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Unformatted text preview: ASCHAM SCHOOL 2002 TRIAL HSC EXAMINATION MATHEMATICS EXTENSION 1 FORM VI General Instructions: 0 Reading Time: 5
minutes a Working Time: 2
hours 0 Write using blue or black pen
@ Approved calculators and
templates may be used. a A table of standard integrals is
provided at the back of this paper
0 All necessary working should be
shown in every question. Collection: 0 Start each question in a new
answer book a Write your name and teacher’s
name on each book. 0 If you use a second book, place
it inside the first. Total Marks : 84
 Attempt Questions 1 — 7
o All questions are of equal value. vm..........................uMcnswn 1 mamemancs Form V1 ...................... ..p2 Question 1 Start a new answer book
57: .
a) Express T2— radians as degrees
[1]
b) Find a primitive of e” [1]
C) Use the table of standard integrals to ﬁnd the exact value of 4 dx .
JVx2+4 [2] d) If a, B and y are roots of the equation 6x3 +7x2 —x _2 = 0 , ﬁnd the value of L+._1_‘+.1_ aﬁy 6) Find the domain and range of f (x) = 43in ‘1 g, and sketch the graph of f(x) .[3] d ____ cosx 2
1) dx 6 [ ]
Question 2 Start an new answer book
F" d ' x d 1
a) 1n (1) I4+x2 x %
(ii) Itan 2 2x dx [3]
%
b) If 22W = 10 , ﬁnd rif reXists. [3] V "v ........................ ..uJ(,LeILSZOTL 1 mamemaucs Form V1 ...................... ..p3 C)
AB is the diameter of the circle With centre 0. TP is a tangent to the circle
at T. EPLAP. Prove:
(i) TBPE is a cyclic quadrilateral [2]
(ii) PT = PE. [3]
Question 3 Start a new answer book
a) Solve : 2x 21 ' [3]
5 —— x
b) Prove by mathematical induction that 6" ~1 is divisible by 5 for all positive
integers.
[5]
c) By substituting t .= tan g, ﬁnd the solutions to the equation: 33in x +4cosx =5 for 0° Sx S360°,
giving your answers correct to the nearest degree. [4] ............................ numeuszuu J. lVluLIL€7naUCS norm Question 4 Start a new answer book
I a) Using the substitution u = x3 +1 , evaluate Ix2(x3 +1) dx [2]
—1 b) (i) Factorise : x3 — 3x + 2 (ii) Hence draw a neat sketch of the polynomial y = x 3 — 3x + 2 without
the use of calculus, showing all intercepts with the coordinate axes. (iii) Hence solve the inequality x3 ~3x +2 > O [4]
c) Find the value of sin (2 sin ‘1 in exact form [3] d) i) Show that the equation f (x) = x3 — 8x + 8 has a zero between —3 and —
4 . ii) Taking x = —3 5 as a first approximation of the solution of the
equation f (x) = 0 , use Newton’s method once to find a closer
approximation, giving your answer to 2 decimal places [3]
Question 5 Start a new answer book
a) A bug is oscillating in simple harmonic motion such that its displacement x
metres from a fixed point 0 at time t seconds is given by the equation J'c'=—4x. When t=0, v=2m/sandx=5. (i) Show that x = a cos(21‘ + B)is a solution for this equation, where a
and B are constants. (ii) Find the period of the motion. (iii) Show that the amplitude of the oscillation is #26 . (iv) What is the maximum speed of the bug? [5] . d (I 2) .. 2 b) (1) Prove that a: 7v = x [ ] (ii) The acceleration of a creature is given by J? = —‘3u2e"‘ Where x is the displacement from the origin, and u is the initial velocity at the
origin. Given that u = 2 m/ s: (a) Show that v2 = 4e”
( ﬂ ) Explain why v > 0 , and ﬁnd x in terms of t.
(y) Describe the subsequent motion of the creature as t —> oo. [5] b) m ........................ "gunman 1 mamemaucs Form V1 ...................... ..p5 Question 6 Start a new answer book A ladder is slipping down a vertical wall. The ladder is 4 metres long. The
top of the ladder is slipping down at a rate of 3 m / s. How fast is the bottom of the ladder moving along the ground when the bottom is 2 metres away
from the foot of the wall? [4] The point P(2ap,ap 2 ) lies on the parabola x2 = 4ay . The focus S is the point
(O,a). The tangent at P meets the Taxis at Q. g (i) Find the equation of the tangent at P and the coordinates of Q. (ii) Prove that SP = SQ
(iii) Hence show that APSQ +24SQP = 180 ° [4] In a town in Mathsland, a ‘ﬂu epidemic is spreading at a rate proportional
to the population that have it, such that is it predicted that the number of people who have the disease will double in 3 weeks, i.e. 9,14— : kA , where A dt
is the number of people with ‘ﬂu in time t weeks. (i) Show that A = Aoe’“ ,where A O is the initial number of people with
‘ﬂu, satisﬁes the above differential equation. (ii) Find It in exact form (iii) In the neighbouring town with a population of 20,000, three people
have the ‘ﬂu. How many weeks (to the nearest week) will it take for the whole population to contract the disease?
[4] . ...._.W.ww. . ‘ .Lv‘uulLolluJleo l'UI "L V 1 . . . . . . . . . . . . . . . . . . . . . . . .pO Question 7 Start a new book a) Make a large neat copy of the diagram in your answer book. AB is the diameter of the circle with centre 0 and radius r. BC = r,
AQ = QP =PC, and AAOQ = 6 (i) Prove that c056 =2? (Hint: use the cosine rule in triangles AQO and QOC) ‘ [5] (ii) Hence prove that QC == “[6— [2] b) ii) mummowu cut/4 ........................ ..Extension 1 Mathematics Form V1 ...................... ..p7 same point. Show that 6 = tan ‘1 x2 +1350 Hence ﬁnd the value of x Which will make 9 a maximum. [6] End of Examination ............................... ..u/LLUILOLUIL 1 mumemaucs rorm Standard Integrals ,, _ 1
Ix dx mn+1 n+1 x , n¢——1;x¢0,ifn<0 Jldx =lnx, x>0
x Jemdx =ie‘”, a¢0 1.
Icosaxdx =~—smax, a¢0
a
. 1
Ismaxdx =——cosax, a¢0
a
2 1
{sec axdx =~—tanax, a¢0
a
1
J'secaxtanaxdx =—secax, a¢0
a a +x a a
1 . _ x
I —dx =sm1—, a>0, ~a<x<a
az—x2 ‘1 NOTE: lnx =log8x, x>O ...
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