SULIT / / Matematik Tambahan Kertas ½ jam 0 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 0 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.. Show your working. It may help you to get marks.. The diagram in the questions provided are not drawn to scale unless stated.. The marks allocated for each question and sub-part of a question are shown in brackets... A list of formulae is provided on pages to.. A booklet of four-figure mathematical tables is provided. 9. You may use a non-programmable scientific calculator. Kertas soalan ini mengandungi halaman bercetak / ZON A KUCHING 0 SULIT

SULIT / The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. x b ± b ac a ALGEBRA 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 mn log a mn log a m + log a n m log a n log a m log a n 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 Volume generated b π y dx or a b π x dy a GEOM ETRY Distance Midpoint x + (x, y) x ( x y x ) + ( y ) y, + y A point dividing a segment of a line nx + mx ny + my (x, y), m + n m + n. Area of triangle r x + y xi + yj r x + y ( ) ( ) x y + x y + x y x y + x y + x y / ZON A KUCHING 0 SULIT

SULIT / STATISTICS x N x σ x fx f ( x x ) N x N x 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) 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 m L + C fm Q I 00 Q 0 σ npq z x µ σ TRIGONOMETRY Arc length, s rθ Area of sector, A sin A + cos A sec A + tan A cosec A + cot A sin A sina cosa r θ cos A cos A sin A cos A sin A tan A tan A tan A 9 sin (A ± B) sina cosb ± cosa sinb 0 cos (A ± B) cosa cosb sina sinb tan (A ± B) a sin A b sin B tan A ± tan B tan Atan B c sin C a b + c bc cos A Area of triangle absinc / ZON A KUCHING 0 SULIT

SULIT / THE UPPER TAIL PROBABILITY Q(z) FOR THE NORMAL DISTRIBUTION N(0, ) z 0 9 9 Subtract 0.0 0. 0. 0. 0. 0.000 0.0 0.0 0. 0. 0.90 0. 0. 0. 0.09 0.90 0. 0.9 0. 0. 0.0 0. 0.090 0.0 0. 0.0 0. 0.0 0.9 0.00 0.0 0.0 0.0 0. 0. 0. 0. 0.9 0.9 0. 0. 0. 0.9 0. 0.9 0. 0. 0.9 0.0 0. 0. 0. 0.9 0. 0. 0 0 9 9 0 9 0. 0. 0. 0. 0.9 0.0 0. 0.0 0.9 0. 0.00 0.09 0.9 0.090 0. 0.0 0. 0. 0.0 0. 0.9 0. 0. 0.0 0. 0.9 0. 0.9 0.00 0. 0.9 0. 0. 0.9 0. 0. 0. 0. 0.99 0. 0. 0. 0.0 0.9 0.0 0.0 0. 0. 0.9 0. 0. 0. 0. 0. 0. 0 0 9 0 0 9 9 0 9.0.... 0. 0. 0. 0.09 0.00 0. 0. 0. 0.09 0.09 0.9 0. 0. 0.09 0.0 0. 0.9 0.09 0.09 0.0 0.9 0. 0.0 0.090 0.09 0.9 0. 0.0 0.0 0.0 0. 0.0 0.0 0.09 0.0 0. 0.0 0.00 0.0 0.00 0.0 0.90 0.00 0.0 0.09 0.9 0.0 0.09 0.0 0.0 9 0 9 0 0 9.....9 0.0 0.0 0.0 0.09 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.00 0.0 0.0 0.0 0.0 0.0 0.00 0.009 0.09 0.0 0.00 0.09 0.00 0.0 0.0 0.09 0.0 0.09 0.0 0.00 0.0 0..0 0.0 0.00 0.0 0.0 0.0 0.0 0.00 0.09 0.09 0.0 0.0 0.09 0.0 0 9.0... 0.0 0.09 0.09 0.00 0.0 0.0 0.0 0.00 0.0 0.00 0.0 0.00 0.0 0.0 0.09 0.00990 0.00 0.0 0.0 0.009 0.00 0.0 0.0 0.0099 0.09 0.0 0.09 0.009 0.09 0.00 0.0 0.009 0.0 0.0 0.0 0.00 0.0 0.0 0.00 0.00 0 0 0 0 0 9 0. 0.000 0.009 0.00 0.00 0.00 9 0.00 0.009 0.00 0.00 0.009 9.....9 0.00 0.00 0.00 0.00 0.00 0.000 0.00 0.00 0.00 0.00 0.00 0.000 0.00 0.000 0.00 0.000 0.00 0.00 0.00 0.009 0.00 0.00 0.000 0.00 0.00 0.009 0.000 0.009 0.009 0.009 0.00 0.009 0.009 0.00 0.00 0.000 0.009 0.000 0.000 0.009 0.009 0.00 0.00 0.0099 0.00 0.000 0.00 0.00 0.009 0.009 0 9 9 9 0 9.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000 0.000 0.0000 0 f ( z) exp z π k f (z) Q ( z) f ( z) dz Q(z) Example: If Z ~ N(0, ), then P(Z > k) Q(k) P(Z >.) Q(.) 0.09 z O k / ZON A KUCHING 0 SULIT

