Created by T. Madas. Candidates may use any calculator allowed by the regulations of this examination.

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1 IYGB GCE Mathematics SYN Advanced Level Snoptic Paper C Difficult Rating: Time: 3 hours Candidates ma use an calculator allowed b the regulations of this eamination. Information for Candidates This snoptic practice paper follows closel the Advanced Level Pure Mathematics Sllabus, suitable for first assessment Summer 018. The standard booklet Mathematical Formulae and Statistical Tables ma be used. Full marks ma be obtained for answers to ALL questions. The marks for the parts of questions are shown in round brackets, e.g. (). There are questions in this question paper. The total mark for this paper is 00. Advice to Candidates You must ensure that our answers to parts of questions are clearl labelled. You must show sufficient working to make our methods clear to the Eaminer. Answers without working ma not gain full credit. Non eact answers should be given to an appropriate degree of accurac. The eaminer ma refuse to mark an parts of questions if deemed not to be legible.

2 Question 1 (**+) Seats in a theatre are arranged in rows. The number of seats in this theatre form the terms of an arithmetic series. row 0 row 5 row 4 row 3 row row 1 The sith row has 3 seats and the fifteenth row has 50 seats. a) Find the number of seats in the first row. The theatre has 0 rows of seats in total. b) Find the number of seats in this theatre. Question (***) a) Use integration b parts to find cos 1 ( ) d. ( 3) b) Hence determine sin 1 ( ) d. ( 3)

3 Question 3 (***) A( 9,40) B( 7,40) O D( 18,0) C ( 36,0) The figure above shows the cross section of a river dam modelled in a sstem of coordinate aes where all units are in metres. The cross section of the dam consists of a circular sector ADB and two isosceles triangles OAD and DBC. The coordinates of the points A, B, C and D are ( 9,40 ), ( 7,40 ), ( ) ( 18,0 ), respectivel. 36,0 and a) Find the length of AD. b) Show that the angle ADB is approimatel radians. c) Hence determine, to the nearest m, the cross sectional area of the dam. Question 4 (****) Use the substitution 3 t = 1 to show that ( )( ) d = C. ( 10)

4 Question 5 (**+) = 1+ e A B S O R C The figure above shows the graph of the curve with equation = 1+ e, R. The point C has coordinates ( 1,0 ). The point B lies on the curve so that BC is parallel to the ais. The point A lies on the ais so that OABC is a rectangle. The region R is bounded b the curve, the coordinate aes and the line BC. The region S is bounded b the curve, the ais and the line AB. Show that the area of R is equal to the area of S. ( 7) Question 6 (***+) The first term of a geometric progression is 100 and its sum to infinit is a) Find the sum of the first five terms of the progression. The th n term of the progression is denoted b u n. b) Evaluate the sum ur r=. 6

5 Question 7 (***+) The curve C 1 has equation = 1, R. The curve C has equation = + 1, R. a) Sketch the graph of C 1 and the graph of C in the same set of aes, indicating the coordinates of an intercepts of the graphs with the coordinate aes. b) Hence, solve the inequalit Question 8 (****) A curve has equation ( ) = e 4ln, > 0. 4 The tangent to the curve, at the point where =, crosses the coordinate aes at the points A and B. Show that the area of the triangle OAB, where O is the origin, is given b ( 16 3e) 16( 4 e). ( 10)

6 Question 9 (****) At the point P, which lies on the curve with equation 3 =, e the gradient is 6 5. Determine the possible coordinates of P. ( 10) Question 10 (****) ( 0,) = f ( ) = 0. O ( 6,0) The figure above shows part of the curve with equation f ( ) The curve meets the coordinate aes at ( 6,0 ) and ( 0,4 ). a) Sketch the graph of = f ( ). =. The sketch must include the coordinates of an points where the graph meets the coordinate aes. b) Sketch on separate diagram the graph of f ( 4 1) =. The sketch must include the coordinates of the point where the graph meets the ais.

7 Question 11 (****) r The figure above shows the design of an athletics track inside a stadium. The track consists of two semicircles, each of radius r m, joined up to a rectangular section of length metres. The total length of the track is 400 m and encloses an area of A m. a) B obtaining and manipulating epressions for the total length of the track and the area enclosed b the track, show that A = 400r π r. In order to hold field events safel, it is required for the area enclosed b the track to be as large as possible. b) Determine b differentiation an eact value of r for which A is stationar. c) Show that the value of r found in part (b) gives the maimum value for A. d) Show further that the maimum area the area enclosed b the track is π m. The calculations for maimizing the area of the field within the track are shown to a mathematician. The mathematician agrees that the calculations are correct but he feels the resulting shape of the track might not be suitable. e) Eplain, b calculations, the mathematician s reasoning.

