BAHAGIAN SEKOLAH BERASRAMA PENUH DAN SEKOLAH KLUSTER ADDITIONAL MATHEMATICS

Transcription

1 BAHAGIAN SEKOLAH BERASRAMA PENUH DAN SEKOLAH KLUSTER ADDITIONAL MATHEMATICS

2 PAGE ABOUT THIS MODULE I WE LEARN II EXAMINATION FORMAT II-III ANALYSIS TABLE OF SPM ADD MATHS QUESTIONS IV 5 LIST OF FORMULAE AND NORMAL TABLE V-VII 6 ADDITIONAL MATHEMATICS NOTES VIII-XVI 7 PROBLEM SOLVING STRATEGY XVII 8 PARTITION XVIII This module is. specially planned for students who will be sitting for SPM.. to provide eposure and to familiarize students with the needs of the actual SPM eam questions.. to prepare students with adequate knowledge prior to the eamination.. comprises challenging questions which incorporate a variety of questioning techniques and levels of difficulty and conforms to the current SPM farmat. That which we persist in doing becomes easier not that the nature of the task has changed, but our ability to do has increased. I

3 Key towards achieving A Read question carefully Follow instructions Start with your favourite question Show your working clearly Choose the correct formula to be used +(Gunakannya dengan betul!!!) Final answer must be in the simplest form The end answer should be correct to S.F. (or follow the instruction given in the question) π. Kunci Mencapai kecemerlangan Proper / Correct ways of writing mathematical notations Check answers! Proper allocation of time (for each question) Paper : - 7 minutes for each question Paper : Sec. A : 8-0 minutes for each question Sec. B : 5 minutes for each question Sec. C : 5 minutes for each question III

4 ANALYSIS TABLE OF SPM ADDITIONAL MATHEMATICS QUESTIONS AMaths (7) SPM Paper Paper Chapter Section A Section B Section C Functions Quadratic Equations Quadratic Functions Simultaneous Equations 5 Indices and Logarithms 6 Coordinate Geometry 7 Statistics 8 Circular Measure 9 Differentiation / / / / / / / 0 Solution of Triangle Inde Number Progressions Linear Law Integration ½ ½ / / / / / Vectors 5 Trigonometric Functions 6 Permutations / Combinations 7 Probability 8 Probability Distributions 9 Motion Along A Straight Line 0 Linear Programming Total IV

5 SULIT 7/ The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. b ± = 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 log a n m = 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 d d d du dv v u u dy y =, = d d, v d v dy d = dy du du d CALCULUS Area under a curve b a = y d or b a = dy 5 Volume generated b a = πy d or b a = π dy GEOM ETRY Distance = Midpoint + (, y) = r = + y i+ yj r = + y ( y ) + ( y ) y, + y 5 A point dividing a segment of a line n + m ny + my (,y) =, m + n m + n 6. Area of triangle = ( y + y + y ) ( y + y + y ) V [ Lihat sebelah 7/ SULIT

6 STATISTICS = N = σ = f f ( ) N = N _ 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 ( ) f = f f p (X=r) = r C p q n r n r, p + q = N F + fm 5 M = L C P 6 I = 00 P 0 Mean, μ = np σ = npq μ z = σ TRIGONOMETRY Arc length, s = rθ Area of sector, A = sin A + cos A = sec A = + tan A 5 cosec A = + cot A 6 sina = sinacosa r θ 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 m sinasinb tan A ± tan B tan (A ± B) = m tan Atan B a sin A = b sin B = c sin C a = b +c - bc cosa Area of triangle = absinc VI [ Lihat sebelah 7/ SULIT

7 7/ SULIT VII [ Lihat sebelah

8 TO EXCEL in You need to set a TARGET familiar with FORMAT of EXAM PAPERS analyse the EXAM QUESTIONS master the TECHNIQUES OF ANSWERING QUESTIONS do EXERCISES VIII

9 ADDITIONAL MATHEMATICS NOTES FUNCTIONS (a) a b c i. Domain = {a,b,c} ii Codomain = {,,,} iii. Range = {,,} iv. Objects of are a and b v. Images of b are, and. (b) Types of Relations i. One-to-one a b c ii. Many-to-one (c) Absolute Value Function The corresponding range of values of f() is 0 f() 5 The corresponding range of values of f() means the range from the smallest value of y to the largest value of y, based on the given domain. (d) Composite Functions f y f() g g[f()] = gf() a b c iii. One-to-many fg() = f[g()] In general, gf () fg() gf a b iv. Many-to-many a b c f = ff, f = fff or ff (e) Determining one of the functions in a given composite function i. Given f and fg, find g. - Substitute g into f() ii. Given f and gf, find g. - Let y= f() (f) To find the Inverse Function : - Let y = f(), then = f - (y). IX

10 . QUADRATIC EQUATIONS (a) a + b + c = 0 y b ± b ac = a Sum of roots: b α + β = a Product of roots c αβ = a (b) Form quadratic equation from given roots: - (sum of two roots) + product of two roots = 0. QUADRATIC FUNCTIONS (a) Types of roots b - ac > 0 different (distinct) roots. b - ac = 0 equal roots b - ac < 0 no real roots. b - ac 0 with real roots y b - ac > 0 b - ac = 0 b - ac < 0 (b) Completing the Squares y = a( - p) + q a +ve minimum point (p, q) a ve maimum point (p, q) (c) Quadratic Inequalities ( a)( b) 0 Range: a, b ( a)( b) 0 Range: a b y y 0 + a _ + _ b INDICES & LOGARITHM (a) = a n log a = n Inde Form Logarithmic Form (b) Laws of Indices. a a = a. a a = a. ( a ) a n m n+ m n m n m = a n m nm Laws of Logarithm. log a y = log a + log a y. log a y = loga log a y. log a n = n log a. log a a = 5. log a = 0 log cb 6. log a b = log ca 7. log a b = log b a b X

11 5. COORDINATE GEOMETRY (a) Distance between A(, y ) and B(, y ) AB = ( ) + ( y y) (b) Midpoint of AB + y + y M =, (c) P divides AB internally in the ratio m : n m : n A(, y ) P B(, y ) n + m ny + my P=, n + m n + m (d) Gradient of AB y y m = y-intercept m = -intercept (e) Equation of a straight line (i) given m and A(, y ) y y = m( ) (ii) given A(, y ) and B(, y ) y y y y = (f) Area of polygon L =... y y y y (g) Parallel lines m = m (h) Perpendicular lines m m = - 6. STATISTICS Measure of Central Tendency (a) Mean = n for ungrouped data = f f for ungrouped data with frequency. = f f i for grouped data, i = midpoint of each class interval (b) Median The centre value of a set of data after the data is arranged in the ascending or descending order. Formula n F M = L + C fm L = Lower boundary of the Median class n = Total frequency F = Cumulative frequency before the median class f m = Frequency of the median class C = Size of the class interval From the Ogive Cumulative Kekerapan Frequency longgokan n n 0 Median Sempadan atas Median Upper Boundary XI

