SULIT 47/ 47/ Matematik Tambahan Kertas ½ jam 009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 009 MATEMATIK TAMBAHAN Kertas Dua jam tiga puluh minit JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU. This question paper consists of three sections : Section A, Section B and Section C.. Answer all question in Section A, four questions from Section B and two questions from Section C.. Give only one answer / solution to each question.. 4. Show your working. It may help you to get marks. 5. The diagram in the questions provided are not drawn to scale unless stated. 6. The marks allocated for each question and sub-part of a question are shown in brackets.. 7. A list of formulae is provided on pages to. 8. A booklet of four-figure mathematical tables is provided. 9. You may use a non-programmable scientific calculator. Kertas soalan ini mengandungi halaman bercetak 47/ ZON A KUCHING 009 SULIT
SULIT 47/ The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. x b ± b 4ac 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 4 (a m ) n a mn 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 ) y uv, dy dx dv u dx du v + du v dx dv u u dy y, dx dx, v dx v dy dx dy du du dx CALCULUS 4 Area under a curve b y dx or a b x dy a 5 Volume generated b π y dx or a b π x dy a GEOM ETRY Distance Midpoint x + (x, y) x 4 r x + y xi + r x yj + y ( 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 ( ) ( ) x y + x y + x y x y + x y + x y 47/ ZON A KUCHING 009 SULIT
SULIT 47/ x N x σ x fx f ( x x ) N x N x STATISTICS 7 8 9 w I I w n n! P r ( n r)! n n! C r ( n r)! r! 0 P(A B) P(A) + P(B) P(A B) 4 σ f ( x x) f fx f x P(X r) Mean µ np r C p q n r n r, p + q N F 5 m L + C fm Q I 00 Q 6 0 σ npq 4 z x µ σ TRIGONOMETRY Arc length, s rθ Area of sector, A sin A + cos A 4 sec A + tan A 5 cosec A + cot A 6 sin A sina cosa r θ 7 cos A cos A sin A cos A sin A tan A 8 tan A tan A 9 sin (A ± B) sina cosb ± cosa sinb 0 cos (A ± B) cosa cosb m sina sinb tan A ± tan B tan (A ± B) m tan Atan B a sin A b sin B c sin C a b + c bc cos A 4 Area of triangle absin C 47/ ZON A KUCHING 009 SULIT
SULIT 4 4 47/ SECTION A [40 marks] Answer all questions in this section. Solve the simultaneous equations x y + 6 0 and x + xy 0 0. Give your answer correct to decimal places. [5 marks] Diagram shows a circle with centre O. 00 O S R DIAGRAM P T Q PTQ is a tangent to the circle at T and PQ OQ 0 cm. Calculate the length of the arc STR, [4 marks] the area of the shaded region. [4 marks] Table shows the frequency distribution of scores of a group of players in a game. Score 0 4 5 9 0 4 5 9 0 4 5 9 0 4 Number of players 0 0 w 6 TABLE It is given that the median of the distribution is 7. Calculate the value of w. [ marks] Hence, calculate the variance of the distribution. [4 marks] 47/ ZON A KUCHING 009 SULIT
