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Unformatted text preview: ID. A straight line through the point R(Mt. (I) has a variable slope m. It intersects the circle
5
x7 + y2 = l at A and B. Let P he the midpoint of A8. (3) Find the coordinates of P in terms of m, (9 marks) (b) The locus of I’ is a part of a curve C. Find the equation of C and name it. (6 marks) (c) Sketch the locus of Pr (5 marks) 11. (a) Showthat Sift—L”: 200529 +1.
srn6 By putting 6 = % + q) in the above identity, show that c0539! — sin3Q = l 2sin2¢2. cosnﬁ + sintﬁ (7 k)
mar S (b) Using the substitution «1) = %  11, show that E
1 cos3Q ddb = o sin3u du.
o cos¢ + sind: g cosu + sinu z llence, or otherwise. show that z a
‘1 c0539 dd? = lJZCOSBQsin39dw
2 I) cosd: + siud) cosda + sind) o
(8 marks) (0) Using the results in (a) and (b), evaluate c0539? dd). 0 cos¢ + sin!»
(5 marks) 1
1 127 Let {(x) beafunction of x and let k and s be constants. (a) By using the substitution y = x + ks, show that 5 (hi):
I [(x + ks)dJr = I f(x)dx.
0 ks Hence show that, (or any positive integer n . HS
[IIHXI + {(x +x) + + ftx + (n ~ llstx = I f(x)dx.
(I 0
(l0 marks)
1
. 2
(b) Evaluate I —d—’—‘: by using the substitution x = sinﬂ.
o l — x1
Using this result together with (a), evaluate
.L
1" 1 + I_ 4 I + , . _ , 1 dx .
i  1 _ r 1  L2 _ "*1 1
a (J— x \A (1+7?) x/r n+2") \fl (x+ In)
(10 marks) END OF PAPER 83CE~Add Maths ”(El—4 “H“
M HONG KONG EXAMINATIONS AUTHORITY HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 1984 WMﬁ$ ae—
ADDITIONAL MATHEMATICS PAPER I 8.30 am—i 0.30 am (2 hours) This paper must be answered in English Answer ALL questions in Section A and any THREE
questions from Section B. All working must be clearly shown. 84CEADD MATHS 1—1
9 ~ Am._. ‘_.u_.~__ , ;~r w _ 4.. ... a... < .._ ...,.. . "mm. mu ‘:~v.,;, SECTION A (39 marks)
Answer ALL questions in this section. 1. Given 0 = 3i ~ 2j, (3) Find the unit vector in the direction of A?. (b) If P is a point such that 14—17: "IA—E, expiess 67” in terms of m.
(7 marks) 2. The surface area of a sphere is increasing at a rate of 8 cm2/s . How
fast is the volume of the Sphere increasing when the surface area is 3611 cm2 ?
(8 marks) 3. Let z=(1—2i)5 (a) Using the binomial theorem, express 2 in the form a + bi ,
where a , b are real. (b) Find the real part of % . Hence write down the real part of z + £ , correct to the nearest
integer.
(6 marks)
4. Solve for x :
1 2xl — 1  < 2 .
(5 marks) 84CEADD MATHS l~——2
10 ) 5. Let or and {3 be the roots of the equation
x2 —2x(m2 —m+1)=0, where m is a real number. (a) Show that (0:  6)2 > 0 for any value of m . (b) Find the minimum value of lo: — 3 . (7 marks) 6. ABC is a triangle in which AB = AC and LBAC = 20 . The median
AD = h . Find a point P on AD so that the product of the
distances from P to the three sides of AABC is a maximum. (6 marks) 84CEADD MATHS I—3
11 i
i SECTION B (60 marks) Answer any THREE questions from this section.
Each question carries 20 marks. 7. In Figure l , ABCD is a square with E = i and AD‘=j . P_ and d
Q are respectively points on AB and BC produced With B)" — k an
CQ = m . AQ and DP intersect at E and LQEP = 0 . A D ‘u
i A i B P Q
C Figure l (a) By calculating AQ  131; , ﬁnd cost) in terms of m and k . (8 marks)
‘ QE = .1.