12 (c) Mode Data with the highest frequency From the Histogram : Kekerapan Frequency For ungrouped data ( ) σ = n = n For grouped data σ = f ( ) f 0 Mod Mode Class Sempadan Boundary kelas Measure of Dispersion (a) Interquartile Range Formulae : n F Q = L + C f Q = L Ogive: + Q n F f Q Cumulative Frequency Kekerapan longgokan n C = f f 7. INDEX NUMBERS (a) Price Inde P I = 00 P0 where P o = price at the base time P = price at a specific time (b) Composite Inde Iw I = w where I = price inde or inde number w = weightage n 0 Q Q Sempadan atas Upper Boundary Interquartile Range = Q Q (b) Variance, Standard Deviation Variance = (Standard Deviation) 8. CIRCULAR MEASURE (a) Radian Degree θ r 0 80 = θ π (b) Degree Radian θ o π = θ rad 80 (c) Arc length s = jθ (d) Area of sector L = j θ = js XII

13 (e) Area of segment L = j (θ r sin θ o ) 9. DIFFERENTIATION (a) Differentiation using the First Principal dy δ y = lim d 0 δ d (b) (a) = 0, a = constant d d (c) ( n ) = n n- d d (d) (a n ) = an n- d (e) Product Rule d dv du (uv) = u + v d d d (f) Quotient Rule du d u v d u = d v v dv d (g) Composite Function d (a+b) n = dy du d du d = an(a+b) n- dy (h) Turning point = 0 d Maimum point: dy d y = 0 and < 0 d d 0. INTEGRATION n+ (a) a n a d = + + c n (b) n+ n ( a + b) ( a + b) d = + c an ( + ) b (c) f ( ) + g( ) d a b = ( ) d + b a f g( ) d (d) f ( ) d f ( ) d f ( ) d a b (e) ( ) d = a a b c b a + = b af f ( ) d b a a (f) f ( ) d = f ( ) d a (h) Area under the curve y 0 a b y b a b c a b A = y d a b A = dy a Minimum point: dy d y = 0 and d d > 0 0 (i) Volume of revolution y (i) Rate of change dy dy d = dt d dt (j) Small change : δ y dy. δ d 0 a b V b = π a y d XIII

14 y b a 0 V b = π dy a (a) s = 0 at the fied point O (b) v = 0 stops momentarily maimum / minimum displacement (c) a = 0 v constant v maimum/ minimum. PROGRESSIONS Arithmetic Progressions (a) T n = a + (n - )d (b) S n = n {a + (n - )d} = n (a + l) (c) d = T - T Geometric Progressions (a) T n = ar n- n a( r ) (b) S n = for r < r a( r n ) S n = for r > r a (c) S = for - < r < r and n T (d) r = T. TRIGONOMETRIC FUNCTIONS (a) y P(, y) y sin θ = r 0 r θ (b) tan θ = sin θ cosθ sec θ = cosθ cosec θ = sinθ cosθ cot θ = = tanθ sinθ y cos θ = r tan θ = y General (a) S = T = a (b) T n = S n S n- (c) Sum of terms from T a to T b = S b S a- (c) Sin +ve All Semua +ve. MOTION ALONG A STRAIGHT LINE ds dv dt dt s v a v dt a dt Tan +ve Cos Kos +ve XIV

15 (d) Special Angles θ 0 o 0 o 5 o 60 o Sin θ 0 Cos θ Tan θ 0 θ 90 o 80 o 70 o 60 o Sin θ 0-0 Cos θ 0-0 Tan θ 0 0 (e) Trigonometric Graphs y = a sin b y a (f) sin θ + cos θ = + tan θ = sec θ + cot θ = cosec θ (g) sin(a ± B) = sina cos B ± cos Asin B cos(a ± B) = cosa cosb m sina sinb tan (A ± B) tan A± tan B = m tan Atan B (h) sina = sina cosa cosa = cos A sin A = cos A = sin A tan A tan A = tan A 0 -a b b b b. VECTORS (a) Addition of Vectors. Triangle Law b a + y = a cos b y a b b b b b a. Parallelogram Law b -a y = a tan b y b a a b b b b. Polygon Law A B C XV E uuur uuuv uuuv uuuv uuuv AE = AB+ BC+ CD+ DE D

16 (b) Subtraction of vectors b a a (c) Vectors in the Cartesan Coordinates y P(, y) r y j - b (b) Combination n C n = n n! C r = ( n r)! r! n n Cr = Cn r (c) Binomial Distribution P(X = r) = n Cr p r q n-r (d) Mean = μ = np Standard deviation σ = npq (e) Converting Normal Distribution to Standard Normal X μ Distribution Z = σ (f) Probability Distribution Graph 0 i r = i + yj r = + y r i+ yj ˆr = = r + y 5. SOLUTIONS OF TRIANGLE (a) Sine Rule a = b = c A B C sin sin sin (b) Cosine Rule a = b + c bc cos A b + c a cos A = bc (c) Area of Triangle L = absin C 6. PROBABILITY DISTRIBUTIONS (a) Permutation n P n = n! n n! P r = ( n r)! 0 a Z P(Z > a). P(Z < a) = P(Z > a) use P. P(Z < -a) = P(Z > a) use P. P(Z > -a) = P(Z > a) use R. P(a < Z < b) = P(Z>a) P(Z > b) 5. P(-a < Z< b) = P(Z >a) P(Z > b ) 6. P(-a < Z < -b) = P(b < Z < a) Eamples: a) P(Z> 0.) b) P ( Z< 0.) c) P ( -. < Z < 0.) Eamples: d) P( Z > a ) = 0., find a e) P (Z > a) = 0.6, find a f) P (Z < a) = 0., find a g) P ( Z < a ) = 0.7, find a h) P(X > a) = 0., given μ = 5, σ = XVI

17 How to Solve a Problem Understand the Problem Plan your Do - Carry out Strategy Your Strategy Answers Check your Which Topic / Subtopic? Choose suitable strategy Carry out the calculations Is the answer reasonable?. PN ZABIDAH BINTI AWANG SM AGAMA PERSEKUTUAN, LABU.. EN AMIRULFAIZAN BIN AHMAD SBP INTEGRASI SELANDAR, MELAKA.. PN ROHANI MD NOR SEKOLAH SULTAN ALAM SHAH, PUTRAJAYA. EN ZUZI BIN SHAFIE SM AGAMA PERSEKUTUAN, KAJANG. 5. PN SARIPAH BINTI AHMAD SM SAINS MUZAFFAR SYAH, MELAKA. XVII

18 Master these questions XVIII

19 Answer all questions. For eaminer s use only. f g - 6 Diagram Set A Set B Set C In Diagram, the function f maps set A to set B and the function g maps set B to set C. Determine (a) f ( ) (b) g(-) (c) gf () [ marks ] Answer : (a).. (b)... (c).... Given function f : and function g :, find (a) f, (b) the value of f g(). [ marks ] Answer : (a).. (b)...