SULIT / SECTION A [0 marks] Answer all questions. Solve the simultaneous equations x y and x xy. [ marks] Given the quadratic function f(x) x x +. (a) By using completing the square method, express f(x) in the form a(x + p) + q where a, p and q are constants. [ marks] State the minimum / maximum point. [ mark] (c) Sketch the graph of f(x) x x + for x. [ marks] Diagram shows several rectangles with a fixed base of cm. The height of the first rectangle is 00 cm, and the height of each subsequent rectangle decreases by cm. cm cm 00 cm cm Diagram (a) Calculate the area, in cm, of the 0 th rectangle. [ marks] Determine how many rectangles can be formed. [ marks] (c) Given that the total area of the first n th rectangles is 0 cm, find the value of n. [ marks] / ZON A KUCHING 0 SULIT

SULIT / (a) Sketch the graph of y sin x for 0 x π. [ marks] Hence, using the same axes, sketch a suitable straight line to find the number of solutions for the equation π ( sin x) x for 0 x π. State the number of solutions. [ marks] Table shows the length of leaves collected from a type of tree. Length (cm) Frequency 0 0 9 0 Table (a) Find the mean lengths of leaves collected from the tree. [ marks] Without drawing an ogive, find the interquartile range of the distribution. [ marks] Diagram shows a sector OABC with centre O and the arc OB with centre C where AOC 00. It is given that OC 0 cm. [Use π.] O C A B Calculate Diagram (a) BCO and AOB in radians, [ marks] the area, in cm, of the sector OCB and the sector AOB, [ marks] (c) the area, in cm, of the shaded region. [ marks] / ZON A KUCHING 0 SULIT

SULIT / Use graph paper to answer this question. SECTION B [0 marks] Answer any four questions from this section. Table shows the value of two variables, x and y, obtain from an experiment. The variables n x and y are related by the equation y ( p +) x, where p and n are constants. x 9 y. 0 0 Table (a) Based on Table, construct a table for the values of log 0 x and log 0 y. [ mark] Plot log 0 y against log 0 x, using a scale of cm to 0.0 unit on the log 0 x - axis and cm to 0.0 unit on the log 0 y - axis. Hence, draw the line of best fit. [ marks] (c) Use the graph in to find the value of (i) y when x., (ii) n, (iii) p. [ marks] Solution by scale drawing is not accepted. Diagram shows a rectangle ABCD. y D C(a, ) O x A(0, ) Diagram B(a, ) / ZON A KUCHING 0 SULIT

SULIT / (a) Find (i) the value of a, (ii) the coordinates of point D. [ marks] A point P moves such that its distance from point A is always units. (i) Find the equation of the locus of P, (ii) Determine whether this locus intersects straight line BC. [ marks] 9 Diagram 9 shows a parallelogram OLMN. The midpoint of MN is P and LP meets OM at Q. M L Q P N Diagram 9 O Given that OL x, ON y, OQ µ OM and LQ λ LP. (a) Express OP in terms of x and y. Express OQ in terms of (i) λ, x and y, (ii) µ, x and y. [ mark] [ marks] Hence, find the value of and of µ. [ marks] (c) Given that the area of triangle OQL is cm, find the area of the parallelogram OLMN. [ marks] / ZON A KUCHING 0 SULIT

SULIT 9 / 0 (a) Water is being poured into an inverted conical tank as shown in Diagram 0(a), at rate of 0. m s. Find the rate of change in the height of the water when the height is m. m 9 m Diagram 0(a) [ marks] Diagram 0 show the curve of x y (y + )(y ). y A(0, h) x y (y + )(y ). O x B(0, k) Diagram 0 Find (i) the value of h and of k. [ mark] (ii) the area of the shaded region. [ marks] / ZON A KUCHING 0 SULIT

