# SEKOLAH MENENGAH ZON A KUCHING LEMBAGA PEPERIKSAAN PEPERIKSAAN PERCUBAAN SPM 2008

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1 SULIT 7/ Matematik Tambahan Kertas Sept 008 Jam Name :.. Form :.. SEKOLAH MENENGAH ZON A KUCHING LEMBAGA PEPERIKSAAN PEPERIKSAAN PERCUBAAN SPM 008 MATEMATIK TAMBAHAN Kertas Dua jam JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU This question paper consists of 5 questions.. Answer all questions.. Give only one answer for each question.. Write your answers clearly in the spaces provided in the question paper. 5. Show your working. It may help you to get marks. 6. If you wish to change your answer, cross out the work that you have done. Then write down the new answer. 7. The diagrams in the questions provided are not drawn to scale unless stated. 8. The marks allocated for each question and sub-part of a question are shown in brackets. 9. A list of formulae is provided on pages to. 0. A booklet of four-figure mathematical tables is provided.. You may use a non-programmable scientific calculator. This question paper must be handed in at the end of the examination. Question For Total TOTAL 80 Obtained Kertas soalan ini mengandungi 5 halaman bercetak 7/ 008 Hak Cipta Zon A Kuching [Lihat Sebelah SULIT

2 SULIT 7/ The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. b x = b ac a ALGEBRA 8 log a b = log log c c b a a m a n = a m + n a m a n = a m n (a m ) n = a nm 5 log a mn = log a m + log a n 6 m log a n = log a m log a n 7 log a m n = n log a m 9 T n = a + (n )d 0 n S n = [a ( n ) d] T n = ar n n n a( r ) a( r ) S n = r r a S r, r <, (r ) dy dv du y = uv, u v dx dx dx du dv v u u dy y, dx dx, v dx v dy dx dy du du dx CALCULUS Area under a curve b = y dx or a b = x dy a 5 Volume generated b = y dx or a b = x dy a GEOMETRY Distance = Midpoint x (x, y) = x ( x x ) ( y y ) y, y 5 A point dividing a segment of a line nx mx ny my (x, y) =, m n m n 6. Area of triangle = r x y ( ) ( ) x y x y x y x y x y x y x i yj r x y 7/ 008 Hak Cipta Zon A Kuching SULIT

3 SULIT 7/ STATISTICS x = N x = = = x fx f ( x x) = N f ( x x) f = x N x fx f x w I I i wi n n! P r ( n r)! n n! C r ( n r)! r! 7 i P(A B) = P(A) + P(B) P(A B) P(X = r) = r C p q n r n r, p + q = N F 5 m = L C fm P 6 I 00 P 0 Mean, = np npq z = x TRIGONOMETRY Arc length, s = r Area of sector, A = sin A + cos A = sec A = + tan A 5 cosec A = + cot A 6 sin A = sinacosa r 7 cos A = cos A sin A = cos A = sin A tan A 8 tana = tan A 9 sin (A B) = sinacosb cosasinb 0 cos (A B) = cos AcosB sinasinb tan A tan B tan (A B) = tan Atan B a sin A b sin B c sin C a = b +c bc cosa Area of triangle = absin C 7/ 008 Hak Cipta Zon A Kuching Lihat sebelah SULIT

4 SULIT 7/ Answer all questions. For Diagram shows the linear function f. x f f(x) n DIAGRAM (a) State the value of n. (b) Using the function notation, express f in terms of x. [ marks ] Answer : (a)... Two functions are defined by f : x x and gf : x x ax b, find the value of a and of b. (b)... g : x x x. Given that [ marks ] Answer :... 7/ 008 Hak Cipta Zon A Kuching SULIT

5 For SULIT 5 7/ x The function of p is defined as p(x), x h. x Find (a) the value of h, (b) p ( x ). [ marks ] Answer : (a).. (b)... Find the range of values of t if the following quadratic equation has no roots (t + ) x + 6x + = 0. [ marks ] Answer :... 7/ 008 Hak Cipta Zon A Kuching [ Lihat sebelah SULIT

6 SULIT 6 7/ 5 Given that and are the roots of the quadratic equation x x 7. Form the quadratic equation whose roots are and. For [ marks ] 5 Answer :... 6 Diagram shows the graph of a curve y = a(x + p)² + q that passes through the point (0, 5) and has the minimum point (, ). Find the values of a, p and q. y [ marks ] (0, 5) O (, ) DIAGRAM x Answer : p =... q =... 6 a =... 7/ 008 Hak Cipta Zon A Kuching SULIT

7 For SULIT 7 7/ 7 Find the range of values of x for which x(x ) 5. [ marks] 7 Answer :... x 8 Solve 7 x 9 [ marks ] 8 Answer :... 9 Given that lg 0 and lg 7, find, without using scientific calculator or mathematical tables, find the value of log. [ marks ] 9 Answer :... 7/ 008 Hak Cipta Zon A Kuching [ Lihat sebelah SULIT

8 SULIT 8 7/ th 0 The n term of an arithmetic progression is given by T n 5n. Find For (a) the first term and the common difference, (b) the sum of the first 5 terms of the progression. [ marks] Answer : (a). (b) The first three terms of a geometric progression are 968, 656, 87,.... Find the three consecutive terms whose product is 576. [ marks ] Answer :... 7/ 008 Hak Cipta Zon A Kuching SULIT

