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Unformatted text preview: CRANBROOK SCHOOL TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION 2000
MATHEMATICS 3 UNIT (Additional)
4 UNIT (First Paper) Time allowed — Two hours 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. * Standard integrals are printed on the back page.
Board—approved calculators may be used. * You may ask for extra Writing Booklets if you need them, * Submit your work in five booklets :
(i) QUESTIONS ‘1 s; 2. (8 page) (ii) QUESTIONS 3 s; 4 (3 page) (iii) QUESTION 5 (4 page) (iv) QUESTION 6 (4 page) (v) QUESTEON 7 (4 page) l. (8 page booklet) (a) If the equation 5x3 — 6x2 — 29x + 6 = O has roots orﬁmy find the value of 052 + ﬂz + 3/2.
' ‘ [3 marks] (b) (i) Show that there exists one value of the constant b for which the polynomial
P( x) = x4 +‘2x3 — x2 — 8x — b is divisible by Q( x ) = x2 — 4. [2 marks]
(ii) Hence or otherwise find the roots of P( x ) for this value of b. [2 marks] . ~ . d . ' (c) (1) Find Ex—(cosecx cotx) in terms of cosecx. [3 marks] 1!
(ii) Use your result in (i) to ﬁnd the exact value of [VA cos ecx(cot2x + cos eczx)dx.
6 [2 marks]
2. y Y
(a) Find the general solutions of sin 26 + case '= O in radian form. [3 marks]
(b) Find the solutions of 3sin6 + 4c0s9 = —4 for O S 9 S 47: , giving your answers in radians,
correct (where necessary) to 3 decimal places. [4 marks]
(c) PR is a tangent to the circle centre 0, at the point T. Prove that AATP = AABT.
(Redraw the diagram below as part of your answer).
[5 marks] p T R 3. (new 8 page booklet please) . 2 9
(21) Find the term independent of x in the expansion of (—3—;— — 3L] [4 marks]
1 . . x . _ (b) Twelve candidates for election to a committee of four' include two well—known geniuses, Mr
GJ. Baker and Mr SK. Blazey. If all candidates have an equal chance of selection, what is the
probability that the committee
(1) includes Mr‘Baker but excludes M: Blazey? (ii) includes at least one of these two geniuses? [4 marks] (c) A weather bureau ﬁnds that it predicts maximum temperatures with about 60% accuracy.
What is the probability that, in a particular week, it is accurate
(i) on every day but Saturday and Sunday? (ii) on exactly ﬁve days? [4 marks] 4., 3 + 2
(a) Solve x > 2 ' [3 marks]
x — l
(b) Prove by Mathematical Induction that
2 X1!+5>< 2!+10x3!+ +(n2 +1) >< n!= n X(n + 1)! [Smarks]
(c) (i) Show that "C, .' "CM 2 (n — r +1) : r
"C 2x"C 3X"C nx"C
" H nce evaluate ‘ +—~——2—+ 3 + + "
(11) 6 "C0 “C1 "C2 nCn—l [4 marks} 5.. (new 4 page booklet please) (a) Find the derivative of cos"(2x +1), stating the values of x for which it is defined. [2 marks]
0 621:
b Differentiate sin“ e“ and hence find ——~—— dx correct to two decimal lac . ( ) ( ) “L m p es [4 marks]
(c) The rate of emission E, in tonnes per year, of chloro—ﬂuorocarbons (CFC's) in Australia from
. . . 30 2 . .
18th July 2000 Will be given by E = 80+(m) , where I IS the time in years.
(i) What is the rate of emission E on 18th July 2000? [1 mark]
(ii) What is the rate of emission E on 18th July 2005 ? [1mm]
(iii) Draw a sketch of E as a function of t. [2 marks] (iv) Calculate the total amount of CFCs emitted in Australia during the years 2000 to 2005.
[2 marks] 6. (new 4 page booklet please)
(a) Evaluate I: 2 sin x cos2 x dx. [2 mm] (b) Integrate the following using, the substitutions given I4 .. 1 «ll—xi"
x2 (i) "[de (u=x5+1) (11) L dx (x = case) [6 marks] 1 (c) Two roads intersect, making an angle of 30° between them. After an argument at the
intersection, George storms off at 6 km/h along one of the roads, and Jerry walks off calmly at
2 km/h along the other Show that the rate at which the distance between them is increasing is
constant. Find this rate of increase correct to three significant figures. [4 marks] 7. (new 4 page booklet please) (a) The rate of change of the volume of water (V kL) in a dam at any given time t (in hours) is given by 53:1: k(V — 5000) , where k is a constant. (i) Show that V: 5000 + Ae’“ is a solution of this differential equation. [2 marks]
(ii) If the initial volume 18 87 000 kL, and after 10 hours the volume ls 129 000 kL, find
the exact values of A and k. . [3 marks]
(iii) Determine how long it will take the volume to reach 4.2 million kL.
[Give your answer in days and hours, correct to the nearest hour.] [2 marks] (b) The inner and outer radii of a cylindrical tube of constant length change in such a way that the
volume of the material forming the tube remains constant. Find the rate of increase of the
outer radius at the instant when the radii are 3 cm and 5 cm, and the rate of increase of the inner radius is 3 g cm/s. [5 marks]
STANDARD INTEGRALS
Ix" (ix: 1 x"“ (11:4; x¢0 if n<0)
n+1
1 [IX 1 (1X
I—dleogex (x>0) ‘ Ia dx=—e (avéO)
x a
J'cosax dx : ~1— sinax (a¢0) Jseczax dx = ~1— tanax (aiO)
a a
1
Isinax dx = e —— cosax ((1:0) Isecax tanax dx = i secax (a¢0)
a a
1 l 1 x
I 2 dx : — tan —— ((1:50)
a +x (1 a
[.__.=L— — {A r sin~ 3: (a\0, ~a<x<a)
[12"‘12 0 ...
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