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11 M13/5/MATME/SP/ENG/TZ1/XX 3 M13/5/MATME/SP/ENG/TZ1/XX Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. In particular, solutions found from a graphic display calculator should be supported by suitable working, for example if graphs are used to find a solution, you should sketch these as part of your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working. Section a Answer all questions in the boxes provided. Working may be continued below the lines if necessary. 1. [Maximum mark: 7] An arithmetic sequence is given by 5, 8, 11,.. [Maximum mark: 6] Consider the following cumulative frequency table. x Frequency cumulative frequency p (a) Find the value of p. [ marks] (a) Write down the value of d. [1 mark] (b) Find (b) Find (i) the mean; (i) u 100 ; (ii) S 100. [4 marks] (c) Given that u n = 150, find the value of n. [ marks] (ii) the variance. [4 marks] turn over

12 4 M13/5/MATME/SP/ENG/TZ1/XX 5 M13/5/MATME/SP/ENG/TZ1/XX 3. [Maximum mark: 5] In the expansion of (3x ), the term in x can be expressed as p q (3 x) ( ). r (a) Write down the value of p, of q and of r. [3 marks] (b) Find the coefficient of the term in 5 x. [ marks] 4. [Maximum mark: 6] Consider the system of equations x y+ z =.5 x+ y = 1 x y+ z = 3 This system can be represented by the matrix equation AX = B, where (a) (i) Write down the matrix A. (ii) Write down the matrix x X = y. z 1 A. [3 marks] (b) Hence, find X. [3 marks] turn over

13 6 M13/5/MATME/SP/ENG/TZ1/XX 7 M13/5/MATME/SP/ENG/TZ1/XX 5. [Maximum mark: 8] 1 The velocity of a particle in ms is given by sin t v = e 1, for 0 t [Maximum mark: 6] Let f and g be functions such that g( x) = f( x+ 1) + 5. (a) On the grid below, sketch the graph of v. [3 marks] (a) The graph of f is mapped to the graph of g under the following transformations: v t p vertical stretch by a factor of k, followed by a translation. q Write down the value of (i) k ; (ii) p ; (iii) q. [3 marks] (b) (i) Write down the positive t-intercept. (b) Let h( x) = g(3 x). The point A(6, 5) on the graph of g is mapped to the point A on the graph of h. Find A. [3 marks] (ii) Find the total distance travelled by the particle in the first five seconds. [5 marks] turn over

14 8 M13/5/MATME/SP/ENG/TZ1/XX 9 M13/5/MATME/SP/ENG/TZ1/XX 7. [Maximum mark: 7] Do NOT write solutions on this page. A random variable X is normally distributed with µ = 150 and σ = 10. Find the interquartile range of X. Section B Answer all questions in the answer booklet provided. Please start each question on a new page. 8. [Maximum mark: 14] The diagram shows a circle of radius 8 metres. The points ABCD lie on the circumference of the circle. B C A D BC = 14 m, CD = 11.5 m, AD = 8 m, ADC ˆ = 104, and BCD ˆ = (a) Find AC. [3 marks] (b) (i) Find ˆ ACD. ii ence find ˆ ACB. [5 marks] (c) Find the area of triangle ADC. [ marks] d ence or otherise find the total area o the shaded reions. [4 marks]

15 10 M13/5/MATME/SP/ENG/TZ1/XX 11 M13/5/MATME/SP/ENG/TZ1/XX Do NOT write solutions on this page. 9. [Maximum mark: 15] 100 Let f( x) = 1+ 50e 0.x ( ) y. Part of the graph of f is shown below. Do NOT write solutions on this page. 10. [Maximum mark: 16] A Ferris wheel with diameter 1 metres rotates clockwise at a constant speed. The wheel completes.4 rotations every hour. The bottom of the wheel is 13 metres above the ground. 1 diagram not to scale 50 x (a) Write down f (0). [1 mark] ground 13 (b) Solve f( x ) = 95. [ marks] (c) Find the range of f. [3 marks] (d) 1000e Show that f ( x) = 1+ 50e 0.x 0.x ( ). [5 marks] (e) Find the maximum rate of change of f. [4 marks] A seat starts at the bottom of the wheel. (a) Find the maximum height above the ground of the seat. [ marks] After t minutes, the height h metres above the ground of the seat is given by h = 74 + acosbt. (b) (i) Show that the period of h is 5 minutes. (ii) Write down the exact value of b. [ marks] (c) Find the value of a. [3 marks] (d) Sketch the graph of h, for 0 t 50. [4 marks] (e) In one rotation of the wheel, find the probability that a randomly selected seat is at least 105 metres above the ground. [5 marks]