20 For eaminer s use only Given the function f () =, 0 and the composite function f g() = (a) g(), 6. Find (b) the value of when g() = 8. [ marks] Answer :.... Give your answer correct to four significant figures. [ marks ] Solve the quadratic equation ( 5) = ( )( + ) Answer :...

21 5 (a) Given = y, find the range of if y > 0. (b) Find the range of if. [ marks] For eaminer s use only Answer :... 6 Diagram below shows the graph of a quadratic function y = f (). The straight line y = 9 is a tangent to the curve y = f (). y y = f () 0 7 y = -9 Diagram a) Write the equation of the ais of symmetry of the curve. b) Epress f () in form of ( + p) + q, where p and q are constants. [ marks ] Answer : (a)... (b) /

22 For eaminer s use only 7 Solve the equation = [ marks] 7 Answer : Given log 5 = 0.68 and log 5 7 =.09. Calculate (i) log 5., (ii) log [ marks] 8 Answer : Solve the equation log 6 log =. [ marks] 9 Answer :...

23 0. The first terms of the series are,, 8. Find the value of such that the series is a (a) an arithmetic progression, (b) a geometric progression. [ marks ] For eaminer s use only 0 Answer : The sum of the first n terms of an arithmetic progression is given by S n = n + n. Find (a) the ninth term, (b) the sum of the net 0 terms after the 9 th terms. [ marks] Answer: a) b)... 7/

24 For eaminer s use only. Given that p = = 0. + a + b +... [ marks ] Find the values of a and b. Hence, find the value of p. Answer: a =.... b =... p =.... Diagram shows a linear graph of y against y (,) (,-5) DIAGRAM y Given that = h + k, where k and h are contants. Calculate the value of h and k. [ marks] Answer : h =... k =......

25 For eaminer s use only. The equation of a straight line PQ is + y =. Find the equation of a straight line that is parallel to PQ and passes through the point ( 6, ). [ marks] Answer :. 5 7 p Given u = dan v =, find the possible values of p for each of the following 9 case: (a) u and v are parallel, [ marks] (b) u = v. [ marks] 5 Answer : a).. b) 7/

26 For eaminer s use only 6 P Q R r O s S The diagram above shows OR = r, OS = s, OP and PQ are drawn in the square grid. Epress in terms of r and s. (i) OP (ii) uuur PQ. [ marks ] 6 uuur Answer: a) OP =.... b) uuur PQ = Solve the equation cos 0 0 θ + sin θ = 0 for 0 θ 60. [ marks ] 7 Answer:

27 8. P For eaminer s use only O θ Q Diagram above shows a length of wire in the form of sector OPQ, centre O. The length of the wire is 00 cm. Given the arc length PQ is 0 cm, find (a) the angle θ in radian, (b) area of the sector OPQ. [ marks] [ marks] Answer: a) b) Find the equation of the tangent to the curve y = ( 5) at the point (, ). [ marks] 9 Answer: 7/

28 For eaminer s use only 0. A roll of wire of length 60 cm is bent into the shape of a circle. When above the wire is heated, its length increases at a rate of 0. cms. (Use π =.) (i) Calculate the rate of change of radius of the circle. (ii) Hence, calculate the radius of the circle after seconds. [ marks] [ marks] 0 5 Answer: Given f ( ) d = 5 and d = 6. 0 g ( ) Find the value (a) 0 f ( d ) + gd ( ), [ marks] (b) k if [ g ( ) k ] d=. [ marks] Answer: a).. k =.... A chess club has 0 members of whom 6 are men and are women. A team of members is selected to play in a match. Find the number of different ways of selecting the team if (a) all the players are to be of the same gender, (b) there must be an equal number of men and women. [ marks] 7/ Answer: p =..

29 7/. (a) Given that the mean for four positive integer is 9. When a number y is added to the four positive integer, the mean becomes 0. Find the value of y. [ marks] (b) Find the standard deviation for the set of numbers 5, 6, 6,, 7. [ marks] For eaminer s use only Answer: a) b) Hanif, Zaki and Fauzi will be taking a driving test. The probabilities that Hanif, Zaki and Fauzi will pass the test are, and respectively. Calculate the probability that (a) only Hanif will pass the test (b) at least one of them will pass the test. [ marks ] Answer: 7/

30 For eaminer s use only 5. Diagram below shows a standard normal distribution graph. 5 -k k z Given that the area of shaded region in the diagram is 0.788, calculate the value of k. [ marks ] Answer: END OF QUESTION PAPER 7/

31 7/ JAWAPAN (a) (b) 6 (c) 6 h =, k = 7 (a) f = (b) 5 y = (a) g() =, 0 (b) = 5 0 (a) (b) 0,.56, (b)(i) r + s (ii) r s 5. (a) < (b) 7 90,, 70, 0 6 a) = b) f ( ) = ( ) 9 8 (a) (b) 00 7 = y 09 = 0 8 ( i) 0.09 (ii).9 0 ( i) cms (ii) = (a) (b) k = 0 a) 5 b) 55 (a) 6 ( b) 50 (a ) (b).00 a = 0.06, b = 0.006, p = 6 (a) 9/5 ( b) 5/6 5 k =. 7/

32

33 THE UPPER TAIL PROBABILITY Q(z) FOR THE NORMAL DISTRIBUTION N(0, ) KEBARANGKALIAN HUJUNG ATAS Q(z) BAGI TABURAN NORMAL N(0, ) z Minus / Tolak f ( z) ep z π Q ( z) = = k f ( z) dz f O k Q(z) Eample / Contoh: If X ~ N(0, ), then Jika X ~ N(0, ), maka P(X > k) = Q(k) P(X >.) = Q(.) = z

