AS Mathematics Assignment 9 Due Date: Friday 22 nd March 2013
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1 AS Mathematics Assignment 9 Due Date: Friday 22 nd March 2013 NAME GROUP: MECHANICS/STATS Instructions to Students All questions must be attempted. You should present your solutions on file paper and submit them with this cover sheet. (Work submitted without a cover sheet complete with name will not be marked). All work is to be submitted either to your module teacher or the Faculty Office by 4.15pm on the due date above. TEACHER ASSESSED QUESTIONS SUBJECT SPECIFIC SKILLS 1. Presentation : written work presented legibly using standard mathematical notation Grade Evidence/Comment : Communication: written solutions presented in a logical and coherent manner ( showing workings clearly) Grade Evidence/Comment : 3. Problem solving: identifying and using appropriate mathematical techniques Grade Evidence/Comment : 4. Competence in using algebraic techniques Grade Evidence/Comment : 5. Competence in using graphical calculator Grade Evidence/Comment : NOTE: GRADE 1 = BRIILIANT /SORTED GRADE 5 = YET TO DEMONSTRATE THIS PARTICULAR SKILL REFERENCES Edexcel AS Core 2 text book: Sine & cosine Rule: pg Radian and Circle measure: pgs Trigonometric graphs: pgs Trigonometric equations and identities: pgs Trig 1 - Powerpoint Trig 2 - Powerpoint Examsolutions.net (See your subject for the login details, click topic to access mymaths lesson) Students Feedback: Please comment on how you think you performed on this assignment and state the questions that you found difficult.
2 SINE AND COSINE RULES 1. C 5 cm 4 cm A B 6 cm The diagram above shows the triangle ABC, with AB = 6 cm, BC = 4 cm and CA = 5 cm. Show that cos A 3. 4 Hence, or otherwise, find the exact value of sin A. (Total 5 marks) 2. In the triangle ABC, AB = 8 cm, AC = 7 cm, ABC = 0.5 radians and ACB = x radians. Use the sine rule to find the value of sin x, giving your answer to 3 decimal places. Given that there are two possible values of x, find these values of x, giving your answers to 2 decimal places. (Total 6 marks) 3. N C B 500 m 700 m 15 A The diagram above shows 3 yachts A, B and C which are assumed to be in the same horizontal plane. Yacht B is 500 m due north of yacht A and yacht C is 700 m from A. The bearing of C from A is 015. Calculate the distance between yacht B and yacht C, in metres to 3 significant figures. Haringey Sixth Form Centre Mathematics Department 2
3 The bearing of yacht C from yacht B is θ, as shown in the diagram. Calculate the value of θ. RADIANS AND CIRCLE MEASURES (Total 7 marks) 4. An emblem, as shown in the diagram above, consists of a triangle ABC joined to a sector CBD of a circle with radius 4 cm and centre B. The points A, B and D lie on a straight line with AB = 5 cm and BD = 4 cm. Angle BAC = 0.6 radians and AC is the longest side of the triangle ABC. Show that angle ABC = 1.76 radians, correct to 3 significant figures. Find the area of the emblem. (Total 7 marks) 5. O r cm L M A major sector LOM of a circle, with centre O and radius r cm, has LOM = radians, as shown in the diagram. The perimeter of the sector is P cm and the area of the sector is A cm 2. Write down, in terms of r and, expressions for P and A. THIS QUESTION IS CONTINUED ON THE NEXT PAGE Haringey Sixth Form Centre Mathematics Department 3
4 Given that r = 2 2 and that P = A, 2 show that =. 2 1 Express in the form a + b 2, where a and b are integers to be found. 6. Figure 1 A 6 cm cm B C Figure 1 shows the cross-section ABC of a metal cutter used for making biscuits. The straight sides AB and AC are both of length 6 cm and BAC is 1.2 radians. The curved portion BC is an arc of a circle with centre A. Find the perimeter of the cross-section of the cutter. Find the area of the cross-section ABC. Figure 2 a cm 1.2 b cm 6 cm 1.2 A pair of these cutters are kept together in a rectangular box of length a cm and width b cm. The cutters fit into the box as shown in Figure 2. Find the value of a and the value of b, giving your answers to 3 significant figures. Haringey Sixth Form Centre Mathematics Department 4