16 3 M09/5/MATME/SP/ENG/TZ1/XX 4 M09/5/MATME/SP/ENG/TZ1/XX Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. In particular, solutions found from a graphic display calculator should be supported by suitable working, e.g. if graphs are used to find a solution, you should sketch these as part of your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working.. [Maximum mark: 7] The circle shown has centre O and radius 3.9 cm. SECTION A Answer all the questions in the spaces provided. Working may be continued below the lines, if necessary. 1. [Maximum mark: 5] diagram not to scale In an arithmetic series, the first term is 7 and the sum of the first 0 terms is 60. (a) Find the common difference. [3 marks] (b) Find the value of the 78 th term. [ marks] Points A and B lie on the circle and angle AOB is 1.8 radians. (a) Find AB. [3 marks] (b) Find the area of the shaded region. [4 marks]

17 5 M09/5/MATME/SP/ENG/TZ1/XX 6 M09/5/MATME/SP/ENG/TZ1/XX 3. [Maximum mark: 6] 3x x Let f( x) 1, g( x) cos 4 1. Let h( x) ( g f)( x) [Maximum mark: 6] A random variable X is distributed normally with mean 450 and standard deviation 0. (a) Find P( X 475). [ marks] (a) Find an expression for h( x). [3 marks] (b) Given that P(X a) 07., find a. [4 marks] (b) Write down the period of h. [1 mark] (c) Write down the range of h. [ marks]

18 7 M09/5/MATME/SP/ENG/TZ1/XX 8 M09/5/MATME/SP/ENG/TZ1/XX 5. [Maximum mark: 6] Two lines with equations r 1 3 s 3 and r t 5 intersect at the 1 1 point P. Find the coordinates of P. 6. [Maximum mark: 7] 1 1 In a geometric series, u 1 and u (a) Find the value of r. [3 marks] (b) Find the smallest value of n for which S n 40. [4 marks]

19 9 M09/5/MATME/SP/ENG/TZ1/XX 10 M09/5/MATME/SP/ENG/TZ1/XX 7. [Maximum mark: 8] In any given season, a soccer team plays 65 % of their games at home. When the team plays at home, they win 83 % of their games. When they play away from home, they win 6 % of their games. The team plays one game. (a) Find the probability that the team wins the game. [4 marks] (b) If the team does not win the game, find the probability that the game was played at home. [4 marks] Do NOT write on this page. SECTION B Answer all the questions on the answer sheets provided. Please start each question on a new page. 8. [Maximum mark: 15] A fisherman catches 00 fish to sell. He measures the lengths, l cm of these fish, and the results are shown in the frequency table below. Length l cm 0l l 0 0 l l l l l 100 Frequency (a) Calculate an estimate for the standard deviation of the lengths of the fish. [3 marks] (b) A cumulative frequency diagram is given below for the lengths of the fish. (This question continues on the following page)

20 11 M09/5/MATME/SP/ENG/TZ1/XX 1 M09/5/MATME/SP/ENG/TZ1/XX Do NOT write on this page. (Question 8 (b) continued) Use the graph to answer the following. (i) (ii) Estimate the interquartile range. Given that 40 % of the fish have a length more than k cm, find the value of k. In order to sell the fish, the fisherman classifies them as small, medium or large. Small fish have a length less than 0 cm. Medium fish have a length greater than or equal to 0 cm but less than 60 cm. Large fish have a length greater than or equal to 60 cm. [6 marks] (c) Write down the probability that a fish is small. [ marks] The cost of a small fish is $4, a medium fish $10, and a large fish $1. Do NOT write on this page. 9. [Maximum mark: 15] Let f( x)ax bx c where a, b and c are rational numbers. (a) The point P( 4, 3 ) lies on the curve of f. Show that 16a4bc 3. [ marks] (b) The points Q(6, 3) and R(, 1 ) also lie on the curve of f. Write down two other linear equations in a, b and c. (c) These three equations may be written as a matrix equation in the form AX B, a where X b. c (i) Write down the matrices A and B. (ii) Write down A 1. [ marks] (d) Copy and complete the following table, which gives a probability distribution for the cost $X. [ marks] (iii) Hence or otherwise, find f( x). [8 marks] Cost $X (d) Write f( x) in the form f( x) a( xh) k, where a, h and k are rational numbers. [3 marks] P( X x) (e) Find E( X ). [ marks] 10. [Maximum mark: 15] 3 Let f( x) x 4x1. (a) Expand ( x h) 3. [ marks] f xh f x (b) Use the formula f( x) lim ( ) ( ) to show that h0 h the derivative of f( x) is 3x 4. [4 marks] (c) The tangent to the curve of f at the point P( 1, ) is parallel to the tangent at a point Q. Find the coordinates of Q. [4 marks] (d) The graph of f is decreasing for p x q. Find the value of p and of q. [3 marks] (e) Write down the range of values for the gradient of f. [ marks]