9 For SULIT 9 7/ Diagram shows the straight line obtained by plotting log 0 y against log 0 x. log 0 y (, h ) (0, 6) 0 DIAGRAM log 0 x The variables x and y are related by the equation y kx, where k is a constant. Find the value of (a) k, (b) h. [ marks ] Answer : (a) (b)... The coordinates of the vertices of a triangle PQR are P(, h), Q(, 0) and R(5, h). If the area of the PQR is 9 units, find the values of h. [ marks ] Answer : h =. 7/ 008 Hak Cipta Zon A Kuching [ Lihat sebelah SULIT

10 SULIT 0 7/ x y If the straight line is perpendicular to the straight line 5 p 0x y 0, find the value of p. [ marks ] For Answer :. 5 Given the vectors a i mj % % %, b % 8 % i % j and c % 5 % i % j. If vector a b is parallel to % % vector c, find the value of the constant m. ~ [ marks ] 5 Answer :.. 7/ 008 Hak Cipta Zon A Kuching SULIT

11 For SULIT 7/ 6 The diagram shows a parallelogram ABCD drawn on a Cartesian plane. y B A O x C D It is given that AB i j % % and BC i % % j. Find DIAGRAM (a) BD, (b). AC [ marks ] 6 Answer : (a)..... (b).. 7 Solve the equation sin 5cos cos for [ marks ] 7 Answer : / 008 Hak Cipta Zon A Kuching [ Lihat sebelah SULIT

12 SULIT 7/ 8 Given that sin x = 5 and 90 < x < 70, find the value of sec x. [ marks ] For Answer : The diagram 5 shows a semicircle of centre O and radius r cm. C A O B DIAGRAM 5 The length of the arc AC is 7 cm and the angle of COB is 69 radians. Calculate (a) the value of r, (b) the area of the shaded region. [Use π =.] [ marks ] Answer : (a).. (b).. 7/ 008 Hak Cipta Zon A Kuching SULIT

13 For SULIT 7/ 0 Find the coordinates of the turning points of the curve y = x + x. [ marks] 0 Answer : Given that y = m and m = x +. Find dy (a) in terms of x, dx (b) the small change in y when x increases from to 0. [ marks ] Answer : (a).. (b).. Find x dx [ marks ] Answer :.. 7/ 008 Hak Cipta Zon A Kuching [Lihat Sebelah SULIT

14 SULIT 7/ Ben and Shafiq are taking driving test. The probability that Ben and Shafiq pass the test are 5 and respectively. For Calculate the probability that at least one person passes the test. [ marks ] Answer :.. A committee of 5 members is to be selected from 6 boys and girls. Find the number of ways in which this can be done if (a) the committee has no girls, (b) the committee has exactly boys. [ marks ] Answer : (a).. (b).. 7/ 008 Hak Cipta Zon A Kuching SULIT

15 For SULIT 5 7/ 5 A random variable X has a normal distribution with mean 50 and variance Given that P[X > 5] = 088, find the value of.. [ marks ] 5 Answer : END OF QUESTION PAPER 7/ 008 Hak Cipta Zon A Kuching [Lihat Sebelah SULIT SULIT

16 SULIT 7/ Additional Mathematics Paper Sept 008 SEKOLAH MENENGAH ZON A KUCHING LEMBAGA PEPERIKSAAN PEPERIKSAAN PERCUBAAN SPM TINGKATAN ADDITIONAL MATHEMATICS Paper MARKING SCHEME This marking scheme consists of 6 printed pages

17 PAPER MARKING SCHEME 7/ Number Solution and marking scheme Sub Full (a) (b) 0 x 5 or f : x x 5 or f(x) = x 5 a = and b = gf(x) = x + x (x ) + (x ) + B (a) (b) x, x x y = x x t > t < or equivalent (6) (t +)() < 0 B 5 x + x + = 0 () = and + = 7 and = B

18 Number Solution and marking scheme Sub Full 6 a = a( )² + = 5 p = and q = B 7 x 5 (x 5)(x + ) 0 x² x 5 0 B 8 x x x or equivalent x x or x x B 9 5 lg lg7 lg lg lg B 0 (a) d 5 T = or T = 9 (b) 585 8, 5, 6 n = or 5 or 8 or equivalent (Solving) a = 968 and r = B

19 Number (a) k Solution and marking scheme Sub Full (b) log0 y log0 x + log 0 k h h 6 0 h p h h h h h h h h B p or equivalent B p 5 5 m = m or m 5 5 m 5 6 B 6 (a) a b 5i m j uuur % % % % BD i 5 j % % uuur uuur uuur uuur uuur BD BA AD or BA BC (b) 50 uuur AC 7i j % %

20 5 Number Solution and marking scheme 7 66.,9.58 Sub o o Full cos 5 5cos or equivalent 5 cos x B B 9 (a) r = 6 (b) AOC = 0.5 or 7. = r (0.5) 576 or 58 or 6 (6) ( 69) 0 (, ) and (0, ) x = 0, B dy = 0 or x(x + ) = 0 dx B dy = x + 6x dx

21 6 Number (a) Solution and marking scheme dy x 6 or equivalent dx Sub Full dy dm 6m and dm dx (b) 08 y [() 6] 0 0 ( x) c ( x) c B ( x) 5 or or equivalent 5 B (a) 6 or 5 (b) 0 6 C C B

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