SULIT 0 /. (a) The probability that a pen drawn at random from a box of pens is defective is 0.. If a sample of pens is taken, find the probability that it will contain (i) (ii) no defective pens, less than defective pens. [ marks] A commuter train is scheduled to arrive at the station at.0 am but the actual times of arrival are normal distributed about a mean of.0 am with a standard deviation of. minutes. Find the probability that the train is (i) (ii) late, late and arrive before. am. [ marks] SECTION C [0 marks] Answer any two questions from this section. Diagram shows the positions and directions of motion of two objects, A and B, moving along a straight line and passing through a fixed point O at the same time. A B O Diagram The velocity of A, v A ms, is given by v A t t + and the velocity of B, v B ms, is given by v B t + t, where t is the time, in seconds, after leaving point O. [Assume motion to the right is positive] Find (a) the initial velocity object A, [ mark] the minimum velocity object B, [ marks] (c) the values of time, t, in seconds, when both the objects stop instantaneously at the same time, [ marks] (d) the distance, in m, of object A from O when it stops for the first time. [ marks] / ZON A KUCHING 0 SULIT

SULIT / Table shows the price indices and respective weightages, in the year 00 based on the year 00, on four materials, A, B, C, D in the production of a type of foaming cleanser. (a) Material Price index in the year 00 Weightage based on the year 00 A B 0 n C 0 D 0 n + Table If the price of material A is in the year 00 was RM 0.00, calculate its price in the year 00. [ marks] Given that the composite index for the production cost of the foaming cleanser in the year 00 based on the year 00 is 0. Find (i) the value of n, [ marks] (ii) the price of the foaming cleanser in the year 00 if the price in the year 00 is RM 0.00. [ marks] (c) Given that the price of material B is estimated to increase by % from the year 00 to the year 009, while the others remain unchanged. Calculate the composite index of the foaming cleanser in the year 009 based on the year 00. [ marks] The diagram shows a quadrilateral PQRS. P cm S cm cm Q o Diagram cm Given that QSR is an obtuse, PQ cm, QR cm, RS cm, PS cm and RQS. Calculate (a) QSR, [ marks] the length QS, [ marks] (c) the area of triangle PQR. [ marks] / ZON A KUCHING 0 SULIT R

SULIT / Use the graph paper provided to answer this question. A factory produces two types of robot P and Q using two machines, A and B. Given that machine A requires hours to produce one unit of robot P and hours to produce one unit of robot Q while machine B requires hour to produce one unit of robot P and hours to produce one unit of robot Q. The machines produce x units of robot P and y units of robot Q in a particular day according to the following constraints : I II III Machine A is function for not more than days. Machine B is function for at least day. The number of robot P produced is not more than three times the number of robot Q produced. (a) Write down three inequalities, other than x 0 and y 0, which satisfy the above conditions. [ marks] (c) By using a scale of cm to units of commodity on both axes, construct and shade the region R that satisfies all the above constraints. [ marks] By using your graph in, find (i) (ii) the maximum profit obtained if the profit from the sale of one unit of robot P and one unit of robot Q are RM 00 and RM 00 respectively, assuming all the robots produced are sold. The maximum number of units of robot Q that can be produced if the factory produced units of robot P. [ marks] END OF QUESTION PAPER / ZON A KUCHING 0 SULIT

SULIT / NO. KAD PENGENALAN ANGKA GILIRAN Arahan Kepada Calon Tulis nombor kad pengenalan dan angka giliran anda pada petak yang disediakan. Tandakan ( ) untuk soalan yang dijawab. Ceraikan helaian ini dan ikat sebagai muka hadapan bersama-sama dengan buku jawapan. Kod Pemeriksa Bahagian Soalan Soalan Dijawab Markah Penuh Markah Diperoleh (Untuk Kegunaan Pemeriksa) A 0 0 B 9 0 0 0 0 0 C 0 0 0 Jumlah / ZON A KUCHING 0 SULIT

/ Matematik Tambahan Kertas ½ jam Sept 0 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 0 MATEMATIK TAMBAHAN Kertas Dua jam tiga puluh minit JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU MARKING SCHEME Skema Pemarkahan ini mengandungi halaman bercetak

QUESTION NO. ADDITIONAL MATHEMATICS MARKING SCHEME TRIAL SPM exam Zon A Kuching 0 PAPER SOLUTION MARKS x y + P ( ) y y y ( + ) + Eliminate x or y ( y )( y ) + 0 y, y @ x, x Solve the quadratic equation by using the factorization @ quadratic formula @ completing the square must be shown Note : OW if the working of solving quadratic equation is not shown. (a) f ( x) [ x + ( ) x + ( ) ( ) ] + or [( x ) ] + or [( x ) ] ( x ) (c) (, ) min Shape y (, ) (, ) and (0, ) (, ) and (, ) O (, ) (, ) x