21 3 M09/5/MATME/SP/ENG/TZ/XX+ 4 M09/5/MATME/SP/ENG/TZ/XX+ Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. In particular, solutions found from a graphic display calculator should be supported by suitable working, e.g. if graphs are used to find a solution, you should sketch these as part of your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working.. [Maximum mark: 6] Consider the graph of f shown below. SECTION A Answer all the questions in the spaces provided. Working may be continued below the lines, if necessary. 1. [Maximum mark: 5] The following diagram is a box and whisker plot for a set of data. The interquartile range is 0 and the range is 40. (a) Write down the median value. [1 mark] (a) On the same grid sketch the graph of y f( x). [ marks] (This question continues on the following page) (b) Find the value of (i) a ; (ii) b. [4 marks]

22 5 M09/5/MATME/SP/ENG/TZ/XX+ 6 M09/5/MATME/SP/ENG/TZ/XX+ (Question continued) The following four diagrams show images of f under different transformations. Diagram A Diagram B Diagram C Diagram D 3. [Maximum mark: 5] Solve the equation e x 4sin x, for 0 x. (b) Complete the following table. [ marks] Description of transformation Horizontal stretch with scale factor 1.5 Maps f to f( x)1 Diagram letter (c) Give a full geometric description of the transformation that gives the image in Diagram A. [ marks]

23 7 M09/5/MATME/SP/ENG/TZ/XX+ 8 M09/5/MATME/SP/ENG/TZ/XX+ 4. [Maximum mark: 8] The diagram below shows a triangle ABD with AB 13 cm and AD 65. cm. Let C be a point on the line BD such that BC AC 7 cm. 13 B 7 C D 7 (a) Find the size of angle ACB. [3 marks] (b) Find the size of angle CAD. [5 marks] A 6.5 diagram not to scale 5. [Maximum mark: 7] 7 r (a) Expand as the sum of four terms. r4 r (b) (i) Find the value of. r (ii) Explain why cannot be evaluated. r4 30 r4 [1 mark] [6 marks]

24 9 M09/5/MATME/SP/ENG/TZ/XX+ 10 M09/5/MATME/SP/ENG/TZ/XX+ 6. [Maximum mark: 7] Consider the curve y ln ( 3x1 ). Let P be the point on the curve where x. (a) Write down the gradient of the curve at P. [ marks] 7. [Maximum mark: 7] The quadratic equation kx ( k3) x10 has two equal real roots. (a) Find the possible values of k. [5 marks] (b) The normal to the curve at P cuts the x-axis at R. Find the coordinates of R. [5 marks] (b) Write down the values of k for which x ( k3) xk 0 has two equal real roots. [ marks]

25 11 M09/5/MATME/SP/ENG/TZ/XX+ 1 M09/5/MATME/SP/ENG/TZ/XX+ Do NOT write on this page. SECTION B Answer all the questions on the answer sheets provided. Please start each question on a new page. 8. [Maximum mark: 14] Let f( x) x( x5), for 0 x 6. The following diagram shows the graph of f. Do NOT write on this page. 9. [Maximum mark: 13] A van can take either Route A or Route B for a particular journey. If Route A is taken, the journey time may be assumed to be normally distributed with mean 46 minutes and a standard deviation 10 minutes. If Route B is taken, the journey time may be assumed to be normally distributed with mean µ minutes and standard deviation 1 minutes. (a) For Route A, find the probability that the journey takes more than 60 minutes. [ marks] R (b) For Route B, the probability that the journey takes less than 60 minutes is Find the value of µ. (c) The van sets out at 06:00 and needs to arrive before 07:00. [3 marks] Let R be the region enclosed by the x-axis and the curve of f. (a) Find the area of R. [3 marks] (b) Find the volume of the solid formed when R is rotated through 360 about the x-axis. (c) The diagram below shows a part of the graph of a quadratic function g( x) xa ( x). The graph of g crosses the x-axis when x a. [4 marks] (d) (i) Which route should it take? (ii) Justify your answer. [3 marks] On five consecutive days the van sets out at 06:00 and takes Route B. Find the probability that (i) it arrives before 07:00 on all five days; (ii) it arrives before 07:00 on at least three days. [5 marks] The area of the shaded region is equal to the area of R. Find the value of a. [7 marks]

