Secondary School Mathematics & Science Competition. Mathematics. Date: 1 st May, 2013

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1 Secondary School Mathematics & Science Competition Mathematics Date: 1 st May, 01 Time allowed: 1 hour 15 minutes 1. Write your Name (both in English and Chinese), Name of School, Form, Date, Se, Language, Subject and Candidate Number in the spaces provided on the MC nswer Sheet and the Part nswer Sheet.. When told to open this question paper, you should check that all the questions are there. Look for the words END OF PPER after the last question.. nswer LL questions in Part. You are advised to use an H pencil to mark your answers on the MC nswer Sheet. 4. You should mark only ONE answer for each question in Part. If you mark more than one answer, you will receive NO MRKS for that question. 5. Part consists of Sections (1), () and (). nswer NY ONE section. nswer any FOUR questions from your chosen section. 6. For Part, answers should be an eact value or mathematical epressions unless otherwise specified. 7. No marks will be deducted for wrong answers. 8. The diagrams in the paper are not necessarily drawn to scale.

2 FORMULS FOR REFERENCE ) ( ) ( ) ( ) ( ) ( ) ( tan tan 1 tan tan ) tan( ) ( ) (

3 PRT nswer all questions. Choose the best answer for each question. 1. If and are the roots of the quadratic equation C. D. or, find the value of. Which of the following equations have two distinct real roots? I. ( 1) 0 II. ( 1)( ) ( 1) III I and II only. I and III only C. II and III only D. I, II and III. Let f be a cubic polynomial with leading coefficient and 1 f 0.5 f 1 then f. ( 1)( 1)( 1).. ( 1)( 1)( 1). C. ( 1)( 1)( 1). D. ( 1)( 1)( 1). f,

4 4. The figure shows the graph of y a b c. Which of the following is true? y O. a 0, c 0, b 4ac 0. a 0, c 0, b 4ac 0 C. a 0, c 0, b 4ac 0 D. a 0, c 0, b 4ac 0 5. Let f () be a linear function in. It is known that f 01 f 014, f 015 f 016 and Which of the following statements is true?. f f C. f D. f f. 6. If log = a and log = b, then log 750 =. a b +.. a + b +. C. a b +. D. a + b +. 4

5 7. Which of the following may represent the graph of y a a, where a 0 and a 1?. y 1 O 1. y 1 O 1 C. y 1 O 1 D. y 1 O 1 5

6 8. If log log 1 log , then C D Find the remainder if the polynomial is divided by C. 4 D Let P be a polynomial. If k is a factor of which of the following polynomial(s)? I. P II.. I only P III. P. III only C. I and II only D. II and III only P, k MUST also be a factor of 6

7 11. Given that a and b are constants. If the polynomial a b 5 is divided by, the quotient is Q () and the remainder is 1( 1). Find Q () C. 7 D If a, b and c are rational numbers, which of the following MUST be correct? I. If ac bc, then a b. II. If III. If. I only. II only C. III only D. II and III only a b, then b a. a b, then b a. 1. Two roots of the equation m n m 0 are 4 and 7. The third root is.. C. D

8 14. If bc a is a non-zero constant, then. a varies directly as b and c.. a varies inversely as b and inversely as c. C. a varies directly as b and inversely as c. D. a varies directly as c and inversely as b. 15. Let a, b, c and d be non-zero real numbers. If a : c = b : c = d : b = : 1, then d : a =. 1 : 1.. : 1. C. 4 : 1. D. 8 : In the figure, a semi-circle with centre O is shown. E and CD are straight lines. EOD 45. If = OD, find ED. E C O D.. C. D

9 17. In the figure, a circle with centre O is shown. and CE intersect at D. It is given that OD, O 6 and CD 0. Find the length of CE. C D O E C. 8 D. 11 9

10 18. In the figure, two different circles touch internally at P. C is tangent to circle PMN at. QR is tangent to circle PC at P. Which of the following is/are correct? R P N M Q C I. P CP II. PC PC III. PM PN. I only. III only C. I and III only D. II and III only 19. Let O be the origin in the rectangular coordinate plane. point is reflected about the y-ais to point. If the slope of O is m, then the slope of O is. 1. m. 1. m C. m. D. m. 10

11 0. In the figure, two straight lines a by c 0 and p qy r 0 intersect on the negative -ais. Which of the following must be true? I. bc 0 II. qr 0 III. IV. ar cp br cq. I and III only. I and IV only C. II and III only D. II and IV only 1. Find the acute angle between two straight lines y 0 and y C. 60 D. 75. There are m values of p and n values of q. It is given that p > q and the mean of these m + n p q values is greater than. Find the median of these m + n values.. p. q C. p q D. Cannot be determined 11

12 . Trapezium CD is shown in the figure, where = 1.5, C = 4, CD = 17.5, D =. Find the area of CD. D C.. 4 C. 5 D circle is inscribed inside a rhombus. The lengths of diagonals of the rhombus are 0 and 40. Find the radius of the circle C. 15 D The figure shows a shaded region in the rectangular plane, with vertices at (, 0), (5, 0), (5, 6), (0, 6), (0, ), and (, ). line y m is to be drawn to divide the area of the shaded region in half. Find the value of m. y C. 6 7 D

13 6. In the figure, CDEFGH is a unit cube. K is a moving point on the diagonal H such that a straight line pasg through K is perpendicular to H. This straight line intersects the faces CGF and DHE at P and at Q respectively. Let K and PQ y. Find the value of when y is maimized. E H F Q G K P D C.. C. 1 D. 1

