ADDITIONAL MATHEMATICS

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1 00-CE A MATH HONG KONG EXAMINATIONS AUTHORITY HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 00 ADDITIONAL MATHEMATICS 8.0 am.00 am ½ hours This paper must be answered in English. Answer ALL questions in Section A and an FOUR questions in Section B.. Write our answers in the answer book provided. For Section A, there is no need to start each question on a fresh page.. All working must be clearl shown.. Unless otherwise specified, numerical answers must be eact. 5. In this paper, vectors ma be represented b bold-tpe letters such as u, but candidates are epected to use appropriate smbols such as u in their working. 6. The diagrams in the paper are not necessaril drawn to scale. 香港考試局保留版權 Hong Kong Eaminations Authorit All Rights Reserved CE-A MATH

2 FORMULAS FOR REFERENCE sin A ± B sin A cos B ± cos Asin B cos A ± B cos Acos B sin Asin B tan A ± tan B tan A ± B tan A tan B sin Acos B sin A B sin A B A B A B sin A sin B sin cos A B A B sin A sin B cos sin A B A B cos A cos B cos cos A B A B cos A cos B sin sin cos A cos B cos A B cos A B sin A sin B cos A B cos A B * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Section A 6 marks Answer ALL questions in this section. d. Find. d marks. Find d. Hint : Let u. marks 00-CE-A MATH

3 . Given a point A, 0. P h, k is a variable point on the circle C :. Let M be the mid-point of AP. a Epress the coordinates of M in terms of h and k. b Find the equation of the locus of M. marks. Find the constant term in the epansion of 8. marks 5. Simplif the following comple numbers : a i i, b i i marks 6. a If sin cos r cos θ for all values of, where r > 0 and π < θ π, find the values of r and θ. b Find the general solution of the equation sin cos. 5 marks 00-CE-A MATH Go on to the net page

4 7. P, 0 is a point on the curve sin. Find the equation of the tangent to the curve at P. 5 marks 5 8. Let a, b be two vectors such that a, b and the angle between a and b is 60. a Find a. b. b Find the value of k if the vectors a kb and a b are perpendicular to each other. 6 marks 8 9. Let p... *, where is real. B epressing * in the form a b c 0, find the 8 range of possible values of. 8 Hence find the range of possible values of. 6 marks 0. Two lines L : 5 0 and L : 0 intersect at a point A. Find the equations of the two lines passing through A whose distances from the origin are equal to. 6 marks 00-CE-A MATH

5 . Solve. Hence, or otherwise, solve. 6 marks. Prove, b mathematical induction, that... n n n n n for all positive integers n. Hence evaluate marks * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Section B 8 marks Answer an FOUR questions in this section. Each question carries marks.. The line m, where m 0, intersects the parabola at two distinct points A and B. a Show that m <. 8 marks b c Find, in terms of m, the coordinates of the mid-point of chord AB. marks Let L be the perpendicular bisector of chord AB. If L passes through the point 0,, show that 6m m m 0. Hence find the equations of L. 6 marks 00-CE-A MATH 5 Go on to the net page

6 . a Q R O P Figure a In Figure a, OPQ is a triangle. R is a point on PQ such that PR : RQ r : s. Epress OR in terms of r, s, OP and OQ. Hence show that if OR mop noq, then m n. marks b B Y G O In Figure b, OAB is a triangle. X is the mid-point of OA and Y is a point on AB. BX and OY intersect at point G where BG : GX :. Let X A OA a and OB b. Figure b i Epress OG in terms of a and b. ii Using a, epress OY in terms of a and b. Hint : Put OY kog. iii Moreover, AG is produced to a point Z on OB. Let OZ hob. Find the value of h. 00-CE-A MATH 6 5 Eplain whether ZY is parallel to OA or not. 9 marks

7 5. a R P O Q Figure a Figure a shows a pramid OPQR. The sides OP, OQ and OR are of lengths, and respectivel, and the are mutuall perpendicular to each other. i Epress cos PRQ in terms of, and. ii Let S, S, S and S denote the areas of OPR, OPQ, OQR and PQR respectivel. Show that S S S S. 6 marks b E H F A D G B C Figure b Figure b shows a rectangular block ABCDEFGH. The lengths of sides AB, BC and AF are, and respectivel. A pramid ABCG is cut from the block along the plane GAC. i Find the volume of the pramid ABCG. ii Find the angle between the side AB and the plane GAC, giving our answer correct to the nearest degree. 6 marks 00-CE-A MATH 7 6 Go on to the net page

8 6. Figure a shows the ellipse E :. The shaded region in the 5 9 first and second quadrants is bounded b E and the -ais. a Using integration and the substitution 5sinθ, find the area of the shaded region. 5 marks b A piece of chocolate is in the shape of the solid of revolution formed b revolving the shaded region in Figure a about the -ais. Using integration, find the volume of the chocolate. marks c Can Figure b Four pieces of the chocolate mentioned in b are packed in a can, which is in the shape of a right non-circular clinder see Figure b. The chocolates are placed one above the other. Their aes of revolution are parallel to the base of the can and in the same vertical plane. For economic reasons, the can is in a shape such that it can just hold the chocolates. Find the capacit of the can. marks 00-CE-A MATH 8 7