34 SECTION A [0 marks] [0 markah] Answer all questions in this section.. Solve the equations y + y = + y = 0. [5 marks] [ Answer =, y = ; = 5, y = 5 ] Given k is the gradient function for a curve such that k is a constant. y = 0 is the equation of tangent at the point (, ) to the curve. Find, (a) the value of k, (b) the equation of the curve. [ marks] [ marks] [ Answer k = 6 ] 7 [ y = ] Diagram Diagram shows a string of length 5π cm is cut and made into ten circle as shown above. The diameter of each subsequent circles are difrent by cm from its previous. Calculate, (i) the diameter of the smallest circle, (ii) the number or a circle if the length of a circle is 00 [6 marks] Answer : (b)(i) 8 (ii) Table shows the frequency distribution of the marks of a group of form students in a test. Mark Number of students k

35 (a) It is given that the first quartile score it.5. Find the value of k. [ marks ] (b) [ Use the graph paper to answer this question] Using the scale of cm to 0 marks on the horizontal ais and cm to 0 students on the vertical ais, draw a histogram based on the given data. Hence, estimate the mode mark [ marks ] (c) Calculate the mean marks. [ Answer k =, mode = 5.5 mean = 55.8 [ marks ] 5 a) Prove that - sec θ = - sin θ tanθ secθ [ marks] (b) Sketch y = sin for 0 π. Hence using the same aes, draw a suitable straight line to find the number of solutions of the equation π sin = 0. State the number of solutions [ line y = +], number of solution ] π [5 marks] 6 5 A B y P D C In the diagram above, AB = 5, AD = y and DC =. (i) Epress, (a) AC (b) BD in terms of and y. (ii) Given AP = h AC and BP = k BD. State AP (a) in terms of h, and y, (b) in terms of k, and y. Hence, or otherwise, prove that h = k. [ marks] [5 marks] Answer (i)(a) y +, (b) 5 + y (ii)(a) h(y + ) (b) ( (5 5k) + ky; k = 6 5 7/ SULIT

36 SECTION B [ 0 Marks ] Answer four equations from this section. 7 Table 7 shows the values of two variables and y,obtained from an eperiment. Variable and y are related by the equations y = ab, where a and b are constants. One of the value of y is wrongly recorded. 5 y (a) Plot log 0 y against. (b) By using your graph find, (i) the value of y which is wrongly recorded and determine the correct value (ii) the value of a and the value of b (iii) the value of y when =.5. 8 y y = a Q y = + O (a) Refer to the diagram above, answer the following question: (i) Calculate the area of the shaded region. (ii) Q is a solid obtained when the region bounded by the curve y = + and the line y = a is revolved through 80 at the y - ais. If the volume of Q is π unit Find the value a. [6 marks] (b) Find the equation of tangent to the curve y = + r at point = k. If the tangent passes through the point (, 0), find r in terms of k. [ marks ] 56 [Answer 6. (a)(i) (ii) a = (b) y (k + r) = k( k); r = k 8k ] 8

37 Solutions to this question by scale drawing will not be accepted. 9. y U(5, 6) T(0, ) O V(p, q) W Diagram above shows the vertices of a rectangle TUVW in a Cartesian plane. (a) Find the equation which relates p and q by using the gradient of UV. (b) Shows that the area of the Δ TUV can be epressed as p 5 q + 0. [maks] [marks] (c) Hence, calculate the coordinates of V given the area of the rectangle TUVW is 5 unit. [marks] (d) Find the equation of the straight line TW in the intercept form. [marks] 0 Diagram above shows a sector MJKL of a circle centre M and two sectors, PJM and QML, with centre P and Q respectively. Given the angle of the major sector JML is.6 radian. Find, (a) the radius of the sector MJKL, [ marks] (b) perimeter of the shaded region, [ marks] (c) the area of sector PJM, [ marks] (d) the area of the shaded region. [ marks] [ Answer. (a).795 (b) 7. (c) 5 ] 7/ SULIT

38 (a) In a centre of chicken eggs incubation, 0% of the eggs hatched are male chickens. If 0 newly born chickens are chosen at random, find the probability (correct to four decimal places) that (i) eggs hatched are male chicken, (ii) at least 9 eggs hatched are female chickens. (b) The mass of guava fruits produced in a farm shows a normal distribution with mean 0 g and standard deviation g. Guava fruits with mass between 06 g and g are sold in market, whereas those with mass 06 g or less are sent to the factory to be processed as drinks. Calculate, (i) the probability (correct to four decimal places) that a guava fruit chosen randomly from the farm will be sold in the market, (ii) the number of guava fruits that has been sent to the factory and also not sold in the market, if the farm produced 500 guava fruits. [ marks] [6 marks] [ Answer (a)(i) 0.00 (ii) 0.9 (b)(i) 0.88 (ii) 00 ] Sections C Answer two questions from this section.. A B P 8 m Q In the diagram above, P and Q are two fied points on a straight line such that PQ = 8 m. At a certain instant, particle A passes the point P with a velocity V A = t 6, whereas particle B passes the point Q with a velocity V B = 5 t where t is time in second after the particle A and the particle B have passed the point P and the point Q. [Assume direction P to Q is the positive.] (a) Find the distance between the particle A and particle B at the instant when particle A stopped momentarily. [marks ] (d) Find the time, t, when the distance between the particle A and particle B is maimum before the two particles meet.. [ marks ] (c) For how long the two particles A and B are moving in the same direction? (d)(i) Find the time, t, when the particles A and B meets. (ii) Hence, find the distance from the point P when the two particles meet.. [ marks ] [Answer (a) 7 m (b) s (c) s (d)(i) 8 s (ii) 6 m ] A small factory produces a certain goods of A model and B model. In a day, the factory produces units of A model and y units of B model where 0 and y 0. Time taken to produce one unit A model and one unit B model is 5 minutes and minutes respectively. The production of these goods in a certain day is

