered marks
10 10 40

• Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers,
sharpeners and rulers.
• Students are NOT permitted to bring into the examination room: any technology (calculators or
software), notes of any kind, blank sheets of paper and/or correction fluid/tape.
Materials supplied
• Question and answer book of 12 pages
• Formula sheet
• Working space is provided throughout the book.
Instructions
• Write your student number in the space provided above on this page.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
• All written responses must be in English.
At the end of the examination
• You may keep the formula sheet.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic
devices into the examination room.
© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2019
2019 SPECMATH EXAM 1 2

THIS PAGE IS BLANK

 3 2019 SPECMATH EXAM 1

Instructions
Answer all questions in the spaces provided.
Unless otherwise specified, an exact answer is required to a question.
In questions where more than one mark is available, appropriate working must be shown.
Unless otherwise indicated, the diagrams in this book are not drawn to scale.
Take the acceleration due to gravity to have magnitude g ms–2, where g = 9.8

Question 1 (4 marks)
dy 2 ye 2 x
Solve the differential equation = given that y(0) = π.
dx 1 + e 2 x

Question 2 (3 marks)
x
Find all values of x for which | x − 4 | = + 7.
2

TURN OVER
2019 SPECMATH EXAM 1 4

Question 3 (3 marks)
A machine produces chocolate in the form of a continuous cylinder of radius 0.5 cm. Smaller
cylindrical pieces are cut parallel to its end, as shown in the diagram below.
The lengths of the pieces vary with a mean of 3 cm and a standard deviation of 0.1 cm.

a. Find the expected volume of a piece of chocolate in cm3. 1 mark

b. Find the variance of the volume of a piece of chocolate in cm6. 1 mark

c. Find the expected surface area of a piece of chocolate in cm2. 1 mark

 5 2019 SPECMATH EXAM 1

Question 4 (3 marks)
The position vectors of two particles A and B at time t seconds after they have started moving are given by
t t
rA (t ) = ( t 2 − 1) ~i +  a +  j and rB (t ) = ( t 3 − t ) ~i +  arccos    j respectively, where a is a real constant
  3 ~    2 ~
and 0 ≤ t ≤ 2.

Find the value of a if the particles collide after they have started moving.

TURN OVER
2019 SPECMATH EXAM 1 6

Question 5 (6 marks)
The graph of  f (x) = cos2 (x) + cos (x) + 1 over the domain 0 ≤ x ≤ 2π is shown below.

y
4

x
O 3 2 5 3
−2 2 2 2
–1

a. i. Find  f ′(x). 1 mark

ii. Hence, find the coordinates of the turning points of the graph in the interval (0, 2π). 2 marks

1
b. Sketch the graph of y = on the set of axes above. Clearly label the turning points and
f ( x)
endpoints of this graph with their coordinates. 3 marks
 7 2019 SPECMATH EXAM 1

Question 6 (3 marks)
Find the value of d for which the vectors a = 2 i − 3 j + 4k , b = −2 i + 4 j − 8k and c = −6 i + 2 j + d k are
linearly dependent.            

TURN OVER
2019 SPECMATH EXAM 1 8

Question 7 (5 marks)
π
a. Show that 3 − 3 i = 2 3 cis  − . 1 mark
 6

( )
3
b. Find 3 − 3 i , expressing your answer in the form x + iy, where x, y ∈ R. 2 marks

( )
n
c. Find the integer values of n for which 3 − 3 i is real. 1 mark

( )
n
d. Find the integer values of n for which 3 − 3 i = ai , where a is a real number. 1 mark
 9 2019 SPECMATH EXAM 1

Question 8 (4 marks)
1 + 2x
Find the volume of the solid of revolution formed when the graph of y = is rotated about the x-axis
over the interval [0, 1]. 1 + x2

TURN OVER
2019 SPECMATH EXAM 1 10

Question 9 (4 marks)
a. A light inextensible string is connected at each end to a horizontal ceiling. A mass of
m kilograms hangs in equilibrium from a smooth ring on the string, as shown in the diagram
below. The string makes an angle α with the ceiling.

α α

Express the tension, T newtons, in the string in terms of m, g and α. 1 mark

Question 9 – continued
 11 2019 SPECMATH EXAM 1

b. A different light inextensible string is connected at each end to a horizontal ceiling. A mass of
m kilograms hangs from a smooth ring on the string. A horizontal force of F newtons is
applied to the ring. The tension in the string has a constant magnitude and the system is in
equilibrium. At one end the string makes an angle β  with the ceiling and at the other end the
string makes an angle 2β  with the ceiling, as shown in the diagram below.

 1− cos ( β ) 
Show that F = mg  . 3 marks
 sin ( β ) 

TURN OVER
2019 SPECMATH EXAM 1 12

Question 10 (5 marks)
dy  π π  3 2
Find at the point  ,  for the curve defined by the relation sin( x 2 ) + cos( y 2 ) = xy .
dx  6 3 π

π −a b
Give your answer in the form , where a, b ∈ Z+.
a (π + b )

END OF QUESTION AND ANSWER BOOK

Victorian Certificate of Education
2019

SPECIALIST MATHEMATICS
Written examination 1

FORMULA SHEET

Instructions

This formula sheet is provided for your reference.