(b) Gwen that E P 4 .
(i) Express E in terms of k .
(ii) Let {LE=r . Express A7; in terms of r and m. AQ (iii) If 6 = 90° , use the above results to find the values of
k ,m and r . (12 marks) B4CEADD MATHS ——4 12 ) 8. Let f(x) = 5x2 + bx + c ,where b and c are real, 0 > 0 and
l
f — < O .
(2) (a) Show that the equation
f(x) = 0 has two distinct real roots.
(6 marks) (b) Let or and {3 (a < H) be the roots of f(x) = 0. (i) By expressing f(x) in factor form, show that
o < a < % < 13. (ii) If la% = '6 %l , ﬁnd the value of b and hence
the range of values of c .
(14 marks)
9. Let 0) (s5 1) be a cube root of l .
(a) (i) Prove that l + w + 0.22 = 0 .
(ii) Prove that for any integer k ,
1+ w3k+1+(w2)3k+1= 0
1+ w3k+2 + (w2)3k+2 .._.. 0 .
(6 marks) (b) Making use of the property of complex numbers: lcrl2 = on; ,
or otherwise, show that for any complex number 2 , I1—m?!= Izwl. (5 marks) (c) If 2 represents a variable point on the Argand diagram and c is
a positive constant, what kind of curves does the equation [1  co?! = c
represent ? Sketch the locus of 2 on the same diagram for each
of the possible values of c.) when 0 = % . (9 marks) ‘ 84CEADD MATHS l5
l3 ~.#F_AWW_M M. . . 10. In Figure 2 , ABCD is a square tin plate of side 2x/T’l m. I’QRS is a
square whose centre coincides with that of ABCD. The shaded parts are cut off and the remaining part is folded to form a right pyramid with
base PQRS . Let PQ = 2x metres and let the volume of the pyramid = V cubic metres. A I \
/ \
\ \\\\\\\\\\\\\\ (a) Show that the height of the pyramid is given by 2V1 — E metres. Hence express V as a function of x . (8 marks) (b) Find the stationary points of the graph of V . Find the equations of the tangents to the graph at the stationary points and at x — Hence sketch the graph for 0 < x < l . (12 marks) 84CEADD MATHS l~6 14 ) 11. in Figure 3 , AB is a railway 50km long. C is a factory )1 kilometres
from B such that LABC = 90° . Goods are to be transported from C
to A . The transportation cost per tonne of goods across the country
by truck is $2 per km, whereas by railway it is $1 per km. C A Figure 3
' l
A P B (a) Let P be a point on the railway , LPCB = 0 , and let $N be the
total transportation cost for l tonne of goods from C to P and then to A . Find N in terms of 6 and h.
(4 marks) (b) If h = 50 , show that the least transportation cost for 1 tonne of
goods from C to A is $50(\/§ + l). l
(7 marks) L
(c) (i) Suppose h > 50¢? Show that tan6 < \/—13 ,and ’
deduce that %% < O for all possible values of 0 . (ii) If h = 200 , what route should be taken so that the transportation cost is the least? §
(9 marks) 3 END OF PAPER ‘ 84CEADD MATHS —7
15 WWW... ~ HONG KONG EXAMINATIONS AUTHORITY HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 1984 Bﬁﬂu¥i$ Eit:
ADDITIONAL MATHEMATICS PAPER II 11.15 am—1.15 pm (2 hours) This paper must be answered in English Answer ALL questions in Section A and any THREE
questions from Section B. All working must be clearly shown. B4CE—AOD MATHS ”—1 16 ) SECTION A (39 marks)
Answer ALL questions in this section. I. In the expansion of (x2 + %)a , where a 95 0 , the coefficient of
x' is denoted by Br . Find the value of a if B7 = 4310 . (5 marks) 2. Prove by mathematical induction that, for all positive integers n ,
4n3 — n is divisible by 3. (6 marks) 3. The slope at any point (x , y) of a curve is given by Lil = 4sin2x + 1 .
