HKCEE_AMath_1983_Paper 1 & 2.pdf - ii HONG KONG...

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Unformatted text preview: fiififiéflfii HONG KONG EXAMINATIONS AUTHORITY ~J‘LAE5’Ffiifi'Iiflfif’9‘ HONG KONG CERTIFICATE OF EDUCATION EXAMINATION I983 III in £5! E ADDITIONAL MATHEMATICS E‘II =3 -- PAPER | :4~Mf'7‘£ Two hours J;-’FAB§E~I<5}E-I—B#E +5? 8.30 a.m.—10.30 a.m. *‘fiififi‘fimfi RIF-2% This paper must be answered in English Answer ALL questions in Sectian A and any THREE questions from Section B. All working must be cleariy shown. SECTION A (40 marks) Answer ALL questions in this section. 1. Determine the range of values of A for which the equation x"+4x+2+ A(2x+ I) = 0 has no real roots. (5 marks) 2. Given that a, b, c are in arithmetic progression and the pusitive numbers x, y, z are in geometric progression; prnve that (b—€)logx + (c-Iz)Iogy +(I1—b)Iogz ‘—‘ 0. (6 marks) 83-CE»ADD MATHS MEI—1 3. Figure 1 show: an isosceles triangle ABC with A BC = 21 and AB = AC. The perimeter of the triangle is 2 metres. The triangle is revolved about BC so as to form a solid consisting of two cones with a common base of radius AD. Express the volume of this solid in terms of x. Hence find the value of x for which this volume is a maximum, I ( 6 marks) x D x Figure l 4. Expand (l + ax)‘(l .. 4x)j in ascending powers of I up to and including the term containing x1. Given that the coefficient of x is zero, evaluate the coefficient of x’. (7 marks) 5. Solve the inequality lx(x - 2)] < l . (8 marks) 6. The complex number 2 satisfies the condition lz — (3 + i)l = lz ~(S + Sill. lf 2 = x + iy, where x and y are real, find and simplify the relation between x and y. Find also the values of x and y for which lzl is a minimum. (8 marks) SECTION B (60 marks) Answer any THREE questions from this section, Each question carries 20 marks. 7. (a) Using De Moivre‘s theorem, or otherwise, show that (i) cos40 = cos”) — 6cos’0 sin’9 + sin‘9 (ii) sin49 = 4cos’ll sinB — 4cos€ sinJH . (5 made) (b) Using (a), or otherwise, show that ”"49 = 4tan0 — 4tan30 . I ~ 6tan’0 + tan‘G (3 marks) (C) By putting x = tan!) and using the result of (b), show that the equation x‘+4x’—6x’—4x+l=0 ..................... (t) can be transformed to tan40 = l. ....................................... (u) Find the general solution of equation (it) in terms of 1r. Hence deduce the four roots of (t) leaving your answers in terms of 1r . (12 marks) 33-CE-ADD MATHS ”El—2 8. Let (a) (b) (C) (b) rm = x’ +ax’ + bx — 72. Given that x = 4 is a double root of :_f = 0I find the values of a and b. (5 marks) x I Show that f(x) can be expressed in the form (x + p)3 + q, and find )2 and (1. Hence find the three roots of f(x) = 0 . (9 marks) Represent the three roots XI, x1, x; of {(x) = 0 on an Argand diagram by the points A, H, C, respectively, x, and Jr2 being complex conjugates and O < arg (x,) < n . By considering triangle ABC, or otherwise, determine arg (x2 — 4) . x, - 4 (6 marks) Prove, by mathematical induction, that for all positive integers n , l><2+2X3 +3x4 +...