QUESTION NO. SOLUTION MARKS (a) T 0 00 + 9( ) or Area 00 + (n )( ) or 00 @ 00 + (n )( ) > 0 n (c) n 0 [ (00) + ( n )( ) ] or 00 (n )(n ) 0 n (a) y y sin x 0 π π - π Shape of sine curve P π y π x x Modulus Amplitude or period Translation P P P y π x Sketch straight line correctly P Number of solutions

QUESTION NO. (a) SOLUTION fx + + + + 9 + + x 0. MARKS (0) Q 0. +. 9 OR (0) Q 0. + P Lower boundary OR (0) 9 Interquartile range. 0. (a) BCO 0.0 rad AOB 0.9 rad (0) (.0) or. (0) (0.9) or.9 (c) Area of segment BC. (0) sin.0 r 9. Area of the shaded region.

log 0 x 0.0 0. 0.0 0. 0. 0.9 log 0 y 0.9.0..9.0. log 0 y Q Correct both axes (Uniform scale) All points are plotted correctly Line of best fit..0. (a) Each set of values correct (log 0 y must be at least decimal places), log 0 y nlog 0 x + log 0 (p + ) where Y log 0 y, X log 0 x, m n and c log 0 (p + ) (c) (i) X log 0. 0. Y. log 0 y y. n gradient.9 0.0 n.0 0. 0 (0.,.9). log 0 (p + ) Y-intercept log 0 (p + ) 0.0 p 0.99...0 0. 0. 0. (0, 0.0) 0. 0 0. 0. 0. 0. 0. 0. 0. 0. log 0 x 0.9.0

QUESTION NO. SOLUTION MARKS (a) (i) a a a a ± a > 0, a (ii) x + y, + 0, + ( ) D(, ) @ Solving the equations y x and y x + D(, ) (i) ( x 0) + ( y + ) x + y + y 0 (ii) Get equation of BC, y x 9 x x + (x 9) b x + 0 ac ( ) + (x 9) 0 ()() > 0 The locus intersects the line BC. 0

QUESTION NO. SOLUTION MARKS 9 (a) OP x + y P (i) OQ λx + λ y (ii) OQ OL+ LQ x + µ LP ( ) x + µ LM + MP x + µ y x µ x + µ y x : λ µ y : λ µ µ µ (c) µ λ Area of triangle OLM Therefore area of parallelogram OLMN 0

QUESTION NO. SOLUTION MARKS 0 (a) r h r h 9 dv dh π h 9 π h dh 0. 9 dt π h dh 0. 9 dt (i) (ii) h and k Area of the shaded region 0 ( ) + ( ) y y dy y y dy 0 y y 0 + y y 0 ( 0 0) + ( ) 0 0 Note: OW once only for correct answer without showing the process of intergration. 0

9 QUESTION NO. SOLUTION MARKS (a) (i) P( X 0) C (0.) (0.) 0 0 0. (ii) P( X < ) 0 C0(0.) (0.) + C(0.) (0.) 0. (i) P( Z > 0.) @ R( 0.) (ii) 0.9 @ 0.9 P( 0. < Z <.0) P(Z 0.) P(Z. 0) @ R( 0.) R(. 0) 0. @ 0. 0

0 QUESTION NO. SOLUTION MARKS (a) Initial velocity v A dv t + 0 dt t v B + (c) v ( t )( t ) 0 A v ( t )( t ) 0 B t (d) s A t t + dt t t + t () + () + () 0

QUESTION NO. SOLUTION MARKS (a) P0 A : 00 0 P 0 RM (i) ( ) + (0 n) + (0 ) + 0( n + ) 0 + n 0 + 0n 0 + 0n n (ii) RM0 00 0 P 0 RM (c) 0 + (0 0.) I 09 / 0 ( ) + ( ) + (0 ) + (0 ) 0

QUESTION NO. SOLUTION MARKS (a) sin sin QSR QSR 0 0 QRS PS + ()()cos @ Sine Rule QS. (c) +.. cos PQS PQS 9 Area of triangle PQR sin 9.999 0

Answer for question (a) I. x + y II. x + y y III. y x Refer to the graph, or graph(s) correct graphs correct Correct area (c) ii) max point (, ) k RM(00x + 00y) Maximum Profit RM 00() + RM 00() RM 00 (ii) units 0 0 R (, ) 0 0 x