39 restricted by the following conditions: I. The number of units of A model is not more than 60, II. The number of units of B model is more than two times the number of units of A model by 0 units or less. III. The total time for the production of A model and B model is not more than 00 minutes. Write an inequality for each of the above condition.. Hence draw the graphs for the three inequalities. Marks and shades the region R which satisfy the above conditions. Use your graph to answer the following questions: (a) Find the range of the number of units of A model which can be produced if 0 units of B model are produced. (b) Find the total maimum profits which can be obtained if the profit gained from one unit of A model and one unit of B model is RM 6 and RM respectively. (c) If the factory intends to produce the same number of units of A model and B model, find the maimum number of units which can be produced for each o A model and B model. Answer 60, y 0, 5 + y 00 (a) 5 8 (b) RM5 (c). Diagram 6 shows a quadrilateral ABCD such that ABC is acute. D 9.8 cm 5. cm C A 0.5. cm 9.5 cm DIAGRAM 6 (a) Calculate, (i) ABC, (ii) ADC, (iii) area, in cm, of quadrilateral ABCD. (b) A triagle A B C has the same measurements as those given for triangle ABC, that is, A C =. cm, C B = 9.5 cm and B A C = 0.5, but which is different in shape to triangle ABC. (i) Sketch the triangle A B C. (ii) State the size of A B C. B [8 marks] [ marks] Answer. (a)(i) (ii) (iii) (b)(i) C (ii) A B 0. Table shows the price indices and percentage usage of four items, P, Q, R, and S, which are the main ingredients of a type biscuits. 7/ SULIT

40 Item Price inde for the year 995 based on the year 99 Percentage of usage (%) P 5 0 Q 0 R 05 0 S 0 0 (a) Calculate, (i) the price of S in the year 99 if its price in the year 995 is RM7.70 (ii) the price inde of P in the year 995 based on the year 99 if its price inde in the year 99 based in the year 99 is 0. [5 marks] (b) The composite inde number of the cost of biscuits production for the year 995 based on the year 99 is 8. Calculate, (i) the value of, (ii) the price of a bo of biscuit in the year99 if the corresponding price in the year 995 is RM. [5 marks] [ Answer (a)(i) RM 9 (ii) 6 (b)(i) 5 (ii) RM 5 ] Section C Alternative Answer two questions from this section.. Diagram 6 shows ΔSTQ such that ST =. cm and TQ = 9.5 cm. T S Diagram 6 Q The area of the triangle is 5 cm and STQ is obtuse. (a) Find (i) STQ [ STQ = 8. 7 or 8 8' ] (ii) the length, in cm, of SQ [9.9 cm] (iii) the shortest distance, in cm, from T to SQ. [.6] [ 5 marks]

41 T cm 5 cm R P Q 5 cm Diagram 7 (b) Diagram 7 shows a pyramid TPQR with a horizontal triangular base PQR. T is vertically above Q. Given that PQ = QT = 5 cm, TR = cm and PRQ = 5.Calculate two possible values of PQR [ PQR = 6.60 o and.0 o ] (c) Using the acute PQR in (i), calculate ( i) the length of PR [7.67] (ii) the value of PTR [9. 0 ] (iii) the surface area of the plane TPR [.58] [ 5marks]. shows the bar chart for the monthly sales of five essential items sold at a sundry shop. Table shows their price in the year 000 and 006, and the corresponding price inde for the year 006 taking 000 as the base year. Cooking Oil Rice Salt Sugar Flour units Diagram Items Price in the year 000 Price in the year 006 Price Inde for the year 006 based on the year 000 Cooking Oil RM.50 5 Rice RM.60 RM.00 5 Salt RM0.0 RM0.55 y Sugar RM0.80 RM.0 50 Flour RM.00 z 0 TABLE 7/ SULIT

42 (a) Find the values of (i), (ii) y (iii) z. [=.00,y=7.5,z=.0] [ marks] (b) Find the composite price inde for cooking oil, rice, salt, sugar and flour in the year 006 based on the year 000. [.7] [ marks] (c) Calculate the total monthly sales for those essential items in the year 006, given that the total monthly sales in the year 000 was RM 50.[ marks] (d) the composite inde for the year 008 based on the year 000 if the monthly sales of those essential items increased by 0% from the year 006 to the year 008. [57.0] [ marks]. Use the graph paper provided to answer this question. The Mathematics Society of a school is selling souvenirs of type A and y souvenirs of type B in a charity project based on the following constraints : [0] I : The total number of souvenirs sold must not eceed 75. II : The number of souvenirs of type A sold must not eceed twice the number of souvenirs of type B sold. III : The profit gained from the selling of a souvenir of type A is RM9 while the profit gained from the selling of a souvenir of type B is RM. The total profit must not be less than RM00. (a) Write down three inequilities other than 0 dan y 0 which satisfy the above constraints. [ Answer + y 75, y and 9 + y 00] [ marks] (b) (c) Hence, by using a scale of cm to 0 souvenirs on both aes, construct and shade the region R which satisfies all the above constraints. [ marks] By using your graph from (b), find (i) the range of number of souvenirs of type A sold if 0 souvenirs of type B are sold. [ 6 number of A type souvenirs sold 5] (ii) the maksimum which may be gained. [Answer RM 500] [ marks] 5. An object, P, moves along a straight line which passes through a fied point O.

43 Figure 8 shows the object passes the point O in its motion. t seconds after leaving the point O, the velocity of P, v m s is given by v = t 8t +. The object P stops momentarily for the first time at the point B. P O (Assume right-is-positive) Figure 8 Find: (a) the velocity of P when its acceleration is ms, [9 ms ] [ marks] (b) the distance OB in meters, [0 m] [ marks] (c) the total distance travelled during the first 5 seconds. [8 m] [ marks] B. (a) (i) Use area formula (.)(9.5) sin STQ = 5 (ii) (iii) STQ = 8. 7 or 8 8' Using cosine Rule SQ = (.)(9.5) cos STQ SQ = 9.9cm TQS = h sin 9.05 = or equivalent 9.5 =.6 cm (b) 5 = sin5 sin p sin p = sin5 = QPR = 8 ', 6'@ 8.0,.60 (c) PQR = 80 o 5 o 8.0 PQR = 80 o 5 o.60 o PQR = 6.60 o and.0 o (i) PR 5 = sin. sin5 5 PR = sin. sin5 = 7.67 cm PR = 7.67 cm 7/ SULIT

44 (ii) Use Cosine Rule + ( 50) (7.67) cos PTR = = ()( 50) PTR = 9. 0 (iii) Area PVR = ()( 50) sin 9. =.58 cm. (a) (i) =.00 (ii) y = 7.5 (iii) z =.0 (b) Use composite inde formula 5(80) + 5(00) + 7.5(50) + 50(60) + 0(0) I = =.7 P =.7 (c) 50 P = RM (d) I I = = 00 = (a) The three inequalities are + y 75, y and 9 + y 00 (b) (c) (i) 6 number of A type souvenirs sold 5 (ii)maimum profit = RM [ 9(50) + (5) ] = RM500. refer by graph