dx If the curve cuts the xaxis at x = 12'— , ﬁnd the equation of the curve. (6 marks) 4. The area of the triangle bounded by the two lines x + y = 4 and
x — y = 2p and the y—axis is 9 . Find the two values of p . (6 marks) 84CEADD MATHS “—2
17 Making use of the derivative of tan30 , ﬁnd Itanzeseczodf). 1! Hence evaluate J 3 tan40 d0 0
(8 marks) Given the equation x2 + y2 — 2kx + 4ky + 6k2 — 2 = 0. (b) (a) Find the range of values of k so that the equation represents a
circle with radius greater than i .
(b) Find the locus of the centre of the circle as k varies within the
range in (a).
(8 marks)
SECTION B (60 marks)
Answer any THREE questions from this section.
Each question carries 20 marks.
(a) Prove that ~1— + ——————3 = ————————~—————3xa + 9x2 *12x + 4
x3 (2 — 3x)2 9x5 —12x4 + 4x3
2 3 2 _
Hence ﬁnd the value of I iii—W dx
1 9x5 —12x4 + 4x3
(7 marks) (i) Using the substitution u = sin (13 , ﬁnd J —cQ%£d¢ .
sin 4) (ii) Using the substitution x — tan¢ and the ”result of (i), evaluate ~. 1
[@d
" x J?
(13 marks) 84CEADD MATHS I—3 18 l 8. If you attempt this question, you should refer to the separate supplementary
leaﬂet provided. (a) Find the general solution of the equation sin20 + sin89 = sin50 . (6 marks) (b) Let y = sinx + 2cosx . Complete Table 1 on the separate answer
sheet provided and use the data to plot the graph of y = sinx + 2cosx . By adding two suitable straight lines to the graph, ﬁnd the
solutions of the equations (i) Ssinx + IOcosx = 11 , (ii) sinx + 2cosx = ﬁ + 2 . Give your answers correct to the nearest __7_r__ 200 '
(14 marks) 84—CEADD MATHS ”—4
l9 ., ﬁ........w.aum m xwam—mmmm , ,WMM ti
i WI Given the curve C : x2 + 4y2 = 4 and the point P(0 , 3). (a) I, is a line of variable slope m through P. If L cuts C at
two distinct real points, ﬁnd the possible range of values of m If I, touches C, what are the possible values of m 7 Hence write down the equations of the two tangents from
P to C . (10 marks) (b) Q(2cosG , sin6) is a point on C . Find by differentiation
the gradient of C at Q and hence show that the equation of the tangent T at Q is xcoso + ZysinB = 2 . Express the distance from P to the tangent T in terms of 0 . Find the distance when ' :21
(1) 9 2: .. . _ 1 11 5116 — — . ( l I 3 Interpret case (ii) geometrically. (10 marks) 84~CEADD MATHS “—5 20 "" '—"—'————————‘—_———V_—’___——————————__ 10. ) (3) Use the substitution x = asindz to show that a 2
Ix/azx2 dx = 1m
a T (5 marks) (b) Figure 1 shows two semicircles APB and AQB with a common
centre C(O , b) and equal radii a . AB is parallel to the xaxis. (i) Show that the equation of APB is and that of AQB is (ii) The region bounded by the two semicircles is revolved
about the x—axis to generate a solid (called an anchor—ring).
Use the result in (a) to prove that the volume of the anchor—
ring is Zuzazb . (8 marks) 84CEvADD MATHS ”—6 21 , ..__... ._._._',' “varM .m « WWW.“ . WV . h m (C) i A sweet has the form of an anchor—ring with a = 2 mm and b = 8mm. Write down its volume in terms of 1r . The sweet is now dropped into water and it dissolves with a rate
of change of volume given by _d_K = —321r2 (2 — t) mm’lh,
dt . 3 . . .
where V is the volume in mm , t 15 the t1me in hours. Find V in terms of t and hence find the time required to dissolve eet corn letely.
the whole sw P (7 marks) 84—CE—ADD MATHS ”7 ll. i In Figure 2 , ABC is a triangle with [.A = 0 . P is a point on AB
such that PA = PB = PC = Q. R and Q are points on AC and BC,
respectively, such that LQPC = LRPC = x . (a) B A R C Figure 2 Show that PR = —,’Z—5i“—‘i—
s1n(x + 0) (4 marks) (b) Find LPCQ in terms of 6 and hence ﬁnd PQ in terms of Q , (C) (d) x and 6 .