26 13 M09/5/MATME/SP/ENG/TZ/XX+ M10/5/MATME/SP/ENG/TZ1/XX+ Do NOT write on this page. 10. [Maximum mark: 18] Let f( x) 3sinx4 cos x, for x. (a) Sketch the graph of f. [3 marks] (b) (c) Write down (i) (ii) the amplitude; the period; (iii) the x-intercept that lies between and 0. [3 marks] Hence write f( x) in the form psin( qx r). [3 marks] (d) Write down one value of x such that f( x) 0. [ marks] (e) Write down the two values of k for which the equation f( x) k has exactly two solutions. (f) Let g( x) ln ( x 1 ), for 0 x. There is a value of x, between 0 and 1, for which the gradient of f is equal to the gradient of g. Find this value of x. [ marks] [5 marks] Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. In particular, solutions found from a graphic display calculator should be supported by suitable working, e.g. if graphs are used to find a solution, you should sketch these as part of your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working. SECTION A Answer all the questions in the spaces provided. Working may be continued below the lines, if necessary. 1. [Maximum mark: 5] 1 3 Let A and B (a) Write down A 1. [ marks] (b) Solve AX B. [3 marks]

27 3 M10/5/MATME/SP/ENG/TZ1/XX+ 4 M10/5/MATME/SP/ENG/TZ1/XX+. [Maximum mark: 6] Consider the arithmetic sequence 3, 9, 15,, (a) Write down the common difference. [1 mark] 3. [Maximum mark: 7] Let f( x) xcos x, for 0 x 6. (a) Find f( x). [3 marks] (b) Find the number of terms in the sequence. [3 marks] (c) Find the sum of the sequence. [ marks] (b) On the grid below, sketch the graph of y f( x) [4 marks]

28 5 M10/5/MATME/SP/ENG/TZ1/XX+ 6 M10/5/MATME/SP/ENG/TZ1/XX+ 4. [Maximum mark: 6] The following frequency distribution of marks has mean 4.5. Mark Frequency x [Maximum mark: 7] The graph of y pcos qx r, for 5 x 14, is shown below. y ( 4, 7) (a) Find the value of x. [4 marks] (b) Write down the standard deviation. [ marks] There is a minimum point at ( 0, 3) and a maximum point at ( 4, 7 ). (a) Find the value of (i) p ; (ii) q ; (iii) r. ( 0, 3) x [6 marks] (b) The equation y k has exactly two solutions. Write down the value of k. [1 mark]

29 7 M10/5/MATME/SP/ENG/TZ1/XX+ 8 M10/5/MATME/SP/ENG/TZ1/XX+ 6. [Maximum mark: 7] The acceleration, a ms, of a particle at time t seconds is given by 1 a 3sin t, for t 1. t The particle is at rest when t 1. Find the velocity of the particle when t [Maximum mark: 7] Evan likes to play two games of chance, A and B. For game A, the probability that Evan wins is 0.9. He plays game A seven times. (a) Find the probability that he wins exactly four games. [ marks] For game B, the probability that Evan wins is p. He plays game B seven times. (b) (c) Write down an expression, in terms of p, for the probability that he wins exactly four games. Hence, find the values of p such that the probability that he wins exactly four games is [ marks] [3 marks]