45 y y = R = y 0 y = 0 ( 50, 5 ) y = 75 O / SULIT

46 For eaminer s use only Answer all questions.. Function f is defined by, f ( ) =, Find the range corresponding to the domain 0 [ marks] Answer :.. +. Given the function f: + 5, g : and fg: 5 where m and n are constants, find the value of m and of n, m + n 5, [ marks ] Answer : m =. n =... Perfect Score 009 [ Lihat sebelah SULIT

47 For eaminer s use only. Diagram shows part of the mapping of to z by the function f : a + b followed by the function g : y, y c. Calculate the values of a, b, c y c and d. 6 d Diagram [ marks] Answer: a= b= c= d=... If the -ais is a tangent to the curve + p = p, find the values of p. [ marks ] Answer : p =... Perfect Score 009 [ Lihat sebelah SULIT

48 5. Given α and β are the roots of + = 0. Form the quadratic equation with roots α and β. [ marks ] For eaminer s use only 5 Answer : Given the quadratic function of f() = 6. a) Epress the quadratic function f() in the form k + m( + n), where k, m and n are constants. b) write the equation of the ais of symmetry [ marks ] 6 Answer : (a).... (b)... Perfect Score 009 [ Lihat sebelah SULIT

49 For eaminer s use only 7. Find the range of values of if ( ) = + 5 f always positive. [ marks] 7 Answer :... n+ n n 8. Simplify and state your answer in the simplest form [ marks] 8 Answer :... y 9. Solve the equation 9 + y = +9. [ marks] 9 Answer :... Perfect Score 009 [ Lihat sebelah SULIT

50 0. Given + log k = log ( m ), epress k in terms of m. 9 + [ marks] For eaminer s use only 0 Answer : Solve the equation log = log9 ( + ) [ marks] Answer: Perfect Score 009 [ Lihat sebelah SULIT

51 For eaminer s use only th. Given that the n term, T n = 0 n for an arithmetic progression. Find the sum of the first terms of the progression. [ marks] Answer: Given the sum of the first terms of a geometric progression is 567 and the sum of the net three terms of the progression is 68. Find the sum to infinity of the progression. [ marks] Answer :. Perfect Score 009 [ Lihat sebelah SULIT

52 . Given that the sum of the first three terms of a geometric progression is times the third term of the progression. If the common ratio, r > 0, find the common ratio. [ marks ] For eaminer s use only Answer :. 5. Diagram shows the graph of log y against log. Values of and values of y are related by the equation y = k Find the value of n and the value of k. log y n, where n and k are constants. *(5, 6) 0 log (, 0) Diagram [ marks] Answer : n=.. k=. Perfect Score 009 [ Lihat sebelah SULIT

53 For eaminer s use only 6. Diagram shows a semicircle KLMN, of diameter KLM, with centre L. y K N (,y) L 0 M Diagram Given that the equation of the straight line KLM is + y = and point N(, y ) lies on the circumference of a circle KLMN, find the equation of the locus of the moving point N. [ marks ] 6 Answer: If a = i + ( p + )j and b = i + 6 j, find the value of p if a + b is parallel to the -ais. [ marks] 7 Answer: Perfect Score 009 [ Lihat sebelah SULIT

54 Given that sin 0 = a and cos0 = b, epress sin 50 in terms of a and For eaminer s use only [ marks] 8 Answer: Diagram below shows two sectors, OAB and OCD with centre O. E D C A B O Given that COD = 0.9 rad, BC = 5 cm and perimeter of sector OAB is 0. cm. Using π =., find the area of the shaded region of ABCED. [ marks ] 9 Perfect Score 009 Answer: [ Lihat sebelah SULIT

55 For eaminer s use only SULIT 7/ dy 0. Given that y= and = g() where g() is a function in. d Find g( ) d. [ marks] 0 Answer: The gradient of the curve y = h + k at the point, 7 is. Find the value of h and the value of k. [ marks] Answer:.. Perfect Score 009 SULIT

56 SULIT 7/. A coach wish to choose a player from two bowlers to represent the nation in a tournament. The following data show the number of pins scored by the two players in si sucessive bowls: Player A: 8, 9, 8, 9, 8, 6 Player B: 7, 8, 8, 9, 7, 9 By using the values of mean and standard deviation, determine the player which qualified to be choosen because the score is consistent. [ marks] For eaminer s use only Answer: In a debate competition, the probability of team A, team B and team C will qualify for the final are,, respectively. Find the probability that at least teams will qualify 5 for the final. [ marks] Answer: Perfect Score 009 SULIT

57 SULIT 7/ For eaminer s use only. The letters of the word G R O U P S are arranged in a row. Find the probability that an arrangement chosen at random (a) begins with the letter P, (b) begins with the letter P and ends with a vowels. [ marks] Answer: ( a) ( b ) The life span of certain computer chip has a normal distribution with a mean of 500 days and a standard deviation of 0 days. a) Calculate the probability that a computer chip chosen at random has a life span of more than 50 days b) Given that 6% of the computer chips have a life span of more than n days, find the value of n. [ marks] 5 Answer : (a) (b)... Perfect Score 009 SULIT

58 SULIT 7/ END OF QUESTION PAPER Perfect Score 009 SULIT

59 Paper Time: Two hours and thirty minutes Instruction : This question paper consists of three sections: Section A, Section B and Section C. Answer all questions in Section A, four questions from Section B and two questions from Section C. Give only one answer/ solution for each question. All the working steps must be written clearly. Scientific calculator that are non-programmable are allowed. Section A [0 marks]. Given that (-, k) is a solution for the simultaneous equation + py 9 = = p y where k and p are constants. Find the value of k and of p. [5 marks] Jawapan: k =, p = ; k =, p = 8. Given function f :. (a) Find, (i) f (), (ii) (f ) (). [ marks] (b) Hence, or otherwise, find (f ) () and show (f ) () = (f ) (). [ marks] (c) Sketch the graph of f () for the domain 0 and find it s corresponding range. [ marks] Jawapan: (a)(i) 9 8 (ii) (c) y y 0 0 8/9

60 . Diagram shows five semicircles. DIAGRAM The area of the semicircles form a geometric progression. Given that area of the smallest semicircle is of the area of the largest semicircle. If the total area of the 6 semicircles is 0 cm, find (a) area of the smallest semicircle (b) area of the largest semicircle [5 marks] Jawapan: (a) 0 (b) 60. Given that tan( y) = and tan y =, show that tan = Sketch the graph of y = tan for Hence, using the same aes, draw a suitable straight line and find the number of solutions for the equation tan + = 6 [6 marks] Jawapan: Number of solutions =

