Girraween High School_Mathematics Extcnsionl_Trial HSC_2008
Girraween High SchootMathematics Extensionl_Trial HSC_2008
Total marks - 84
Question 2 (12 marks). Start on a SEPARATE page.
Attempt Questions 1 -7 All questions are of equal value. Find the term independent of x in the expansion of
Question 1 (12 marks). Start on a SEPARATE page.
(2X
3
.;
r
2
. x
SlTI-
Marks
Find the acute angle between the lines 2x + y
= 17
and 3x - y
=3 .
(b)
Evaluate: lim
x~o
2
If I(x) = 2sin-
(b)
2
2x
, find
The point P(17,36) divides the line joining A(2,1) and B(5,8) externally
(i)
in the ratio m : n. Find m and n.
2
the domain and range of
2
(ii)
(c)
2x 3
Solve for x: --:::: I
x-2
(iii)
2
(d)
Differentiate y
tan -
3
2
AD and CD are tangents to a circle. B is a
on the circle such
that LCBA and LCDA are equal and are both double LBCA. Prove
that BC is a diameter of the circle.
3
x
3
Use the substitution u = cosx to evaluate fcos 3 xs in x dx
3
o
D
·2
·3­
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•
(Iinllw<cn High School_Mathematics Extensionl_Trial HSC_2008
Girraween High School_Mathematics Extensionl_Trial HSC-.:2008
Question 3 (12 marks). Start on a SEP ARATE page.
Question 4 (12 marks). Start on a SEPARATE page.
(a) Leon walks on level ground, in a northerly direction, away from an electricity
(a) tower. When he arrive at a point A, the angle of elevation to the top of the
•
Prove the following by the Principle of mathematical induction.
52" - I is divisible by 24 for n ~ I .
3
tower is 23 ° . Luke walks on level ground on a bearing of 032°T from the same
tower, until he reaches point B, and notices that the angle of elevation is 17°.
The distance between A and B is SSm. Let h be the height of the tower and
(b) assume that the tower base C, is perpendicular to the ground.
P(2ap, ap') is a variable point on the parabola x' '" 4ay . M is the foot
of the perpendicular from P to the x -axis. Q is the point on MP such
that MP = PQ . Find the equation of the locus of Q.
NOT TO
SCALE
Electricity
Tower (h)
N
(c) For the function y
C
.S
(i) (i) Copy the diagram above onto your booklet and clearly mark
(iii) Find expressions for AC and BC in terms of h.
2
Hence,or otherwise, find the height h ofthe tower to the nearest metre.
(b) "'-+­
x -9
Write down the equations of horizontal and vertical
2
asymptotes. on it all the information given.
(ii) (ii) Find any stationary points and determine their nature.
2
(iii) Sketch the graph showing the above features.
2
3
Find how many arrangements can be made by taking all the letters
of the word
(c) (i) MATHEMATICS
(ii) In how many ofthem do the vowels occur together?
3
An archer finds that in the long run, he scores a bull' s eye on 3 out of 5
occasions. He fires 8 rounds at a target. Assuming that each trial is an
independent event, find the probability of
(i) exactly 5 bull's eyes.
(ii) at least 7 bull's eyes.
3
3
·4·
·5­
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Girraween High School_Mathematics Extension I_Trial HSC_2008
Girr.ween High SchootM.thematics Extensionl_Trial HSC_ZOOS
Question 6 (12 marks). Start on a SEPARATE page,
Question 5 (12 marks)
(b)
.J3 cos x
(i)
Express 3 sin x
(ii)
Hence find the general solution to 3sinx -
in theform A sin(x
2
a)
==.J3
(a)
2
Find the roots of the equation x' -15x + 4 == 0,
that two of its
roots are reciprocals,
3
N is the number ofKagaroos in a certain population at time t years,
(b)
The population size N satisfies the equation
dN
-
:=
dt that N == 500 + Ae
kt
where A is a constaut, is a solution
of the equation.
(ii)
(iii)
x in
Initially, there are 3500 Kangaroos but after 3 years there are
2
2
vmls ofa
The
v
2
only 3300 left. Find the value of A aud the exact value of k ,
to, find au expression for
terms of t.
-keN - 500), for some constant k, (c)
(i)
If dx == x + 6 and x == -5 when
dt
64 -16x
moving in a straight line is given by
2
8x where the displacement from a fixed point
a
is
x metres.
(i)
FifI au expression for the acceleration aud show that the motion
2
is simple harmonic,
2
Find when the number of Kangaroos begin to fall below 2300.
2
Between which two points is the particle oscillating?
2
Sketch the
2
(iii)
Find the period and amplitude of the motion.
2
(iv)
Find the maximum speed of the particle.
of the population size against time.
t
-7­
- 6­
...... Girraween High School_Mathematics Extensionl_Trial HSC_200S
'Ih,."rrll Illg.1t Sd,ool ~MlIlltomaticg Extellsionl_Trial HSC_200S
Question 6 (12 marks). Start on a SEPARATE page.
Question 5 (12 marks)
(i)
2
Express 3sinx-,/3 cosx in the fonn Asin(x-a)
(a)
(ii)
(b)
Hence fmd the general solution to 3sinx -,/3 cos X""
,/3
2
roots are reciprocals.
3
1'1 is the number ofKagaroos in a certain population at time t years.
(b)
The population size 1'1 satisfies the equation
(c)
(i)
Verify that 1'1
500 + Ae -" where A is a constant, is a solution
of the equation.
Oi)
If dx = x + 6 and x
dt
-5 when
to, find an expression for
x in
tenns of t.
dN = -keN _ 500), for some constant k.
dt
-
Find the roots of the equation x l -15x + 4 =0, given that two of its
2
The speed v m I s of a particle moving in a straight line is given by
v' = 64 -16x - 8/ where the displacement from a fixed point 0 is
2
Initially, there are 3500 Kangaroos but after 3 years there are
x metres.
(i)
only 3300 left. Find the value of A and the exact value of k .
2
(iii)
Find when the number of Kangaroos begin to fall below 2300.
2
(iv)
Sketch the graph of the population size against time.
2
Fi~d
an expression for the acceleration and show that the motion
is simple hannonic.
2
(ii)
Between which two points is the particle oscillating?
2
(iii)
Find the period and amplitude of the motion.
2
(iv)
Find the maximum speed of the particle.
~
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Girraween High SchooUJathematics Extensionl_Trial HSC _2008
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Question 7 (12 marks), Start on a SEP ARATE page.
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2
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