30 9 M10/5/MATME/SP/ENG/TZ1/XX+ 10 M10/5/MATME/SP/ENG/TZ1/XX+ Do NOT write on this page. Do NOT write on this page. SECTION B Answer all the questions on the answer sheets provided. Please start each question on a new page. 8. [Maximum mark: 14] 9. [Maximum mark: 16] Let f( x) Ae kx 3. Part of the graph of f is shown below. y The diagram below shows a quadrilateral ABCD with obtuse angles ABC and ADC. B 13 5 x 4 C A 30 4 y D 4 diagram not to scale The y-intercept is at ( 0, 13 ) x AB 5 cm, BC 4 cm, CD 4 cm, AD 4 cm, BAC 30, ABC x, ADC y. (a) Use the cosine rule to show that AC 4140cos x. [1 mark] (b) Use the sine rule in triangle ABC to find another expression for AC. [ marks] (c) (i) Hence, find x, giving your answer to two decimal places. (ii) Find AC. [6 marks] (d) (i) Find y. (ii) Hence, or otherwise, find the area of triangle ACD. [5 marks] (a) Show that A 10. [ marks] (b) Given that f ( 15) 349. (correct to 3 significant figures), find the value of k. [3 marks] (c) (i) Using your value of k, find f( x). (ii) Hence, explain why f is a decreasing function. (iii) Write down the equation of the horizontal asymptote of the graph f. Let g( x)x 1x4. [5 marks] (d) Find the area enclosed by the graphs of f and g. [6 marks]

31 11 M10/5/MATME/SP/ENG/TZ1/XX+ 3 M1/5/MATME/SP/ENG/TZ1/XX Do NOT write on this page. 10. [Maximum mark: 15] The weights of players in a sports league are normally distributed with a mean of 76.6 kg, (correct to three significant figures). It is known that 80 % of the players have weights between 68 kg and 8 kg. The probability that a player weighs less than 68 kg is (a) Find the probability that a player weighs more than 8 kg. [ marks] (b) (i) Write down the standardized value, z, for 68 kg. (ii) Hence, find the standard deviation of weights. [4 marks] Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. In particular, solutions found from a graphic display calculator should be supported by suitable working, e.g. if graphs are used to find a solution, you should sketch these as part of your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working. SecTion a Answer all questions in the boxes provided. Working may be continued below the lines if necessary. 1. [Maximum mark: 6] The first three terms of an arithmetic sequence are 36, 40, 44,. To take part in a tournament, a player s weight must be within 1.5 standard deviations of the mean. (c) (i) Find the set of all possible weights of players that take part in the tournament. (ii) A player is selected at random. Find the probability that the player takes part in the tournament. Of the players in the league, 5 % are women. Of the women, 70 % take part in the tournament. (d) Given that a player selected at random takes part in the tournament, find the probability that the selected player is a woman. [5 marks] [4 marks] (a) (i) Write down the value of d. (ii) Find u 8. [3 marks] (b) (i) Show that S = n n + 34 n. (ii) Hence, write down the value of S 14. [3 marks]

32 4 M1/5/MATME/SP/ENG/TZ1/XX 5 M1/5/MATME/SP/ENG/TZ1/XX. [Maximum mark: 7] Let f( x)= x 8x 9. (a) (i) Write down the coordinates of the vertex. (ii) Hence or otherwise, express the function in the form f( x) = ( x h) + k. [4 marks] 3. [Maximum mark: 6] Let M = 3 1 x x and N = x (a) Find det M. [ marks] (b) Solve the equation f( x)= 0. [3 marks] (b) Write down det N. [1 mark] (c) Find the value of x for which detm = det N. [3 marks]

33 6 M1/5/MATME/SP/ENG/TZ1/XX 7 M1/5/MATME/SP/ENG/TZ1/XX 4. [Maximum mark: 7] 5π The graph of y = ( x 1 )sin x, for 0 x, is shown below. y 5. [Maximum mark: 6] p Let M = 1 1 and M 1 1 = q (a) Find the value of p and of q. [3 marks] (b) Solve the system of linear equations. 0 1 π k x px y z = 7 x+ y z = x+ qy z = 3 [3 marks] The graph has x-intercepts at 0, 1, π and k. (a) Find k. [ marks] The shaded region is rotated 360 about the x-axis. Let V be the volume of the solid formed. (b) Write down an expression for V. [3 marks] (c) Find V. [ marks]

