SPECIALIST MATHEMATICSWritten examination 1
Thursday 8 June 2017 Reading time: 2.00 pm to 2.15 pm (15 minutes) Writing time: 2.15 pm to 3.15 pm (1 hour)
QUESTION AND ANSWER BOOK
Structure of bookNumber of questions
Number of questions to be answered
Number of marks
11 11 40
• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpenersandrulers.
• StudentsareNOTpermittedtobringintotheexaminationroom:anytechnology(calculatorsorsoftware),notesofanykind,blanksheetsofpaperand/orcorrectionfluid/tape.
Materials supplied• Questionandanswerbookof10pages.• Formulasheet.• Workingspaceisprovidedthroughoutthebook.
Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.• AllwrittenresponsesmustbeinEnglish.
At the end of the examination• Youmaykeeptheformulasheet.
Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2017
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2017
STUDENT NUMBER
Letter
2017SPECMATHEXAM1(NHT) 2
THIS PAGE IS BLANK
3 2017SPECMATHEXAM1(NHT)
TURN OVER
InstructionsAnswerallquestionsinthespacesprovided.Unlessotherwisespecified,anexactanswerisrequiredtoaquestion.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.Taketheacceleration due to gravitytohavemagnitudegms–2,whereg=9.8
Question 1 (3marks)A5kgmassonasmoothplaneinclinedat30°isheldinequilibriumbyahorizontalforceofmagnitudePnewtons,asshowninthediagrambelow.
P
30°
a. Onthediagramabove,showallotherforcesactingonthemassandlabelthem. 1mark
b. FindP. 2marks
2017SPECMATHEXAM1(NHT) 4
Question 2 (3marks)
Findagiventhat 816
622 −=
−∫ xdx
a
elog ( ),a ∈(–2,4).
Question 3 (3marks)Findthegradientofthecurvewithequation x
yx=
=sin .
1514
when Giveyouranswerintheform a b ,wherea,b ∈ Z+.
5 2017SPECMATHEXAM1(NHT)
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Question 4 (4marks)Findthevaluesofaandbgiventhatz – 1 – i isafactorofz3+(a + b)z2+(b2 – a)z–4=0,whereaandb arerealconstants.
Question 5 (4marks)Partofthegraphof y x
x=
+
−
1
1 24isshownbelow.
0.5
1
1.5
–0.5 0 0.5
y
x
Findthevolumegeneratediftheregionboundedbythegraphof y x
x=
+
−
1
1 24,thelines x x= − =
12
12
and , andthex-axisisrotatedaboutthex-axis.
2017SPECMATHEXAM1(NHT) 6
Question 6 (3marks)Findallrealsolutionsoftan(2x)=–tan(x).
7 2017SPECMATHEXAM1(NHT)
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Question 7 (5marks)
Let dydx y= −( )4 2.
a. Expressyintermsofx,wherey(0)=3. 3marks
b. Expressd ydx
2
2 intermsofy. 2marks
2017SPECMATHEXAM1(NHT) 8
Question 8 (3marks)A3kgmasshasvelocityvms–1,where v x= 2arctan whenithasadisplacementxmetresfromtheorigin,x>0.
Findthenetforce,Fnewtons,actingonthemassintermsofx.
9 2017 SPECMATH EXAM 1 (NHT)
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Question 9 (4 marks)The random variables X and Y are independent with μX = 4, var(X) = 36 and μY = 3, var(Y) = 25.
a. The random variable Z is such that Z = 2X + 3Y.
i. Find E(Z). 1 mark
ii. Find the standard deviation of Z. 1 mark
b. Researchers have reason to believe that the mean of X has decreased. They collect a random sample of 64 observations of X and find that the sample mean is X = 3 8.
i. State the null hypothesis and the alternative hypothesis that should be used to test that the mean has decreased. 1 mark
ii. Calculate the mean and standard deviation for a distribution of sample means, X , for samples of 64 observations. 1 mark
2017SPECMATHEXAM1(NHT) 10
END OF QUESTION AND ANSWER BOOK
Question 10 (4marks)Considerthevectors
a i j k and b i j k.= − − + = + +2 3 2 c
Findthevalueofc,c ∈ R,iftheanglebetween� �a and b is π
3.