9 7. P is a point in an Argand diagram representing the comple number such that i. In Figure, the circle C is the locus of P. Imaginar C O Real Figure a Write down the coordinates of the centre and the radius of C. marks b c Q is the point on C such that the modulus of the comple number represented b Q is the greatest. Find the comple number represented b Q. 5 marks R is the point on C such that the principal value of the argument of the comple number represented b R is the least. Find the comple number represented b R. 5 marks 00-CE-A MATH 9 8 Go on to the net page

10 8. Let f be a polnomial, where 0. Figure 5 a shows a sketch of the curve f, where f denotes the first derivative of f. a i Write down the range of values of for which f is increasing. ii iii Find the -coordinates of the maimum and minimum points of the curve f. In Figure 5 b, draw a possible sketch of the curve f. 6 marks b In Figure 5 c, sketch the curve f. marks c Let g f, where 0. i In Figure 5 a, sketch the curve g'. ii A student makes the following note : Since the functions f and g are different, the graphs of f and g should be different. Eplain whether the student is correct or not. marks 00-CE-A MATH 0 9

11 Candidate Number Centre Number Seat Number Total Marks on this page If ou attempt Question 8, fill in the first three boes above and tie this sheet into our answer book. O 8 0 f Figure 5 a a iii continued O 8 0 b continued Figure 5 b O 8 0 Figure 5 c END OF PAPER 00-CE-A MATH 0

12 00 Additional Mathematics Section A.. c, where c is a constant h. a, k b a i b 6. a i, π b π nπ, where n is an integer a 6 b

13 or , 0. or < or <. 575

14 Section B Q. a m m m m * Since the line intersects the parabola at two distinct points, discriminant m m > 0 6m m < 8 8m 6m > 0 b Let the coordinates of A and B be, and, respectivel. m From *, m coordinate of mid-point m m coordinate of mid-point m m m m m m the coordinates of the mid-point are,. m m c Slope of the perpendicular bisector of AB m m m 0 m m m m m m m m 6m m m 0 m

15 m 6m 5m 0 m m m 0 m, or. From a, m < m 8 The equation of L is [ ].

16 Q. a sop roq OR r s Comparing coefficients with the epression OR mop noq, s r m and n r s r s s r m n r s r s b i!! a b OG!! a b 8 ii OY kog k! k! a b 8 Using a, k k 8 8 k 7 8! 8! OY a b 8 7 7! 6! a b 7 7 iii!! From b i, OG a b 8! a OZ 8 h Using a, 8 h 6 h 7

17 ZY OY OZ! 6! 6! a b b a! 7 As ZY λoa, so ZY is parallel to OA.

18 Q.5 a i, RP, PQ QR cos QR RP PQ QR RP PRQ ii, S, S S sin PRQ PRQ RP QR S sin S S S S b i Volume of ABCG height area base P Q R O A B C D E H G F

19 ii Using a, area of GAC 6 Let h be the perpendicular distance from B to plane GAC. Considering the volume of ABCG, 6 h h 6 Let θ be the angle between AB and plane GAC. h sinθ AB / 6 θ correct to the nearest degree

20 Q.6 a Area d 5 Put d 5 5sinθ. d 5 cosθ dθ π 5sin Area θ 5 cosθ dθ π 5 5 π π π cos θ dθ 5 cos d θ θ π 5 θ sin θ π π 5 π π sin π sin π 5π

21 b Volume π d 5 5 π d 5 9π π π c Capacit of the can Base area height 5π 6 60π

22 Q.7 a The centre is, The radius is. b The position of Q is shown below : Imaginar C Q O θ A, Real Let A denote the centre of C and θ be the angle between OA and the real ais. OA OQ OA AQ tan θ π θ 6 the comple number represented b Q is π π cos i sin. 6 6

23 c The position of R is shown below OR is a tangent to C : Imaginar C S O R π 6 A Real Let S be the point of contact between C and the imaginar ais. OR OS π π AOR AOS 6 π Angle between OR and the real ais π π 6 π 6 the comple number represented b R is π π cos i sin. 6 6

24 Q.8 a i f is increasing when 0 and 8 0. ii From Figure 5 a, f 0 at 0, and 8. As f changes from ve to ve as increases through 0 and 8, f attains a minimum at 0 and 8. As f changes from ve to ve as increases through, f attains a maimum at. iii f O 8 0 b O 8 0 f

25 c i g f. g f g O 8 0 f ii Differentiate with respect to : g f. Since f g, the graphs of f and g are identical. The student is incorrect.

IB SL REVIEW and PRACTICE

IB SL REVIEW and PRACTICE Topic: CALCULUS Here are sample problems that deal with calculus. You ma use the formula sheet for all problems. Chapters 16 in our Tet can help ou review. NO CALCULATOR Problems