34 8 M1/5/MATME/SP/ENG/TZ1/XX 9 M1/5/MATME/SP/ENG/TZ1/XX 6. [Maximum mark: 6] 3 b Consider the expansion of x + = 56x + 307x + + kx +. x 8 (a) Find b. [3 marks] (b) Find k. [3 marks] 7. [Maximum mark: 7] The probability of obtaining tails when a biased coin is tossed is The coin is tossed ten times. Find the probability of obtaining (a) at least four tails; [4 marks] (b) the fourth tail on the tenth toss. [3 marks]

35 10 M1/5/MATME/SP/ENG/TZ1/XX 11 M1/5/MATME/SP/ENG/TZ1/XX Do NOT write solutions on this page. Do NOT write solutions on this page. SecTion B Answer all questions on the answer sheets provided. Please start each question on a new page. 8. [Maximum mark: 13] The histogram below shows the time T seconds taken by 93 children to solve a puzzle. f Frequency (Question 8 continued) Consider the class interval 45 T < 55. (c) (i) Write down the interval width. (d) (ii) Write down the mid-interval value. [ marks] Hence find an estimate for the (i) mean; (ii) standard deviation. [4 marks] John assumes that T is normally distributed and uses this to estimate the probability that a child takes less than 95 seconds to solve the puzzle. (e) Find John s estimate. [ marks] Time The following is the frequency distribution for T T Time 45 T < T < T < T < T < T < T < 115 Frequency 7 14 p 0 18 q 6 (a) (i) Write down the value of p and of q. (ii) Write down the median class. [3 marks] (b) A child is selected at random. Find the probability that the child takes less than 95 seconds to solve the puzzle. [ marks] (This question continues on the following page)

36 1 M1/5/MATME/SP/ENG/TZ1/XX 13 M1/5/MATME/SP/ENG/TZ1/XX Do NOT write solutions on this page. 9. [Maximum mark: 15] The following diagram shows a triangle ABC. C Do NOT write solutions on this page. 10. [Maximum mark: 17] The following diagram shows two ships A and B. At noon, ship A was 15 km due north of ship B. Ship A was moving south at 15 km h 1 and ship B was moving east at 11 km h 1. A 5p 6 15 A 0.7 4p B B BC = 6, CAB = 07. radians, AB = 4 p, AC = 5p, where p > 0. (a) (i) Show that p ( 41 40cos 07. ) = 36. (ii) Find p. [4 marks] Consider the circle with centre B that passes through the point C. The circle cuts the line CA at D, and ADB is obtuse. Part of the circle is shown in the following diagram. C (a) Find the distance between the ships (i) at 13:00; (ii) at 14:00. [5 marks] Let st () be the distance between the ships t hours after noon, for 0 t 4. 6 (b) Show that st ()= 346t 450t+ 5. [6 marks] (c) Sketch the graph of st (). [3 marks] A 0.7 D B (d) Due to poor weather, the captain of ship A can only see another ship if they are less than 8 km apart. Explain why the captain cannot see ship B between noon and 16:00. [3 marks] (b) Write down the length of BD. [1 mark] (c) Find ADB. [4 marks] (d) (i) Show that CBD =19. radians, correct to decimal places. (ii) Hence, find the area of the shaded region. [6 marks]

37 3 M09/5/MATME/SP/ENG/TZ/XX+ 4 M09/5/MATME/SP/ENG/TZ/XX+ Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. In particular, solutions found from a graphic display calculator should be supported by suitable working, e.g. if graphs are used to find a solution, you should sketch these as part of your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working.. [Maximum mark: 6] Consider the graph of f shown below. SECTION A Answer all the questions in the spaces provided. Working may be continued below the lines, if necessary. 1. [Maximum mark: 5] The following diagram is a box and whisker plot for a set of data. The interquartile range is 0 and the range is 40. (a) Write down the median value. [1 mark] (a) On the same grid sketch the graph of y f( x). [ marks] (This question continues on the following page) (b) Find the value of (i) a ; (ii) b. [4 marks]

38 5 M09/5/MATME/SP/ENG/TZ/XX+ 6 M09/5/MATME/SP/ENG/TZ/XX+ (Question continued) The following four diagrams show images of f under different transformations. Diagram A Diagram B Diagram C Diagram D 3. [Maximum mark: 5] Solve the equation e x 4sin x, for 0 x. (b) Complete the following table. [ marks] Description of transformation Horizontal stretch with scale factor 1.5 Maps f to f( x)1 Diagram letter (c) Give a full geometric description of the transformation that gives the image in Diagram A. [ marks]