Question 11 (4marks)Findthelengthofthecurvespecifiedparametricallybyx = aθ – asin(θ),y = a – acos(θ)from
θπ
θ π= =23
2 to ,wherea ∈ R+.Giveyouranswerintermsofa.
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 Certificate of Education 2017
© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2017
SPECMATH EXAM 2
Specialist Mathematics formulas
Mensuration
area of a trapezium 12 a b h+( )
curved surface area of a cylinder 2π rh
volume of a cylinder π r2h
volume of a cone 13π r2h
volume of a pyramid 13 Ah
volume of a sphere 43π r3
area of a triangle 12 bc Asin ( )
sine ruleaA
bB
cCsin ( ) sin ( ) sin ( )
= =
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 ( ) tan ( ) tan ( )tan ( ) tan ( )
x y x yx y
+ =+
−1tan ( ) tan ( ) tan ( )
tan ( ) tan ( )x y x y
x y− =
−+1
cos (2x) = cos2 (x) – sin2 (x) = 2 cos2 (x) – 1 = 1 – 2 sin2 (x)
sin (2x) = 2 sin (x) cos (x) tan ( ) tan ( )tan ( )
2 21 2x x
x=
−
3 SPECMATH EXAM
TURN OVER
Circular functions – continued
Function sin–1 or arcsin cos–1 or arccos tan–1 or arctan
Domain [–1, 1] [–1, 1] R
Range −
π π2 2
, [0, �] −
π π2 2
,
Algebra (complex numbers)
z x iy r i r= + = +( ) =cos( ) sin ( ) ( )θ θ θcis
z x y r= + =2 2 –π < Arg(z) ≤ π
z1z2 = r1r2 cis (θ1 + θ2)zz
rr
1
2
1
21 2= −( )cis θ θ
zn = rn cis (nθ) (de Moivre’s theorem)
Probability and statistics
for random variables X and YE(aX + b) = aE(X) + bE(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 )
approximate confidence interval for μ x z snx z s
n− +
,
distribution of sample mean Xmean E X( ) = µvariance var X
n( ) = σ2
SPECMATH EXAM 4
END OF FORMULA SHEET
Calculus
ddx
x nxn n( ) = −1 x dxn
x c nn n=+
+ ≠ −+∫ 11
11 ,
ddxe aeax ax( ) = e dx
ae cax ax= +∫ 1
ddx
xxelog ( )( ) = 1 1
xdx x ce= +∫ log
ddx
ax a axsin ( ) cos( )( ) = sin ( ) cos( )ax dxa
ax c= − +∫ 1
ddx
ax a axcos( ) sin ( )( ) = − cos( ) sin ( )ax dxa
ax c= +∫ 1
ddx
ax a axtan ( ) sec ( )( ) = 2 sec ( ) tan ( )2 1ax dxa
ax c= +∫ddx
xx
sin−( ) =−
12
1
1( ) 1 0
2 21
a xdx x
a c a−
=
+ >−∫ sin ,
ddx
xx
cos−( ) = −
−
12
1
1( ) −
−=
+ >−∫ 1 0
2 21
a xdx x
a c acos ,
ddx
xx
tan−( ) =+
12
11
( ) aa x
dx xa c2 2
1
+=
+
−∫ tan
( )( )
( ) ,ax b dxa n
ax b c nn n+ =+
+ + ≠ −+∫ 11
11
( ) logax b dxa
ax b ce+ = + +−∫ 1 1
product rule ddxuv u dv
dxv dudx
( ) = +
quotient rule ddx
uv
v dudx
u dvdx
v
=
−
2
chain rule dydx
dydududx
=
Euler’s method If dydx
f x= ( ), x0 = a and y0 = b, then xn + 1 = xn + h and yn + 1 = yn + h f (xn)
acceleration a d xdt
dvdt
v dvdx
ddx
v= = = =
2
221
2
arc length 1 2 2 2
1
2
1
2
+ ′( ) ′( ) + ′( )∫ ∫f x dx x t y t dtx
x
t
t( ) ( ) ( )or
Vectors in two and three dimensions
r = i + j + kx y z
r = + + =x y z r2 2 2
� � � � �ir r i j k= = + +ddt
dxdt
dydt
dzdt
r r1 2. cos( )= = + +r r x x y y z z1 2 1 2 1 2 1 2θ
Mechanics
momentum
p v= m
equation of motion
R a= m