SULIT 5 47/ 4 If the volume of a cube decreases from 5 cm to 4.4 cm, find the small change in the sides of the cube. [ marks] x + 4 Given that f ( x), find the value of f (). x [ marks] 5 Prove that sin x sin x cot x. [ marks] Sketch the graph of y sin x for 0 x π. Hence, using the same axes, sketch a suitable straight line to find the number of solutions of the equation x π for 0 x π. State the number of solutions. sin x [6 marks] 6 cm 5 cm 8 cm... DIAGRAM A piece of wire is cut into 5 parts which are bent to form circles as shown in Diagram. The radius of each circle increases by cm consecutively. Calculate (i) the radius of the last circle, [ marks] the area of the last circle. [ mark] Diagram shows a rectangular geometric pattern. A M B P N D Diagram The first rectangle is ABCD and followed by MBNP and so on. The length and width of the next rectangle is half of the length and width of the previous rectangle. Given that AB 0 cm and BC 0 cm. Find the perimeter of the seventh rectangle. [ marks] 47/ ZON A KUCHING 009 SULIT C DIAGRAM
SULIT 6 47/ SECTION B [40 marks] Answer four questions from this section. 7 Use graph paper to answer this question. Table shows the values of two variables x and y which are related by and q are constants. y pq x+, where p x 4 5 6 y 5.6 5.9 640 6 5849 6096 TABLE Convert x+ y pq to a linear form of mx c Y +. [ marks] Plot log 0 y against ( x + ) by using a scale of cm to unit on the Y-axis and cm to unit on the X-axis. Hence, draw the line of best fit. [4 marks] (c) From the graph in, find the value of p and of q. [4 marks] 8 Diagram 4 shows a triangle OPQ. The point R lies on OP and the point S lies on PQ. The straight line QR intersects the straight line OS at point T. Q T S O R P DIAGRAM 4 Given OP : OR 4 :, PQ : PS :, OP x % and OQ 9y. % Express, in terms of x and / or y, (i) QR, OS. [ marks] If OT hos and QT k QR, where h and k are constants, find the values of h and k. (c) Given that x % units, y 5 % units and POQ 90 o, find PQ. 47/ ZON A KUCHING 009 SULIT [5 marks] [ marks]
SULIT 7 47/ 9 In a certain area, 0% of the trees are rubber trees. (i) If 8 trees in the area are chosen at random, find the probability that at least two of the trees are rubber trees. If the variance of the rubber trees is 5, find the number of rubber trees in the area. [5 marks] The masses of the children in the Primary One in the school have a normal distribution with mean.5 kg and variance 5 kg. 50 of the children have masses between 0 kg and 6.5 kg. Calculate the total number of children in Primary One in that school. [5 marks] 0 Solution by scale drawing is not accepted. In Diagram 5, point T lies on the perpendicular bisector of AB. y D A (, 4) T B(, ) C O DIAGRAM 5 x Find the equation of straight line AB. [ marks] A point P moves such that PA AB. Find the equation of locus of P. [ marks] (c) Locus of P intersects the x-axis at points Y and Z. State the coordinates of Y and Z. [ marks] (d) Find the x-intercept of CD. [ marks] 47/ ZON A KUCHING 009 SULIT
SULIT 8 47/ Given that a curve has a gradient function px + x such that p is a constant. y 6 x is the equation of tangent to the curve at the point (, q). Find the value of p and of q. [ marks] Diagram 6 shows the curve y (x ) and the straight line y x + intersect at point (, 4). y y x + Calculate O (, 4) y (x ) DIAGRAM 6 x (i) the area of the shaded region, [4 marks] the volume of revolution, in terms of π, when the region bounded by the curve, the x-axis, the y-axis and the straight line x is revolved through 60 about the x-axis. [ marks] 47/ ZON A KUCHING 009 SULIT