39 7 M09/5/MATME/SP/ENG/TZ/XX+ 8 M09/5/MATME/SP/ENG/TZ/XX+ 4. [Maximum mark: 8] The diagram below shows a triangle ABD with AB 13 cm and AD 65. cm. Let C be a point on the line BD such that BC AC 7 cm. 13 B 7 C D 7 (a) Find the size of angle ACB. [3 marks] (b) Find the size of angle CAD. [5 marks] A 6.5 diagram not to scale 5. [Maximum mark: 7] 7 r (a) Expand as the sum of four terms. r4 r (b) (i) Find the value of. r (ii) Explain why cannot be evaluated. r4 30 r4 [1 mark] [6 marks]

40 9 M09/5/MATME/SP/ENG/TZ/XX+ 10 M09/5/MATME/SP/ENG/TZ/XX+ 6. [Maximum mark: 7] Consider the curve y ln ( 3x1 ). Let P be the point on the curve where x. (a) Write down the gradient of the curve at P. [ marks] 7. [Maximum mark: 7] The quadratic equation kx ( k3) x10 has two equal real roots. (a) Find the possible values of k. [5 marks] (b) The normal to the curve at P cuts the x-axis at R. Find the coordinates of R. [5 marks] (b) Write down the values of k for which x ( k3) xk 0 has two equal real roots. [ marks]

41 11 M09/5/MATME/SP/ENG/TZ/XX+ 1 M09/5/MATME/SP/ENG/TZ/XX+ Do NOT write on this page. SECTION B Answer all the questions on the answer sheets provided. Please start each question on a new page. 8. [Maximum mark: 14] Let f( x) x( x5), for 0 x 6. The following diagram shows the graph of f. Do NOT write on this page. 9. [Maximum mark: 13] A van can take either Route A or Route B for a particular journey. If Route A is taken, the journey time may be assumed to be normally distributed with mean 46 minutes and a standard deviation 10 minutes. If Route B is taken, the journey time may be assumed to be normally distributed with mean µ minutes and standard deviation 1 minutes. (a) For Route A, find the probability that the journey takes more than 60 minutes. [ marks] R (b) For Route B, the probability that the journey takes less than 60 minutes is Find the value of µ. (c) The van sets out at 06:00 and needs to arrive before 07:00. [3 marks] Let R be the region enclosed by the x-axis and the curve of f. (a) Find the area of R. [3 marks] (b) Find the volume of the solid formed when R is rotated through 360 about the x-axis. (c) The diagram below shows a part of the graph of a quadratic function g( x) xa ( x). The graph of g crosses the x-axis when x a. [4 marks] (d) (i) Which route should it take? (ii) Justify your answer. [3 marks] On five consecutive days the van sets out at 06:00 and takes Route B. Find the probability that (i) it arrives before 07:00 on all five days; (ii) it arrives before 07:00 on at least three days. [5 marks] The area of the shaded region is equal to the area of R. Find the value of a. [7 marks]

42 13 M09/5/MATME/SP/ENG/TZ/XX+ M10/5/MATME/SP/ENG/TZ/XX+ Do NOT write on this page. 10. [Maximum mark: 18] Let f( x) 3sinx4 cos x, for x. (a) Sketch the graph of f. [3 marks] (b) (c) Write down (i) (ii) the amplitude; the period; (iii) the x-intercept that lies between and 0. [3 marks] Hence write f( x) in the form psin( qx r). [3 marks] (d) Write down one value of x such that f( x) 0. [ marks] Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. In particular, solutions found from a graphic display calculator should be supported by suitable working, e.g. if graphs are used to find a solution, you should sketch these as part of your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working. SECTION A Answer all the questions in the spaces provided. Working may be continued below the lines, if necessary. 1. [Maximum mark: 7] The following table gives the examination grades for 10 students. Grade Number of students Cumulative frequency p 4 q (e) Write down the two values of k for which the equation f( x) k has exactly two solutions. (f) Let g( x) ln ( x 1 ), for 0 x. There is a value of x, between 0 and 1, for which the gradient of f is equal to the gradient of g. Find this value of x. [ marks] [5 marks] (a) Find the value of (i) p ; (ii) q. [4 marks] (b) Find the mean grade. [ marks] (c) Write down the standard deviation. [1 mark]

