Angelfire · 2011. 12. 26. · 2008 . TRIAL ffiGHER SCHOOL CERTIFICATE. GIRRAWEEN HIGH SCHOOL . Mathenlatics Extension . 2. General Instructions Total marks - 120 • Reading time

2008 TRIAL

ffiGHER SCHOOL CERTIFICATE

GIRRAWEEN HIGH SCHOOL

Mathenlatics Extension 2

General Instructions Total marks - 120 • Reading time - 5 minutes Attempt Questions 1 - 8 • Working time - 3 hours All questions are of equal value • Write using black or blue pen • Board - approved calculators may

be used • A table of standard integrals is

provided at the back of this paper • All necessary working should be

shown in every question

Girraween High School Trial Higher School Certificate Mathematics Extension 2

Total marks-120 Attempt Questions 1-8 All questions are of equal value

Answer each question in a SEPARATE piece ofpaper clearly marked Question 1, Question 2, etc. Each piece of paper must show your name.

Question 1 (15 Marks)

J xdxa) 2

~9-4x2

b) J dx 2 ~9-4x2

e

c) Use integration by parts to evaluate Jx3 lnxdx 3 1

d) (i) Find real numbers a,b and c such that 5x2 -4x-9 a bx+c =--+-

(x-2)(x2-3) x-2 x2-3 2

4

J5x2 -4·x-9 52

(ii) Hence show that 2 =In- 2 3 (x - 2)(x - 3) 3

3e) Jsec x tan xdx 2

2

Page 1


Question 2 (15 marks)

a) Let z = 2 + i and w = 3 - 4i , find

(i) 1

1(ii) 1

z

(iii) wz 1

b) (i) Express 1- J3i in mod arg form 2

(ii) Hence find (1- J3i)5 1

(iii) Express (1- J3i)5 in the form x + yi where x and y are real 2

c) If u and v are two non zero complex numbers. Show that if:: = ik for some k E R v

(i) uv+vu =0 2

(ii) If uv + vu = 0 what is the relationship between arg v and arg u 2

d) If co is a complex root of the equation Z3 =1

(i) Show that 1+co+co2 =0 1

(ii) Find the value of (1 + co )(1 + co 2 )(1 + co 4 )(1 + COB) 2

Page 2


(4,9)Question 3 (15 Marks) y

y = fex)

--+--hl-----------~~x

(-1,-5)

a) The graph of y == f(x) is shown above. It has been reproduced for you on pages 9 and 10,

detach these pages and draw neat sketches of the following. Include these pages in your

solutions. The point of intersection of f(x) and the asymptote is ( 1 , 2 ).

1(i) 2

y == f(x)

(ii) y == f(lxl) 2

(iii) y==f'(x) 2

1(iv) y == f(-) 2

x

A

K

l------I:L-----.-l(-------lf3

b) The circle above has diameter AB and centre O. KH is a chord to the circle

and X is a point on the circumference such that KX == XH. XP is the perpendicular

from P to AB. Prove that PNMO is a cyclic quadrilateral. 3

c) (i) Find the square root of -8 - 8.J3i

(ii) Hence solve the quadratic equation x2 - 2,{iix + 2.J3i == a

2

2

Page 3


Question 4 (15 Marks)

a) The solid shown has a semicircular of radius 2 units. Vertical cross sections perpendicular to the

diameter of the circle are quarter circles.

-2

(i) By slicing at right angles to the x-axis show that the volume is given by

V:=-n 2

S4-x 2dx 2 2

0

(ii) Find the volume 2

b) The region bounded by the curve y:= sin-1 x and the x-axis in the first quadrant is

rotated about the line y = - 1. Using the method of cylindrical shells find the volume

of the shape fonned. 4

c) Let a, 13 and y be the roots of the cubic equation x3 - 5x2 + l3x -7 := 0 .

1 1 1(i) Find the polynomial with roots - - and - 2

a'j3 y

(ii) Find the polynomial with roots a 2 , 132 and y2 2

d) (i) Prove the identity sin(a + b)e + sin(a - b)e =2sinae cosbe 1

(ii) Hence find Ssin4ecos28 de 2

Page 4


Question 5 ( 15 Marks )

x2 y2a) Given the ellipse - + - =1. Find:

9 4

(i) the eccentricity. 2

(ii) The coordinates of the foci 1

(iii) The equation of the directrices 1

(iv) Sketch the ellipse showing all essential features. 2

2 2

b) Given the hyperbola ~_L =1 9 4

(i) Show that the point P with coordinates (3 sec (), 2 tan ()) lies on the hyperbola 1

(ii) Find the equation of the nOIDlal to the hyperbola at P. 2

(iii) Find the equation of the tangent to the hyperbola at P. 2

(iv) The tangent at P cuts the asymptotes at L and M. Find the coordinates of

L and M. 2

(v) Show that P is the mid point of LM. 2

Page 5


Question 6. ( 15 Marks)

a)

D

21em

(

Gem

F 100gm

E

A light inelastic string of length 27cm is attached to two points D and E on the vertical. shaft DE,

distance 21cm apart, E being vertically below D. F is a smooth ring ofmass lOOgms threaded

on the string. The system is such that F moves with constant speed in a horizontal circle 6cm above E.

