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Unformatted text preview: (.3 man :51 Lunﬁiiur
5m nursuh‘i FORT STREET HIGH SCHOOL
TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION 2002 MATHEMATICS EXTENSION I (ADDITIONAL) Time allowed : Two Hours
(Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions. ALL questions are of equal value. All necessary working should be shown in every question. Marks may be deducted for
careless or badly arranged work. A table of standard integrals is included. Boardapproved calculators may be used. Each question is to be started on a NEW PAGE and solutions are to be written on
ONE side only. . You may ask for extra Writing Paper if required. You must STAPLE each question in a separate bundle clearly labeled with your
Name. Question 1: (start a new sheet of paper) [12 marks]
a) For the polynomial P(x) = 2x3 — 3x2 — 3x + 2 , i) Show that (x +1) is a factor. I [1] ii) Hence, or otherwise, ﬁnd all the factors of P(x). [3] b) Determine the ratio in which the point C(6,9) divides the interval AB, given A is the point (1,5) and B is (3,3). [2]
c) Find the acute angle between the lines 2x + y = 2 and y — x = 4 . [2]
4
(1) Evaluate Iydx if xy = l. [2]
' l
2100] + 2l000
e) F actorise 2”+l + 2" , and hence write ————3— as a power of 2. [2]
Question 2: (start a new sheet of paper) [12 marks] a) Given f (x) = x sin 2x , i) Differentiate f (x) . [1] ii) Hence ﬁnd I (x cos 2x)dx. [2] b) Solve the inequality x +1 2 1 I . [3]
x _ c) Using the substitution u = 2 + x , ﬁnd x(2 + x)“dx. i [3] d) The arc of the curve y = cos 2x between x = O and x = gis rotated a ﬁlll revolution about the x axis. Find the exact volume of the solid formed. [3] Question 3: (start a new sheet of paper) [12 marks] a) Use the Principle of Mathematical Induction to prove that 23" —3” is
divisible by 5 for all positive integers n. [4] A. .
.m b) Use one iteration of Newton’s method to ﬁnd an approximation to the root of the equation xlnx — 2x = 0 near x = 7 (answer to 1 dp)  [3]
c) If t = tan?— ,
2
2
i) Showthat sin0= 2’2 and cos6l=l ’2. [2]
1=t 1+1
ii) Using these results, show that 1_.cosg = [2]
smﬁ 2 iii) Hence ﬁnd the exact value of tan15°. [1]
Question 4: (start a new sheet of paper) [12 marks] a) Newton’s Law of Cooling states that the rate of change in the temperature, T , of a body is proportional to the difference between the temperature of the body and the surrounding temperature, P. i) If A and k are constants, show that the equation T = P + Ae’“
satisﬁes Newton’s Law of Cooling. [2] ii) A cup of tea with a temperature of 100°C is too hot to drink. Two minutes later, the temperature has dropped to 93°C . If the
surrounding temperature is 23°C , calculate A and k. [2] iii) The tea will be drinkable when the temperature has dropped to
80°C. How long, to the nearest minute, will this take? [2] b) P(2t,t2) is a variable point which moves on the parabola x2 = 4y. The
tangent to the parabola at P cuts the xaxis at T. M is the midpoint of PT. i) Show that the tangent PT has equation tx — y — t2 = O. [2] 2
ii) Show that M has the coordinates . [2]
iii) Hence ﬁnd the Cartesian equation of the locus of M as P moves
on the parabola. [2]
Question 5: (start a; new sheet of paper) [12 marks] a) Given f(x) = \/2x——3 , i) Draw a neat sketch of f (x) , stating its domain and range. [3] ii) Find the inverse function f “(x), and sketch it on the same
diagram as f (x) , stating its domain and range. [3] b) A pebble is projected from the top of a vertical cliff with velocity 20ms'I at an angle of elevation of % radians. The cliff is 40m high and overlooks a lake. (Assume g = —10m/ 5) i) Taking the origin 0 to be at the base of the cliff immediately
below the point of projection, draw a neat diagram showing the
path of the pebble. Mark all data given above on the diagram. [1] ii) Derive expressions for the horizontal component x(t) and the
vertical component y(t) of the pebbles displacement from 0
after t seconds. [2] iii) Calculate the time that elapses before the pebble hits the lake
and the distance of the point of impact from the foot of the cliff.
Give your answer to one decimal place. [3] Question 6: (start a new sheet of paper) [12 marks]
a) If a,,6,y are the roots of x3 — 2x2 + 3x + 7 = 0, ﬁnd the values of
i) —2 + 3 + [2]
a [3 7
ii) a2 + ,62 +72 [2] b) The elevation of a hill at a point P due east of it is 38° and at a point Q
south of it the elevation is 23°. If the distance from P to Q is 420m, ﬁnd the height of the hill. [4]
C)
i) Express 2cos(2x) + 2J§ sin( 2x) in the form R c0s(2x + a) ,
where 0 < o: (g. [2]
ii) _ Hence or otherwise ﬁnd all positive solutions of
2c0s(2x) + 2J3 sin(2x) = o. [2]
Question 7: (start a new sheet of paper) [12 marks] a) A particles motion is deﬁned by the equation v2 =16 + 6x — x2 , where x is its displacement from the origin in metres and v its velocity in ms".
Initially, the particle is 8 metres to the right of the origin. i) Show that the particle is moving in Simple Harmonic Motion. [1]
ii) Find the centre, the period and the amplitude of the motion. [3] iii) The displacement of the particle at any time t is given by the
equation x=asin(nt+0)+b. Find the values of 0 and b, given OSQSZII. [2] b) Without using a calculator, show that tan'l + tan"1 = .3. c) The acceleration of a particle moving in a straight line is given by x = e'z". The particle is at rest at the origin. i) Prove that v2 =1—e‘z". ii) What is the maximum speed the particle can reach? [3] [2]
[1] STANDARD INTEGRALS
andx =an‘”, n¢—1; x¢0,ifn<0
n+1
1
—dx =lnx, x>0
x
ax 1 ax
Je dx =32 , a¢0
1 .
cosaxcix =Zsmax, a¢0
. 1
Ismaxdx =—Zcosax, a¢0
2 1
sec axdx =Ztanax, a¢0
1
fsecaxtanaxdx =Zsecax, a¢0
1 dx —ltan‘1£, a¢0
a2+x2 ‘1 ‘1
—l——dx =sin'li, a>0, ~a<x<a
a2_x2 a
1 dx =ln(x+\/x2—a2), x>a>0
2 2 '
x —a
1 dx =ln(x+\/xz+t12)~
x2+a2 NOTE: lnx=logex, x>0 © Board of Studies NSW 2001 ...
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