43 3 M10/5/MATME/SP/ENG/TZ/XX+ 4 M10/5/MATME/SP/ENG/TZ/XX+. [Maximum mark: 6] An arithmetic sequence, u1, u, u3..., has d 11 and u (a) Find u 1. [ marks] (b) (i) Given that u n 516, find the value of n. (ii) For this value of n, find S n. [4 marks] 3. [Maximum mark: 5] Jan plays a game where she tosses two fair six-sided dice. She wins a prize if the sum of her scores is 5. (a) Jan tosses the two dice once. Find the probability that she wins a prize. [3 marks] (b) Jan tosses the two dice 8 times. Find the probability that she wins 3 prizes. [ marks]

44 5 M10/5/MATME/SP/ENG/TZ/XX+ 6 M10/5/MATME/SP/ENG/TZ/XX+ 4. [Maximum mark: 6] Find the term in x 4 in the expansion of 3x x [Maximum mark: 7] Consider f( x) x x, for x and gx ( ) sin e, for x. The graph of f is given below. y x 1 (a) On the diagram above, sketch the graph of g. [3 marks] (b) Solve f( x) gx ( ). [ marks] (c) Write down the set of values of x such that f( x) g( x). [ marks]

45 7 M10/5/MATME/SP/ENG/TZ/XX+ 8 M10/5/MATME/SP/ENG/TZ/XX+ 6. [Maximum mark: 6] x Let f( x) e sin x 10, for 0 x 4. Part of the graph of f is given below. y 15 M 7. [Maximum mark: 8] The number of bacteria, n, in a dish, after t minutes is given by n 800e 013t.. (a) Find the value of n when t 0. [ marks] (b) Find the rate at which n is increasing when t 15. [ marks] 10 5 (c) After k minutes, the rate of increase in n is greater than bacteria per minute. Find the least value of k, where k Z. [4 marks] 0 A x N 5 There is an x-intercept at the point A, a local maximum point at M, where x p and a local minimum point at N, where x q. (a) Write down the x-coordinate of A. [1 mark] (b) Find the value of (i) p ; (ii) q. [ marks] (c) q Find f( x)dx. Explain why this is not the area of the shaded region. p [3 marks]

46 9 M10/5/MATME/SP/ENG/TZ/XX+ 10 M10/5/MATME/SP/ENG/TZ/XX+ Do NOT write on this page. Do NOT write on this page. SECTION B Answer all the questions on the answer sheets provided. Please start each question on a new page. 9. [Maximum mark: 16] In this question, distance is in metres. 8. [Maximum mark: 15] The diagram below shows a circle with centre O and radius 8 cm. y 10 D C E B 8 A F 10 O 10 x diagram not to scale Toy airplanes fly in a straight line at a constant speed. Airplane 1 passes through a point A. x 3 Its position, p seconds after it has passed through A, is given by y 4 p 3. z 0 1 (a) (i) Write down the coordinates of A. (ii) Find the speed of the airplane in m s 1. [4 marks] (b) After seven seconds the airplane passes through a point B. (i) Find the coordinates of B. (ii) Find the distance the airplane has travelled during the seven seconds. [5 marks] 10 The points A, B, C, D, E and F are on the circle, and [AF] is a diameter. The length of arc ABC is 6 cm. (c) Airplane passes through a point C. Its position q seconds after it passes x 1 through C is given by y 5 q, a R. z 8 a (a) Find the size of angle AOC. [ marks] The angle between the flight paths of Airplane 1 and Airplane is 40. Find the two values of a. [7 marks] (b) Hence find the area of the shaded region. [6 marks] The area of sector OCDE is 45 cm. (c) Find the size of angle COE. [ marks] (d) Find EF. [5 marks]

47 11 M10/5/MATME/SP/ENG/TZ/XX+ Do NOT write on this page. 10. [Maximum mark: 14] Consider f( x) xln ( 4 x ), for x. The graph of f is given below. y x 3 4 (a) Let P and Q be points on the curve of f where the tangent to the graph of f is parallel to the x-axis. (i) Find the x-coordinate of P and of Q. (ii) Consider f( x) k. Write down all values of k for which there are exactly two solutions. [5 marks] Let gx ( ) x 3 ln ( 4 x ), for x. (b) Show that 4 x g ( x) 3x ln ( 4x ). [4 marks] 4 x (c) Sketch the graph of g. (d) Consider g( x) w. Write down all values of w for which there are exactly two solutions. [ marks] [3 marks]