+n(n +1) = 1311(n+l)(n+2). (6 marks) On a battle field, cannon-balls are stacked as shown in Figure 2. For a stack with n layers, the balls in the bottom layer are arranged as shown in Figure 3 with n balls on each side. For the second bottom layer, the arrangement is similar but each side consists of (n - 1) balls; for the third bottom layer, each side has (n — 2) balls, and so on. The top layer consists of only one ball. ‘ Figure 3 (i) Find the number of balls in the r-th layer counting from the top. (ii) Using the result of (a), or otherwise, find the total number of cannonvballs in a stack consisting of n layers. (iii) If the time required to deliver and fire a cannon-ball taken from the nth layer is 2 7 layer. minutes, find the time required to deliver and fire all the cannon-balls in the r-th Hence find the total time needed to use up all the cannon-balls in a stack of 10 layers. (14 marks) Bfl-CE—ADD MATHS llEl—J ‘~— g... 10. In Figure 4, PQR is an isosceles triangle with P i is fi' E m we QR = 2" N ‘5 the mld‘mm of QR' HONG KONG EXAMINATIONS AUTHORITY L and M are variable points on PQ and PR , respectively, such that LM I QR . Let _‘ ‘ #J‘LA:$§7€:I’P$Q% HONG KONG CERTIFICATE OF EDUCATION EXAMINATION I983 LM = x. [if-t 1m 51 2% ADDITIONAL MATHEMATICS 1 gig: PAPER II 2' 1 l 1 3 (a) Find x such that the area of ALMN is a maximum. i 1 s . . (8 marks) i (b) If the figure is revolved about PN, find 1 . I so that the volume of the cone Q N R l :4‘5‘1’24. generated by ALMN is a maximum. In Two hours (6 mks) flsL‘L‘E L’Fi‘mflfi't‘i’i}EET’r-~Ew’%-I—IL‘/} 11.15 a.m.—l.15 p.m. *aemamauteas This paper must be answered in English (c) Show that the volume of the cone generated by revolving the ALMN specified in (a) about PN is only -§—; of the volume generated in (b). (6 marks) I]. Figure 5 shows a rail POQ with LPOQ = 120°. A rod AB of length Wm is free to slide on the rail with its end A on 01’ and end B on Ca. Let 0A = x metres and OH = y metres. Answer ALL questions in Section A and any THREE questions from Section B. All working must be clearly shown. (a) (i) Find a relation between x and y and hence find the value of y when x = 2. (11) Find dx . Figure 5 Given that x and y are functions of time t (in seconds), show that SECTION A (40 marks) Answer ALL questions in this section. u:-2x+gsi_x_ d1 .r+2ydt' (10 marks) 1. A triangle has vertices P(k, ~l), Qt7, 1]) and R(1, 3). Given that the area of the triangle is (b) The end A is pushed towards 0 with a uniform speed of %m/s. When A is at a distance 20 Units. find the two Values of k. of 2 metres from 0. find the speed of the end B. (5 marks) (4 marks) 2. Use the substitution :1 = x2 to find the indefinite integral (6) Suppose the perpendicular distance from 0 to the rod is p metres. Show that [x sin1(x1)dx . = L! .1 _ (5 marks) p 2 J; Hence find fig when X = 2. 3. Use the substitution :1 = l + 3x’ to evaluate (6 marks) 1 J x3 1 + 3x1 dx . o (5 marks) END OF PAPER 93 CE ADD MATHS ”5,4 BS-CE-Add Maths IIIE)—1 Figure I shows the curve y = x1 - 4x. A straight line L intersects the curve at the points P0, '3) and Q(S, 5). Find (a) the equation of L . and (b) the area of the shaded regions Figure l (6 marks) ' — = 0 and Find the eqUations of the two lines which are both parallel to the line 3:: 2y tangent to the ellipse 4x2 + y2 = “3- (6 marks) A circle C passes through the point P0, 2) and the points of intersection of the circles C.:x’+y’-3X+2y-2=0 and C1zx2+y’+x+3y—10=0. Find the equations of (a) the circle C. t C at l’. and (b) the ”“3““ ° (6 marks) Show that sin’nfl -- sin’mt} = sin (n + m)65in(r: ~ mm. Hence, or otherwise, solve the equation sin239 - sinlm - sinO = 0 for 0 < 9 < 1!. (7 marks) 83-CE-Add Maths "(El—2 sccrron n (60 marks) Answer any THREE questions from this section. Each question carries 20 marks. 