(4 marks)
_ Ezsinﬂcosﬂsian
Sh tl t th f AP R — —_——
ow 1a 6 area 0 Q 25in(x + 0)cos(x  0) ’
and show that it can be expressed as
stin26 ( sin20 )
—— 1 — ———————— ............................
2 sin2x + 51112!) (*)
(4 marks) (i) If 0 =18: , find the possible range of values of x . Hence use (as) to deduce the maximum area of APQR and
express it in terms of $2 . (ii) If 0 = ﬂ , what is the possible range of values of x ? Express the maximum area of APQR in terms of 32 . END or PAPER (8 marks) 84CEADD MATHS ”—8 23 ﬁAgﬁ—q: awﬂvﬂw%wwmﬂ_hw  s. . E] If you attempt this question, fill In the datails in the ﬁrst three boxs above and tie this aheat into your answer book. Total Mnlks
on [hl‘i page candid'te Number m M 3.0:) HONG KONG EXAMINATIONS AUTHORITY
HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 1984 Table I IVE—:(I‘ﬁﬁ)
ADDITIONAL MATHEMATICS PAPER II E If! 7m £5! E (SUPPLEMENTARY LEAFLET) y = sinx + 2cosx I. :IIIIIIIIIIIIIIIII Answers (0 (ii) BACEADD MATHS ll—SUPP. 2 24 MCEADD MATHS I«SUPP. I 25 1984 Addilimml Mathemarirs I In 101 11. (a) 44;“ %1 (b) (3 — 4m)i + (3m ~2),' 12cm3/s
(a) 41+381‘ l = 4.l_
(b) RHZ) 3125 Reg + %) = 41 (correct to the nearest integer) *1 < x ~< 3
0») \ﬁ
= 2
AP 3h
(a) ,1‘.‘_'" ﬁ__w
Wm"? 2m + 2)(kr+ k + t (h) (i) Lgk—i + §1 (ii) ri +r(l +m)j =2
(111) r 5
m=k=l
(b) (ii) b=S
5
0<c<4 (a) V=§xz 1—17 3
g 123
(h)(0,0),(5175\/§)
= =128 =1
V 0,V 75%.): (a) N = Zhsecﬂ + (50  htana) (c) (ii) Goods should be transported
directly from C to A by
truck . 102 1984 A dililimml Math ematic‘s ll 10. =3xsin2x~§l'—
2 =1 or 5 tarr39+c (a) 1 < k < 1
(b) The locus is a line segment with end points ('I , 2) and (l . ‘2)
excluded. (20% (b) (i) —35i’n3¢ +c
(11) 82x/i (a) 0=% or 611:1")
n=0,il,:t2...t (b) x=% , y=2V206
x=%.y=2.121
(ii) 0 , 23—3 (3) m > J2 or m < —\/2 m=£c\/§
y=ﬁx+3 , y=—\/§x+3 (b) d = Gsiiid A
«3371111 a'+ 1
(i) 4 (ii) 0 . 1’ lies on the tangent (c) 64"2 mm3 V = 1613:1 — 641121 + 64"2
2 hours ll. (b) LPCQ=£ — .9 PQ =
(d) (i) (ii) 2
Qcosﬂ
cos(x  0) 0<x<1r~20 21nd
0<x<1r~2¢ . 21
Maxunum area = —
2(1 + J?)
o < < l
x 6
Maximum area = —22_¢§_
4N? + 1) 103 ...
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 Marketing, Natural number, HONG KONG EXAMINATIONS AUTHORITY

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