(i) Find the lengths ofDF, FE and r. 3

(ii) Find the tension in the string. 2

(iii) Find the angular speed ofF about DE 2

b) A bullet is fired vertically into the air with a speed of 800m/s. In the air the bullet experiences

" 1 mv 11 .arr reSIstance equa to - as we as graVIty.5

(i) Find the height reached to the nearest metre. 2

(ii) The time taken to achieve this height. 2

(iii) As the bullet returns to the ground it is subject to the same forces,

Find the terminal velocity. 2

c) Solve for x tan-1 3x-tan-1 2x=tan-l ! 2 5

Page 6


Question 7 ( 15 Marks)

a) The cubic equation x3 - ~x -1 =0 is solved in two steps. Firstly let x = u+v and secondly

solve the quadratic equation ;{,2 -;{, +1 =0 the roots of which are u3 and v3

•

(i) Solve the quadratic equation for u 3 and v3 . 1

(ii) Use De Moivre's theorem to find the cube roots with the arguments( of least magnitude. 3

(iii) Find the value of x leave in trigonometric form. 1

1

b) Let In = Jxn.J1-xdx n=O,1,2,3 ... o

2n(i) Show that In =-- In-1 2

2n+3

1

(ii) Henceevaluate Jx 3 .Jl-xdx 2 o

("') Sh th I n !(n +1)! 4"+1111 OW at n =-'---'-- 2

(2n +3)!

c) The curves y = cos x and y = tan x intersect at a point P whose x coordinate is ex

(i) Show that the curves intersect at right angles at P. 2

(.. ) Sh h 2 1+ J511 owt at sec ex =-~ 2 2

Page 7

Girraween High School Trial H1gher School Certificate Mathematics Extension 2

Question 8 ( 15 Marks)

a) If U\ =J2 and Un =~2 + Un_I Prove by Mathematical Induction that

3Un < f2 +1 for all n.

b) (i) Sketch the graph of y =~ ,With the aid of your sketch, show that for any x

1+11 1 positive number u, _u_ < I-dx 1 1+ r r r

1 1 1 1 1 1 1 '(' ') h h(iii) Let an =1+-+-+-+-+-+ .... +-- n n, Byusmg 11 s ow t at 2 3 4 5 6 n

1 -< all < 1 2 n

(l (l

c) (i) Show that Ii(x)dx =Ii(a - x)dx 2 o 0

!!

(ii) Hence find the value of fx sin x dx 2 o

2d) (i) Show that the gradient function for x + / + xy = 12 is

dy -(2x+ y) -= 2 dx 2y+x

(ii) Find the coordinates of the stationary points ofthis function

(iii) Find the coordinates of the points of contact of any vertical tangents, 1

Page 8

Girraween High School Trial Higher School CertlTlCate Mathematics Extension 2

y (4,9)

y = f (x)

--~~---t-+--~-----------~X

(-1 , -5 )

i y

(4,9)

y =f (x)

_~_-\-__-+--jL- ~ X

(-1 , -5 )

Page 9

------------------------------------- ---


y (4,9)

y =f(x)

2

-----l,r----+--+---~-----------!'bx

(-1,-5)

y (4,9)

y = f(x)

2

-----\----I---+-------------~----!l!:.x

(-1,-5)

Page 10

STANDARD INTEGRALS

= _l_xl<+1 Jl::j::. -1: x¢. 0, If J1 <.0 n+!

'!dX = lnx, x> 0·xf

Jf' cosGX(Lr: = .Ls1n ax. a =F 0 (l

J I ' 0sluaxdx = --COSCLt' a '* a '

J = -t.anm:. a*,Oa'

("

1J$cca.r tan a.x: d,"( == -seca.:r, a:;f; 0 a

' 1 1. -1 X=-t.<m - a;=.O." .' 'J dr J

a a>f ao- +X'

x = sin -1 a' (1 > 0, - a < x < a

,... '), .~--)

= Ir{X+ ,,/xl, -a~, x> a> 0

{' 1 : r""~"~'--'-':"f"\

j <Lx =In!x + ~JXL + {/~ I • '1 " ",Ix'" +,Q!.

., /

NOTE In.\' = log~x, x> 0 '-'t:

Mathematics Trial HSC Examination

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Trial Higher School Certificate Mathematics Extension 2 GilTaween High School Trial Higher School Certificate Mathematics EXlension 2 Girraween High School

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Documents

Angelfire · 2011. 12. 26. · 2008 . TRIAL ffiGHER SCHOOL CERTIFICATE. GIRRAWEEN HIGH SCHOOL . Mathenlatics Extension . 2. General Instructions Total marks - 120 • Reading time