SULIT 9 47/ SECTION C [0 marks] Answer two questions from this section. A particle moves along a straight line and passes through a fixed point O. Its velocity, v ms, is given by v t 6t + 5, where t is the time, in seconds, after passing through O. (Assume motion to the right is positive). Find the initial velocity, in ms, the minimum velocity, in ms, [ mark] [ marks] (c) the range of values of t at which the particles moves to the left, [ marks] (d) the total distance, in m, travelled by the particle in the first 5 seconds. [4 marks] In Diagram 7, ABC is a triangle. BMC and AM are straight lines. A B M DIAGRAM 7 C Calculate (i) AMB, the area, in cm, of triangle ABC. [7 marks] A new triangle A B M is formed with A B AB, B M BM and B A M BAM, find the length of A M. [ marks] 47/ ZON A KUCHING 009 SULIT
SULIT 0 47/ 4 Use the graph paper provided to answer this question. A factory produces two types of school bags M and N using two types of machines A and B. Given that machine A requires 0 minutes to produce a bag M and 0 minutes to produce a bag N while machine B requires 5 minutes to produce a bag M and 40 minutes to produce a bag N. The machines produce x units of M and y units of N in a particular day according to the following conditions. I : Machine A is operated for not more than 8 hours. II : Machine B is operated for at least 4 hours. III : The number of units of bag M produced is not more than twice the number of units of bag N. Write the three inequalities for the above conditions. [ marks] (c) Using a scale of cm to units for both axes, construct and shade the region R which satisfies all the above conditions. [ marks] Use the graph constructed in 4, to find (i) the maximum number of units of bag M that can be produced if the factory produces units of bag N. the maximum profit obtained if the profit from one unit of bag M and bag N are RM 8 and RM 0 respectively. [4 marks] 47/ ZON A KUCHING 009 SULIT
SULIT 47/ 5 Table shows the price indices and percentage of usage of four components, P, Q, R and S, which are the number of parts in the making of an electronic device. Item Price index for the year 000 based on the year 997 Percentage of usage (%) P 5 0 Q 40 0 R x 0 S 0 40 Calculate TABLE (i) the price of Q in the year 997 if its price in the year 000 is RM 50.40, the price index of P in the year 000 based on the year 994 if its price index in the year 997 based on the year 994 is 0. [5 marks] The composite index number for the cost of production in the year 000 based on the year 997 is. Calculate (i) the value of x, the price of an electronic device in the year 997 if the corresponding price in the year 000 was RM 88. [5 marks] END OF QUESTION PAPER 47/ ZON A KUCHING 009 SULIT
47/ Matematik Tambahan Kertas ½ jam Sept 009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 009 MATEMATIK TAMBAHAN Kertas Dua jam tiga puluh minit JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU MARKING SCHEME Skema Pemarkahan ini mengandungi 4 halaman bercetak
QUESTION NO. ADDITIONAL MATHEMATICS MARKING SCHEME TRIAL SBP 009 PAPER SOLUTION MARKS x y 6 P 5 (y 6) + (y 6) y 0 0 Eliminate x or y y 5y + 8 0 Solve the quadratic equation by using the factorization @ quadratic formula @ completing the square must be shown y 0.6070, y 4.9 @ x 4.786, x.786 Note : OW- if the working of solving quadratic equation is not shown. 5 OT 7.05 cm π 4 s STR π 7.05 Use the formula s rθ s 8.80 cm STR