14 7. In the figure, circles marked,, C, D, E, F and G are joined by line segments to form a star. Peter fills the numbers 7, 8, 9, 10, 11, 1 and 1 in the circles in a certain order. He calculates the sum of two numbers at the end of each line segment and find that the sums obtained from, C, CD, DE, EF, FG and G form an arithmetic sequence. Find the sum of this arithmetic sequence. F C D E G C. 140 D The graph of y a b c d, where a, b, c, d are real numbers, has four distinct -intercepts. If one of the -intercept is 0, which of the following must be non-zero?. a. b C. c D. d n 9. Let n be a positive integer. If log 18 a, then log log 9 4 log n. na.. na. C. D. na 1 a. na. 1 a END OF PRT 14

15 PRT nswer NY ONE SECTION from Sections (1), () or (). SECTION (1) nswer any FOUR questions. 1. [rithmetic and Geometric Sequences and their Summations] (a) Find the sum to infinity of geometric series ( 1) ( 1) Epress the answer in the form a b, where a and b are integers. ( marks) (b) If the geometric series 4 4 k ( 4 k) ( 4 k) can be summed to infinity, find the range of values of k. ( marks). [rithmetic and Geometric Sequences and their Summations] Consider an arithmetic sequence a, a, 1 a, a,, 8 where all terms are distinct. The terms a 1,a5 and a 8 form the first three terms of a geometric sequence, and a a a 148. Let d be the common difference of the arithmetic sequence (a) Epress a 1 in terms of d. (b) Find the 5 th term of the geometric sequence. (1 marks) ( marks). [Equations of Circles] circle touches both -ais and y-ais, and passes through the point (4, ). Find the two possible equations of the circle. (4 marks) 15

16 4. [Locus] point P on the rectangular plane is equidistant from the straight line L : y = 1 and a fied point (, 1). (a) Find the equation of locus of P. (b) Hence, find the range of values of y in the above equation. ( marks) ( marks) 5. [Probability, Permutation and Combination] (a) fair die is rolled si times. Find the probability that each of the si different numbers will appear eactly once. (Epress the answer correct to significant figures.) ( marks) (b) fair die is rolled seven times. Find the probability that each of the si different numbers will appear at least once. (Epress the answer correct to significant figures.) ( marks) 6. [Probability, Permutation and Combination] large shipment of tablets are divided into batches. Each batch is subject to inspection by the following double sampling inspection plan: I. random sample of 10 tablets is drawn from a batch and inspected. If none of the tablets is defective, the batch is accepted. II. If or more tablets are defective, the batch is rejected. If 1 tablet is defective, another sample of 10 tablets is drawn from the batch and inspected. If none of the tablets from the nd sample is defective, the batch is accepted; otherwise it is rejected. It is known that the shipment contains 5% defective tablets. Find the probability that (a) a batch is accepted. (Epress the answer correct to 4 significant figures.) (b) the 4 th batch is the first to be rejected if the tablets are inspected batch by batch. ( marks) (Epress the answer correct to 4 significant figures.) ( marks) END OF SECTION (1) 16

17 SECTION () nswer any FOUR questions. 7. [inomial Epansion, Eponential and Logarithmic Functions] Let n be a positive integer. If the epansion of possible values of n. 1 n has a constant term, find all (4 marks) 8. [inomial Epansion, Eponential and Logarithmic functions] Solve the following equations: (a) e e 4 0 ( marks) (b) log log ( marks) 9. [Differentiation] 5 The graph of y is shown below. Points and are turning points. (a) Find the coordinates of and. (b) The equation k. 5 4 k ( marks) has three distinct real roots. Find the range of values of 17

18 ( marks) 40. [Differentiation] solid circular cylinder with height h cm and radius r cm has a total surface area of Its volume is V cm. (a) Epress V in terms of r. (b) Find the maimum possible value of V. (Epress the answer in terms of.) 150 cm. ( marks) ( marks) 41. [Integration] d Suppose F d correct to 4 decimal places.) 4. [Integration] 5 ln. If F , find the value of F. (Epress the answer (4 marks) Let (a) Find the values of and. ( marks) (b) Evaluate. 1 d ( 1) ( marks) END OF SECTION () 18

19 SECTION () nswer any FOUR questions in this section. 4. [inomial epansion, Eponential and logarithmic functions] Let n be a positive integer. If the epansion of possible values of n. 1 n has a constant term, find all (4 marks) 44. [Matri] Consider the following system of homogenous linear equations in, y and z: ky 5z 0 y z 0, 5 y kz 0 where k is a real number. If the system has non-trivial solutions, find the value(s) of k. 45. [Differentiation] (4 marks) solid circular cylinder with height h cm and radius r cm has a total surface area of Its volume is V cm. (a) Epress V in terms of r. (b) Find the maimum possible value of V. (Epress the answer in terms of.) 150 cm. ( marks) ( marks) 19

20 46. [Differentiation] statue of 0 m high stands on a pedestal. Nelson stands at m from the base of the pedestal and looks at the statue. He wants to find out how far away he should stand from the base of the pedestal to make the statue appear the largest, i.e., the angle subtended by his lines of sight to the bottom and the top of the statue is the largest. 0 m 6 m m It is given that Nelson s eye-level is 6 m below the bottom of the statue. (a) Epress tan in terms of. ( marks) (b) Find the value of when tan is maimized. (Epress the answer in surd form.) ( marks) 47. [Integration] Let F be an anti-derivative of values of a, b, and c. e, where 0 7. If d c F( b) F( a) e, find the (4 marks) 0

21 48. [Integration] Let (a) Find the values of and. ( marks) (b) Evaluate 1 d. ( 1) ( marks) END OF SECTION () END OF PPER 1