HKCEE_AMath_1984_Paper 1 & 2.pdf - ID A straight line...

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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 mid-point 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? = lJZCOSBQ-sin39dw 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 83-CE~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. 84-CE-ADD 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 2|xl — 1 | < 2 . (5 marks) 84-CE-ADD 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) 84-CE-ADD 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) B4-CE-ADD 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?!= Iz-wl. (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) ‘ 84-CE-ADD MATHS l-5 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) 84-CE-ADD 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 $50(\/§ + l). l (7 marks) L (c) (i) Suppose h > 50¢? Show that tan6 < \/—13 ,and ’ deduce that %% < O for all possible values of 0 . (ii) If h = 200 , what route should be taken so that the transportation cost is the least? § (9 marks) 3 END OF PAPER ‘ 84-CE-ADD MATHS |—7 15 WWW... ~ HONG KONG EXAMINATIONS AUTHORITY HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 1984 Bfiflu¥i$ 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. B4-CE—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 x-axis 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) 84-CE-ADD 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 —c-Q%£d¢ . sin 4) (ii) Using the substitution x — tan¢ and the ”result of (i), evaluate ~. 1 [@d " x J? (13 marks) 84-CE-ADD 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—CE-ADD 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~CE-ADD MATHS “—5 20 "" '—"—'————————-‘—-_—-——V_—’___———-———————-__ 10. ) (3) Use the substitution x = asindz to show that a 2 Ix/az-x2 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 x-axis. (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) 84-CEvADD MATHS ”—6 21 , ..__... ._._._',' “var-M .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) 84-CE-ADD 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) BA-CE-ADD MATHS ll—SUPP. 2 24 M-CE-ADD 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) Lg-k—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. =3x-sin2x~§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) 8-2x/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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