8. Figure 2 shows a tent consisting of two inclined square planes ABCD and EFCD I) standing on the horizontal ground ABFE. The length of each side of the inclined planes is a. N is a point on CF such that AN J. CF. Let NF = )c(ae 0), LCFB = 6 and M be a point on BF such that NM 1 BF. (a) By considering AABM, express AM in terms of a, x and 0. (4 marks) A (h) By considering AANF, express AN in terms of a, x and 0. Figure 2 (5 marks) (c) Using the results of (a) and (b), or otherwise, show that x = 2acos’9 . (5 marks) (d) Given that x = % , find (correct to the nearest degree) the inclination of AN to the horizontal . (6 marks) 9. A(1, -2) and 3(4, 4) are two points on the parabola y2 = 4):. P is a point on the line AB such that AP : PB = l : k. A line L, through A is perpendicular to the tangent at A. Another line L; through E is perpendicular to the tangent at B. L. and L, intersect at N. Let 0 be the origin. (a) Find the coordinates of the point N and the slope of 0N , (8 marks) (b) (i) Express the slope of 01’ in terms of k. (ii) Express tan LPON in terms of k when (l) LPON is acute, (2) LPON is obtuse. (7 marks) (c) Find the value of k in each of the following cases : (i) when LPON = 45°; ii when OPN is a strai t line. ( ) Eh (5 marks) aa-cE-Add Maths "(El-3 \crozwam«wuwmn m... Mum ....,..._ __._... —s M m. . ID. A straight line through the point R(_1, —l) has a variable slope m. It intersects the circle x1 + y’ = l at A and B. Let P be the mid-point of AB. (21) 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 I’. (5 marks) 11‘ (a) Show that M = 2c0529 + I. stnfl By putting 9 = i— + t» in the above identity, show that cosBQ - sinSQ = 1, Zsin2¢. cost» + sin¢ (7 marks) (b) Using the substitution at = 3% -> u, show that 5 1 c0539 d¢= a sin3u du. o cosdh + sin¢ 5 cost: 1‘ sinu 2 Hence, or otherwise, show that I a 1 cosSQ M, = L 1cosSQ—sin39ddj' 0 coszfi + sino 2 a cosd) + sind) (8 marks) (0) Using the results in (a) and (b), evaluate n 5 cos} 9 dd? . a cosnb + sind: (5 marks) 12. let {(x) be a function of x and let k and s be constants. (a) By using the substitution y = x + ks, show that r (kn): {(x +ks)dx = J I(x)dx. o ks Hence show that, for any positive integer n, ‘- HS [[Kx) + I(x +5) + + {(x + (n ~ Ilstx = [ f(x)dx. o o (10 marks) 1 - 2 dx . . . = . (h) Evaluate -—=—.—_—. by ustng the substitution x smfi. o \/l_ _ X, Using this result together with (a), evaluate 1 1" 1 + 1 + _-_l.__—— + + ‘ dx #5 T"f#7 —’JI _ 7 . I (1“ x/l—-(x+il;) \/l“(x+—Z—) \fl‘(x+"2"l) 0 2n (10 marks) END OF PAPER 83-CE-Add Maths II(EI~4 HONG KONG EXAMINATIONS AUTHORITY HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 1984 Ififlfl¥i$ Effig— ADDITIONAL MATHEMATICS. PAPER | 8.30 Elm—10.30 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. 84-CE-ADD MATHS I——1 9 1983 NUMERICAL ANSWERS Addilional Malhe’malirs I 1. 3. 10. ll. -2<)\<—1 —42 x351 and 1—fi<x<l+\/7 X+2)'—10=0 v=2.y=4 (c) a=—(——)—4”121",n=0,il.t2. 1r 5n 91r 131r _ _‘ __ t ._._ t'c1n16,tan16,anl6,anl6 (a) a=—12,b=48 (b) p=-4.q=-8 x=6 or sin/3 (c) an; (X1 ::)= 120° 11 (b) (i) gun) (ii) é—n(n+1)(n+2) (iii) (r + 1) minutes 65 minutes (a) x = I (b) x = %r (a) (n x1 +1;1 +xy = Whenx=2,y=l. (ii) in = _2x + 2 dx x +2y (b) i m/S 8 (c) a; 1983 1. 2. 10. ll. 12. k: Additional Murhenmlits ll 30! —7 1 sin 2):1 + c 8 (b) 13 y = g—x i 5 (a) x2+y1+5x+4y—18=0 (b) 7x+8y—23=0 = _1r_ 1 21 8 0,10,2,10011r (3) AM = Vai + (2n - x)’ 00529 (b) AN = V112 + 4a,j 00529 ~Jri (d) 19° (8) N = (5 . 2) Slope 01' ON =% . = 4 - 2k (1)) (I) Slope of GP 4 + k .. _ 12 -12k [’0 — 1r _.___ (n) tanL N 28+k according as LPON is acute or obtuse . ___ 39 (c) (1) M (ii) k = 1 a = m — m2 m -1 () (1+m7'l+m1) (b) The circle 1:2 + y1 + x + y = 0 (C) ‘ 1 (b) 0‘1: N: N: 101 ...
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