5 7. S P 6 3m R 6m 6m Q The diagram above shows a plan of a patio. The patio PQRS is in the shape of a sector of a circle with centre Q and radius 6 m. Given that the length of the straight line PR is 6 3m, find the exact size of angle PQR in radians. Show that the area of the patio PQRS is 12 m 2. (d) (e) Find the exact area of the triangle PQR. Find, in m 2 to 1 decimal place, the area of the segment PRS. Find, in m to 1 decimal place, the perimeter of the patio PQRS. (Total 11 marks) Haringey Sixth Form Centre Mathematics Department 5
6 8. B 7 cm R A 0.8 rad D C The diagram above shows ABC, a sector of a circle with centre A and radius 7 cm. Given that the size of BAC is exactly 0.8 radians, find the length of the arc BC, the area of the sector ABC. The point D is the mid-point of AC. The region R, shown shaded in the diagram above, is bounded by CD, DB and the arc BC. Find (d) the perimeter of R, giving your answer to 3 significant figures, the area of R, giving your answer to 3 significant figures. (Total 12 marks) Haringey Sixth Form Centre Mathematics Department 6
7 9. C A 2a B A flat plate S, which is part of a child s toy, is shown in the diagram above. The points A, B and C are the vertices of an equilateral triangle and the distance between A and B is 2a. The circular arc AB has centre C and radius 2a. The circular arcs BC and CA have centres at A and B respectively and radii 2a. Find, in terms of and a, the perimeter of S. Prove that the area of the plate S is 2a 2 ( 3 ). TRIGONOMETRIC GRAPHS 10. Sketch, for 0 x 360, the graph of y = sin (x + 30 ). (6) Write down the coordinates of the points at which the graph meets the axes. Solve, for 0 x < 360, the equation 1 sin (x + 30 ) =. 2 Haringey Sixth Form Centre Mathematics Department 7
8 11. y 3 O p q 360 x The diagram above shows the curve with equation y = k sin (x + 60), 0 x 360, where k is a constant. The curve meets the y-axis at (0, 3) and passes through the points (p, 0) and (q, 0). Show that k = 2. Write down the value of p and the value of q. (1) The line y = 1.6 meets the curve at the points A and B. Find the x-coordinates of A and B, giving your answers to 1 decimal place. (5) 12. The curve C has equation y = cos x, 0 x 2. 4 Sketch C. Write down the exact coordinates of the points at which C meets the coordinate axes. Solve, for x in the interval 0 x 2, cos x = 0.5, 4 giving your answers in terms of p. (Total 9 marks) Haringey Sixth Form Centre Mathematics Department 8
9 13. Sketch, for 0 x 2, the graph of y = sin. 6 x Write down the exact coordinates of the points where the graph meets the coordinate axes. Solve, for 0 x 2, the equation sin x 0.65, 6 giving your answers in radians to 2 decimal places (5) (Total 10 marks) TRIGONOMETRIC EQUATIONS AND IDENTITIES 14. Find all the values of θ, to 1 decimal place, in the interval 0 θ < 360 for which 5 sin(θ + 30 ) = 3. Find all the values of θ, to 1 decimal place, in the interval 0 θ, < 360 for which tan 2 θ = 4. (5) (Total 9 marks) 15. Solve, for 0 x 180, the equation 3 sin(x + 10 ) =, 2 cos2x = 0.9, giving your answers to 1 decimal place. 16. Solve, for 0 x < 360, o sin( x 20 ) cos3x 2 (6) (Total 10 marks) 17. Given that sin θ = 5cos θ, find the value of tan θ. (1) THIS QUESTION IS CONTINUED ON THE NEXT PAGE Haringey Sixth Form Centre Mathematics Department 9
10 Hence, or otherwise, find the values of θ in the interval 0 θ < 360 for which sin θ = 5cos θ, giving your answers to 1 decimal place. 18. Find all the values of in the interval 0 < 360 for which cos( 10 ) = cos 15, tan 2 = 0.4, 2 sin tan = 3. (5) (6) (Total 14 marks) 19. Solve, for 0 < 2, the equation sin 2 = 1 + cos, giving your answers in terms of. (Total 5 marks) 20. Show that the equation 5 sin x = cos 2 x can be written in the form 2 sin 2 x + 5 sin x 3 = 0 Solve, for 0 x < 360, 2 sin 2 x + 5 sin x 3 = Find, in degrees to the nearest tenth of a degree, the values of x for which sin x tan x = 4, 0 x < Show that the equation (Total 6 marks) 4 sin 2 x + 9 cos x 6 = 0 can be written as 4 cos 2 x 9 cos x + 2 = 0. Hence solve, for 0 x 720, 4 sin 2 x + 9 cos x 6 = 0, giving your answers to 1 decimal place. (6) Haringey Sixth Form Centre Mathematics Department 10
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