QUESTION NO. SOLUTION MARKS Area OPQ 0 0 sin 60 @ 7.05cm o 4 Area OSTR 7.05 π @ 57.0795 cm 0 0 sin 60 o 7.05 57.0795 7.05 π @ 6.56 cm 8 4 + w 5 7 4.5 + (5) 0 w 7 P Lower boundary OR 4 + w 5 (5) 0 Score Frequency, f Midpoint, x fx fx 0-4 4 8 5-9 7 47 0-4 0 0 440 5-9 0 7 40 5780 0-4 7 54 88 5-9 6 7 6 474 0-4 64 048 Variance f 50 P fx 865 fx 785 785 50 865 50 OR P 4 44.4 7
4 QUESTION NO. SOLUTION MARKS 4 δv 0.6 OR dv dx x OR x 5 0.6 (5) δ x δ x 0.008 f ( x) ( x )() (x + 4)( x) ( x ) f () [ ] + () () () (() 4)( ()) 7 6 5 sin cos x x sin x sin x cos x sin x y x π Sketch straight line correctly P Sine curve P 6 y y sin x x y π Period Amplitude P P O π x Modulus P Number of solutions 4 8
5 QUESTION NO. SOLUTION MARKS 6 (i) T 5 + 4() 44 cm Area of the fifteenth circle 96π cm 00, 50, 5, a 00 AND r T 7.56 6
6 log 0 y x + 4 5 6 7 8 log 0 y.408.00.806.500 4.00 4.800 5 Each set of values correct (log 0 y must be at least decimal places), Y mx + c log 0 y (x + ) log 0 q + log 0 p where Y log 0 y, X (x + ), m log 0 q and c log 0 p (c) log 0 q gradient 4. ( 0.6) log 0 q 7 0 q 4.85 Correct both axes (Uniform scale) All points are plotted correctly Line of best fit 4 log p Y-intercept log p 0.6 p 0 5 0 0 4 5 6 7 8 x +
7 QUESTION NO. SOLUTION MARKS 8 (i) uuur QR 9x 9y % % uuur OS 9y + 9y + x % % % 9 6x + y % % ( ) uuur uuur OT hos 9 6hx + hy % % OR uuur uuur QT kqr 9kx 9ky % % uuur uuur uuur QT QO + OT 9 9y + 6hx + hy % % % 9 6hx + 9 + h y % uuur % Comparing QT, 9k 6h k h --------------------------() OR 9 9k 9 + h k h --------------() 5 (d) Solving the simultaneous equations 6 4 h, k 7 7 uuur PQ x + 9y uuur % % PQ () + 9(5) ( ) ( ) 57.6 units 0
8 QUESTION NO. SOLUTION MARKS 9 (i) p 0., q 0.7, n 8 (0.7) 8 8 C (0.)(0.7) 7 5 0.7447 n(0.)(0.7) 5 n 500 (i) 0.5 6.5.5 50 P < Z < 5 5 n OR 50 P[ 0.7 < Z < 0.6] n 5 P[Z > 0.7] P[Z > 0.6] 50 n 0.40 0.74 50 n n 0. Total number 0 0
9 QUESTION NO. SOLUTION MARKS 0 Get m ( x ) y 4 y + x 5 0 ( x ) + ( y 4) ( 4 ) + ( ) + + 8 + 6 4( 8) x x y y x y x y + 8 5 0 (c) x x 5 0 ( x )( x ) 5 + 0 x 5, x Get both ( ) ( ) Y / Z 5,0 and Z / Y,0 (d) ( ) GetT,, x intercept 0
0 QUESTION NO. SOLUTION MARKS px + x p() + p, q (i) Area of the shaded region ( x 8x + 7) dx 0 ( x 6x + 9) dx (x + ) dx 0 0 x 8x + 7x 8 + 7 0 unit. 0 7 Volume of revolution π y dx 0 π 0 ( x ) 4 dx 5 π ( x ) 5 0 5 5 ( ) (0 ) π 5 5 48 π unit. 5 0
QUESTION NO. SOLUTION MARKS v 5 ms P a t 6 and a 0 or dv dt 0 t v min 4 ms (c) (t )(t 5) < 0 < t < 5 (d) t s t + 5t 4 s s 0 + s 5 s OR 5 v dt + 0 v dt 7 5 7 0 + OR 5 () + 5() 0 + (5) + 5(5) () + 5() m 0
QUESTION NO. SOLUTION MARKS (i) 5 + 8 cos AMB (5)(8) AMB.4 @ 6' sin ACM sin 46.57 5 4 ACM 65.0 @ 65 4' 5 MAC 80 46.57-65.0 68. Area of AMB Area of ACM (5)(8)sin.4 (5)(4)sin 68. OR 9.87 cm 4.5 cm Area of ABC 4.5 + 9.87.807 cm Get B 'M 'M 46.57 @ B 'MM ' 46.57 @ M 'B 'M 86.86 MM ' sin 86.86 8 sin 46. 57 MM ' 0.999 cm @ cm A'M ' 5 + 0.999 5.999 cm @ 6 cm 0
Answer for question 4 I. x + y 48 II. 5x + 8y 48 III. y x Refer to the graph, or graph(s) correct graphs correct Correct area 6 (c) i) 6 ii) max point (, 8) 4 k 8x + 0y Maximum Profit RM 8() + RM 0(8) RM 76 0 0 8 R (, 8) 6 4
4 QUESTION NO. SOLUTION MARKS 5 (i) RM 50.40 40 00 Q 97 5 Q 97 RM 6 I 00,97 I00,94 00 I 97,94 I00,94 5 00 0 (i) I 00, 94 50 5 0 + 40 0 + 0x + 0 40 00 5 0x + 800 00 x 0 RM 88 00 Q 997 Q 997 RM 6.07 0