8 Question 1 (****) A circle C 1 has equation = 0. a) Determine the coordinates of the centre of C 1 and the size of its radius. A different circle C has its centre at ( 14,8 ) and the size of its radius is 10. The point P lies on C 1 and the point Q lies on C. b) Determine the minimum distance of PQ. ( 3) Question 13 (***+) A cubic curve C 1 has equation ( 7)( 3) = +. A quadratic curve C has equation ( 5)( 7 ) = +. a) Sketch on separate set of aes the graphs of C 1 and C. The sketches must contain the coordinates of the points where each of the curves meet the coordinate aes. 4 b) Hence, find in eact form where appropriate, the three solutions of the following equation. ( ) ( 7)( 3) ( 5)( 7 ) + = +. ( 7)

9 Question 14 (****) = 3e P O = 1+ e + e The figure above shows the graphs of = 3e and = 1+ e + e. The graphs meet at the point P. Show that the coordinate of P is e + 3e +. e + 3e + 1 ( 6) Question 15 (****) Use trigonometric algebra to solve the equation sin arcsin 1 + arccos = 1. 4 You ma not use an calculating aid in this question.

10 Question 16 (****) f ( ) O The figure above shows the graph of a function f ( ), defined b f ( ) The function is continuous and smooth. 3 a +, R, = b, R, > Find the value of a and the value of b. ( 7) Question 17 (****) Solve the following simultaneous logarithmic equations. 3log ( ) = 4log 8 log = 1+ log ( 8)

11 Question 18 (****) Consider the following identit for t. 1 ( + 1) t t t At + B C t a) Find the value of each of the constants A, B and C. In a chemical reaction the mass, m grams, of the chemical produced at time t, in minutes, satisfies the differential equation dm m = dt t t +. ( 1) b) Find a general solution of the differential equation, in the form m f ( t) =. ( 6) Two minutes after the reaction started the mass produced is 10 grams. c) Calculate the mass which will be produced after a further period of minutes. 4 d) Determine, in eact surd form, the maimum mass that will ever be produced b this chemical reaction. ( ) ( )

12 Question 19 (****) P O Q R = + 3 The figure above shows the curve C with equation = + 3. The points P( 1,6 ), ( 0,3) Q and R all lie on C. Given that PQR = 90, determine the eact coordinates of R. ( 9) Question 0 (*****) A curve has equation f ( ) 4 a + b, R, where a and b are non zero constants. Find the value of a and the value of b, given further that 3 f 1 ( ) = 4 and ( ) 3 4 f 3 = 1. ( 8)

13 Question 1 (****+) B C A 1 6 π O P Q R D The figure above shows a smmetrical design for a suspension bridge arch ABCD. The curve OBCR is a ccloid with parametric equations a) Show clearl that ( ) ( ) = 6 t sin t, = 6 1 cos t, 0 t π. d cot t d =. b) Find the in eact form the length of OR. c) Determine the maimum height of the arch. ( 5) The arch design consists of the curved part BC and the straight lines AB and CD. The straight lines AB and CD are tangents to the ccloid at the points B and C. The angle BAO is 1 6 π. d) Find the value of t at B, b considering the gradient at that point. e) Find, in eact form, the length of the straight line AD. ( 3) ( 6)

14 Question (****+) B h 1 A π 6 D θ C The figure above shows a triangle ABC, where BAC = 1π, BCA = θ and 6 BC = 1. The straight line segment BD, labelled as h, is perpendicular to AC. Let AD = and DC =. a) B epressing h in terms of θ, and in terms of h, show that + = 3 sinθ + cosθ, and hence deduce that the area of the triangle ABC is given b 1 ( ) sinθ sin θ + π. 6 ( 7) b) B using the trigonometric identities for 1 ( ) cos θ + θ + π 6 and cos θ 1 ( θ + π ) 6, write a simplified epression for the area of the triangle ABC. ( 5) The value of θ can var. c) B using part (b), deduce that the maimum value of the area of the triangle ABC is ( + ) and this maimum value occurs when θ = 5 π. 1 ( 3)