61 5. Diagram 5 shows a parallelogram OABC. O A P D C B DIAGRAM 5 Given that APD, OPC and DCB are straight lines. Given that OA = 6a, and OP : PC = :. (i) (ii) Jawapan: ( a) 6a+ 9c % % ( b) OC = c Epress AP in terms of a and/or c. Given the area of the ΔADB = unit and the perpendicular distance from A to DB is units, find a. [5 marks] 6. Cumulative frequency (5.5, 80) (0.5, 7) (5.5, 58) (0.5, 6) O 0.5 (5.5, 6) DIAGRAM 6 Length of fish in cm

62 Diagram 5 shows an ogive for the distribution of 80 fishes in a tank when the cumulative frequency is plotted against upper boundaries for a certain classes. O is the origin. (a) Construct a frequency table with a uniform class interval from the information given in the ogive. [ marks] (b) Draw a histogram and determine the mode. [ marks] (c) From the frequency table, find (i) (ii) the variance, the median for the length of fish in the tank. [ marks] 7. Use the graph paper provided to answer this question. An eperiment which involves samples of red blood cell used to trace the percentage, P, of the red blood cell which eperience creanation when it is added by drops of sodium chloride solution with different concentration, K mol dm. Table below shows the results of the above eperiment. Sodium chloride concentration (K) Percentage of red blood cells which eperience creanation (P) TABLE 7 Variables P and K are related by the equation P = μ (K + A) where μ and A are constants. (a) Draw the graph of P against K. [5 marks] (b) From your graph, find the value of μ and the value of A. [ marks] (c) Find the value of P when K =.? [ mark] Jawapan: ( b) μ = 0., A= 0.0 ( c)7. 8.

63 5 8. y y = ( )( + ) O (a) Diagram above shows the curve y = ( )( + ). Calculate the area bounded by the curve, -ais, line = and line =. [6 marks] (b) Diagram below shows the shaded region bounded by the curve y = +, line = and line = k. When the region is revolved 60 at the -ais, the volume generated is 8π unit. Find the value of k. [ marks] 7 [ answer (a) (b) k = ] 9. y P(0, β) Jawapan: ( a) α + β = 8 ( b)( ii),5.85 ( c)0.5 Wall R(, y) O Q(α, 0) Floor Diagram 9 shows the -ais and the y-ais which represent the floor and the wall. The end of a piece of wood PQ with length 9 m touches the wall and the floor at the point P(0, β) and point Q(α, 0). (a) Write the equation which relates α and β. [ mark] (b) Given R is a point on the piece of wood such that PR : RQ = :. (i) Show that the locus of the point R when the ends of the wood is slipping along the -ais and the y-ais is + y = 8. (ii) Find the coordinates of R when α =. (iii) Find the value of tan ORQ when α =. [9 marks]

64 6 0. L J P Q α rad R K O T Jawapan: ( a) π 0 ( b) π ( c)6.50 DIAGRAM 0 Diagram 0 shows a circle PQRT, centre O and radius 5 cm. JQK is a tangent to the circle at Q. The straight lines, JO and KO, intersect the circle at P and R respectively. OPQR is a rhombus. JLK is an arc of a circle, centre O. Calculate (a) the angle α, in terms of π [ marks] (b) the length, in cm, of the arc JKL [ marks] (c) the area, in cm, of the shaded region. [ marks]. (a) A study on post graduate students, revealed that 70% out of them obtained jobs si months after graduating. (i) If 5 post graduates were chosen at random, find the probability of not more than students not getting jobs after si months. (ii) It is epected that 80 students will succeed in obtaining jobs after si months. Find the total number of students involved in the study. [5 marks] (b) The mass of 5000 students in a college is normally distributed with a mean of 58kg and variance of 5 kg. Find (i) the number of students with the mass of more than 90 kg. (ii) the value of w if 0% of the students in the colleges are less than w kg. [5 marks] Jawapan: (a) (i) 0.68 (ii)00 (b) (i) 8or 8 (ii) 8.77 or 8.79

65 7. Diagram shows the position and direction of motion for two objects, P and Q, which move along a straight line and passes through two fied points, A and B respectively. At the instant when P passes through the fied point A, Q passes through the fied point B. Distance AB is 8 m. P Q A C B 8 m DIAGRAM The velocity of P, v p ms, is given by v p = 6 + t t, where t is the time in seconds, after passing through A, whereas Q moves with a constant velocity of ms. Object P stops instantaneously at the point C. (Assume towards the right is positive.) Find, (a) the maimum velocity, in ms, for P, [ marks] (b) the distance, in m, C from A, [ marks] (c) the distance, in m, between P and Q at the instant when P is at the point C. [ marks] Jawapan: (a) 8 m/s (b) 8 (c)

66 8.. Diagram shows a quadrilateral ABCD such that ABC is acute. 9.8 cm D 5. cm C A cm 9.5 cm B (a) Calculate, (i) ABC, (ii) ADC, DIAGRAM [8 marks] (iii) area, in cm, of quadrilateral ABCD. (b) A triangle A B C has the same measurements as those given for triangle ABC, that is, A C =. cm, C B = 9.5 cm and B A C = 0.5, but which is different in shape to triangle ABC. (i) Sketch the triangle A B C. (ii) State the size of A B C. [ marks] Jawapan: (a) (i) (ii) (iii) (b) (ii)

67 9. Use the graph paper provided to answer this question. Cloth Preparation time (minutes) Sewing time (minutes) T-shirt 5 50 Slack 0 70 A tailor shop received payment only for sewing T-shirt and slack. Preparation time and sewing time for each T-shirt and slack are shown in the table above. The maimum preparation time used is0 hours and the sewing time must be at least 5 hours 50 minutes. The ratio of the number of T-shirt to slack is not more than : 5. In a certain time, the shop is able to complete pieces of T-shirt and y pieces of slack. (a) Write three inequalities, other than 0 and y 0, which satisfy the above conditions. [ marks] (b) By using a scale of cm to I unit on the -ais and cm to units on the y-ais, draw the graphs for the three inequalities. Hence, shades the region R which satisfies the above conditions. [ marks] (c) Based on your graph, find (i) the minimum number of slacks which can be sewn in that time if pieces of of T-shirt has been sewn.. (ii) maimum total profit received in that time if the profit gained from each piece of T-shirt and slack are RM6 and RM 0 respectively. [ marks] Jawapan: ( a)5+ y y 50 5 y () c (6,), RM 06