A question and answer book is provided with this formula sheet.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic
devices into the examination room.

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2019

SPECMATH EXAM 2

Specialist Mathematics formulas

Mensuration

1 a+b h
area of a trapezium
2
( )

curved surface area of a cylinder 2π rh

volume of a cylinder π r2h

1 2
volume of a cone π r h
3

volume of a pyramid 1 Ah
3

4 π r3
volume of a sphere
3

1 bc sin (A )
area of a triangle
2

a b c
sine rule = =
sin ( A) sin ( B) sin (C )

cosine rule c2 = a2 + b2 – 2ab cos (C )

Circular functions

cos2 (x) + sin2 (x) = 1

1 + tan2 (x) = sec2 (x) cot2 (x) + 1 = cosec2 (x)

sin (x + y) = sin (x) cos (y) + cos (x) sin (y) sin (x – y) = sin (x) cos (y) – cos (x) sin (y)

cos (x + y) = cos (x) cos (y) – sin (x) sin (y) cos (x – y) = cos (x) cos (y) + sin (x) sin (y)

tan ( x) + tan ( y ) tan ( x) − tan ( y )

tan ( x + y ) = tan ( x − y ) =
1 − tan ( x) tan ( y ) 1 + tan ( x) tan ( y )

cos (2x) = cos2 (x) – sin2 (x) = 2 cos2 (x) – 1 = 1 – 2 sin2 (x)

2 tan ( x)
sin (2x) = 2 sin (x) cos (x) tan (2 x) =
1 − tan 2 ( x)
 3 SPECMATH EXAM

Circular functions – continued

Function sin–1 or arcsin cos–1 or arccos tan–1 or arctan

Domain [–1, 1] [–1, 1] R

 π π  π π
Range − 2 , 2  [0, �] − , 
   2 2

Algebra (complex numbers)

z = x + iy = r ( cos (θ ) + i sin (θ ) ) = r cis (θ )

z = x2 + y 2 = r –π < Arg(z) ≤ π

z1 r1
z2 r2 ( 1 2 )
z1z2 = r1r2 cis (θ1 + θ2) = cis θ − θ

zn = rn cis (nθ)  (de Moivre’s theorem)

Probability and statistics

E(aX + b) = aE(X) + b
for random variables X and Y E(aX + bY ) = aE(X ) + bE(Y )
var(aX + b) = a2var(X )

for independent random variables X and Y var(aX + bY ) = a2var(X ) + b2var(Y )

 s s 
approximate confidence interval for μ x −z , x+z 
 n n

mean ( )
E X =µ
distribution of sample mean X σ2
variance var ( X ) =
n

TURN OVER
SPECMATH EXAM 4

Calculus
d n
dx
( )
x = nx n − 1
∫ x dx = n + 1 x
n 1 n +1
+ c, n ≠ −1

d ax
dx
( )
e = ae ax
∫e
ax
dx =
1 ax
a
e +c

d 1 1
dx
( log e ( x) ) =
x ∫ x dx = log e x +c

d 1
dx
( sin (ax) ) = a cos (ax) ∫ sin (ax) dx = − a cos (ax) + c
d 1
dx
( cos (ax) ) = −a sin (ax) ∫ cos (ax) dx = a sin (ax) + c
d 1
( tan (ax) ) = a sec2 (ax) ∫ sec (ax) dx = a tan (ax) + c
2
dx
d
dx
( )
sin −1 ( x) =
1
1 − x2
1
2
x
∫ a − x dx = sin  a  + c, a > 0
2
−1

d
dx
(
cos −1 ( x) =) −1
1 − x2 ∫
−1 x
dx = cos −1   + c, a > 0
2 a
2
a −x
d a x
dx
(
tan −1 ( x) =) 1
1 + x2 ∫a +x2
dx = tan −1   + c
2
a
1
∫ (ax + b) n dx =
a (n + 1)
(ax + b) n + 1 + c, n ≠ −1

1
∫ (ax + b) −1 dx =
a
log e ax + b + c

d dv du
product rule ( uv ) = u + v
dx dx dx
du dv
v −u
quotient rule d u dx dx
 = 2
dx  v  v
dy dy du
chain rule =
dx du dx
dy
Euler’s method If = f ( x), x0 = a and y0 = b, then xn + 1 = xn + h and yn + 1 = yn + h f (xn)
dx
d 2x dv dv d  1 
acceleration a= 2
= = v =  v2 
dt dt dx dx  2 
x2 t2

∫ ∫
2
arc length 1 + ( f ′( x) ) dx or ( x′(t ) )2 + ( y′(t ) )2 dt
x1 t1

Vectors in two and three dimensions Mechanics

r = x i + yj + zk momentum p = mv
     
r = x2 + y 2 + z 2 = r equation of motion R = ma
  
i d r dx dy dz
r =  = i+ j+ k
 dt dt  dt  dt 
r 1 . r 2 = r1r2 cos (θ ) = x1 x2 + y1 y2 + z1 z2
 
END OF FORMULA SHEET