68 0 5. (a) Inde number, I i Weightage, W i 5 The composite inde number for the data in the table above is 08. Find the value of. [ marks] (b) (i) In the year 995, price and price inde for one kilogram of certain grade of rice is RM.0 and 60 respectively. Based on the year 990, calculate the price per kilogram of rice in the year 990. [ marks] Item Price inde in Change of price inde from the Weightage the year 99 year 99 to the year 996 Timber 80 Increased 0 % 5 Cement 6 Decreased 5 % Iron 0 No change Steel No change (ii) Table above shows the price inde in the year 99 based on the year 99, the change in price inde from the year 99 to the year 996 and the weightage respectively. Calculate the composite price inde in the year [ marks] Jawapan: (a) = (b) (i).50 (ii) RM5.90 End of question paper

69 - - FUNCTIONS. Given that f : + m and f : n +, find the values of m and n. Answer:- m = ; n =. Given that f :, g : and fg : a + b, find the values of a and b. Answer:- a = 8 ; b =. Given that f : +, g : a + b and gf : , find the values of a and b. Answer:- a = ; b = 6. Given that g : m + and g : k, find the values of m and k. 5. Given the inverse function f ( ) =, find (a) the value of f(), (b) the value of k if f (k) = k. 6. Given the function f : and g :, find Answer:- k = 6 ; m = Answer:-(a) (b) (a) f (), (b) f g(), (c) h() such that hg() = 6. Answer:-(a) + (b) (c) Diagram shows the function p + g:,, where p is a constant. 7 g Diagram p + 5 Find the value of p.

70 - - Answer:- p = 8. y z Diagram Diagram shows the mapping of y to by the function g : y ay + b and mapping 6 to z by the function h : y y b, y b. Find the, (a) value of a and value of b, (b) the function which maps to y, (c) the function which maps to z. Answer:- (a)a= 6, b=0 (b) 0 y 8 (c) 6 y 0 9. In the Diagram, function h mapped to y and function g mapped y to z. h y g z 8 5 Diagram Determine the values of, (a) h (5), (b) gh() Answer:- (a) (b)8 0. Given function f : and function g : k + n. If composite function gf is given as gf : + 8, find (a) the value of k and value of n, (b) the value of g (0). Answer:-(a) k =,n = (b)

71 - -. The following information refers to the functions f and g. Find f (). Answer:-. (a) Function f, g and h are given as f : g :, h : 6. (i) Determine the function fh(). At the same ais, sketch the graphs of y = g() and y = fh(). Hence, determine the number of solutions for g() = fh(). (ii) Find the value of g ( ). (b) Function m is defined as m : 5. If p is another function and mp is defined as mp :, determine function p. p = +. Given function f :. (a) Find (i) f (), (ii) (f ) (). g () = fg () = + 5 Answer:-(a)(i) (b) ( ) (b) Hence, or otherwise, find (f ) () and show (f ) () = (f ) (). (c) Sketch the graph of f () for the domain 0 and find it s corresponding + 8 range. Answer:-(a)9 8 (b) 9. A function f is defined as f : are constants. p+, for all values of ecept = h and p + (a) Determine the value h. (b) Given value is mapped to itself by the function f. Find the (i) value p, (ii) another value of which is mapped to itself, (iii) value of f ( ). Answer:-(a) h = (b)(i)p =(ii) = (iii) 5

72 - - QUADRATIC EQUATIONS. One of the roots of the quadratic equation is twice the other root. Find the possible values of p. Answer ; p = 5, 7. If one of the roots of the quadratic equation is two times the other root, find an epression that relates. Answer : b = 9ac. Find the possible value of m, if the quadratic equation has two equal roots. Answer ;. Straight line y = m + is tangent to the curve + y + y = 0. Find the possible values of m. Answer : or 5. Given α and β are roots of the equation k ( ) = m. If α + β = 6 and α β =, find the value of k and of m. Answer : k =, m = 6 6. Find the values of λ such that the equation ( λ) (λ + ) + λ + = 0 has equal roots. Hence, find the roots of the equation base on the values of λ obtained. Answer : λ = ± ; roots: λ =, = ; λ =, = 0

73 - 5 - QUADRATIC FUNCTIONS. Diagram shows the graph of the function ( ) y y = p + 5, where p is constant. 0 ( 0, ) (, ) Find, (a) the value of p, Diagram (b) the equation of the ais of symmetry, (c) the coordinate of the maimum point. Answer:- (a) p = (b) = (c) (, 5 ). y 0 ( 0, ) ( ) ( ) f = p + + q Diagram shows the graph of the function ( ) ( ) (a) State the value of q. (b) Find the range of values of p. Diagram f = p + + q. Answer:-(a) q = (b) p <

74 y ( 0, 9) y = + b+ c K Diagram Diagram shows the graph of the function y = + b+ cthat intersects the y- ais at point ( 0, 9 ) and touches the - ais at point K. Find, (a) the value of b and c, (b) the coordinates of point K. Answer:-(a) b = 6, c = 9 (b) ( 0, ). y ( 0, ) (, ) 0 Diagram In Diagram above point (, ) is the turning point on the graph which has equation of the form y = p( + h) + k. Find the, (a) values of p, h and k, (b) equation of the curve formed when the graph as shown is reflected at the ais. (c) equation of the curve formed when the graph as shown is reflected at the y ais. Answer :- (a) p = 5, h =, k = (b) y = 5( ) (c) y = 5( + ) +

75 Function ( ) are constants. f = 8k+ 0k + has a minimum value of r + k, where r and k (a) By using the method of completing the square, show that r = k. (b) Hence or otherwise, find the values of k and r if the graph of the function is symmetrical about = r. 6. The function ( ) ( 6 )( ) (a) Find the value of h. Answer:-(b) k =, and r =, 5 f = + + h has a maimum value of 0 and h is a constant. (b) Sketch the graph of ( ) ( 6 )( ) in (a) above. (c) Write the equation of the ais of symmetry. f = + + h for the value of h that is determined Answer:- ( a) h = --6 (c) = 7. Given y = + k + k has minimum value. (a) Without using the method of differentiation, find the two possible values of k. (b) With these values of k, sketch on the same ais, two graphs for y = + k + k. (c) State the coordinates of the minimum point for y = + k + k. Answer:- (a) k =, (c) (, ), (, ) ****************************************************************************** SIMULTANEOUS EQUATIONS. Solve the simultaneous equations + y = and y = 5 + y.. Solve the simultaneous equations Answer: = -7/, y = ; =, y = - y 5 and + y = = y 5 Answer: = /0, y = 6/ ; =, y = -5. Solve the simultaneous equations - y = and + y - = 7. Give your answers correct to three decimal places. Answer: =.6, y = 0.9; = -0.7, y = -5.