A2 Assignment zeta Cover Sheet. C3 Differentiation all methods. C3 Sketch and find range. C3 Integration by inspection. C3 Rcos(x-a) max and min

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1 A Assignment zeta Cover Sheet Name: Question Done Backpack Ready? Topic Comment Drill Consolidation M1 Prac Ch all Aa Ab Ac Ad Ae Af Ag Ah Ba C3 Modulus function Bb C3 Modulus function Bc C3 Modulus function Ca C3 Sketch and find range Cb C3 Sketch and find range Cc C3 Sketch and find range Da C3 Integration by inspection Db C3 Integration by inspection Dc C3 Integration by inspection 1a C3 Rcos(-a) ma and min 1b C3 Rcos(-a) ma and min 1c C3 Rcos(-a) ma and min a C3 Rcos(-a) min b C3 Rcos(-a) min c C3 Rcos(-a) min d C3 Rcos(-a) min 3 C3 Rcos(-a) with solving 4a C3 Trig proof 4b C3 Trig proof 5a C3 Inverse 5b C3 Domain and range of inverse 5c C3 Solve function 6 C3 Find normal 7 C3 Minimum points 8a C3 Trig solve 8b C3 Trig solve 9 M1 Impulse/Momentum vectors 10 Challenge!

2 α β γ δ ε ζ η θ ι κ λ µ ν ξ ο π ρ σ τ υ ϕ χ ψ ω Perhaps the greatest parado of all is that there are paradoes in mathematics J Newman A Maths with Mechanics Assignment ζ (zeta) (14 questions including drill and challenge plus a C3 past paper to do in the week s break found at the end of this assignment) Due in after the half term break w/b /11 Drill Part A Differentiate the following functions with respect to : (a) (c) e 1 ( + 1 ) (c) sin( e ) (d) ln (e) ln(1+ 4) (f) tan( 3 ) (g) e cos (h) ( 3 e ) 6 Part B For each of these functions sketch f ( ) and f( ) (a) y = ln( + 1) (b) y = 1 e (c) 1 y = 1 + Part C Sketch the following functions where each function is defined on domain, R.. State the range of each function. (a) f( ) = ( + 1) + 4, 0 (b) f( ) = 1 e, 0 (c) f( ) = 3ln, > 0 Part D Integrate the following functions by working out what has been differentiated: 1 (a) cosec cot d (b) d (c) 3 e 3 d 1 Current Work: 1. Write down the maimum and minimum values of the following functions (a) f ( ) = 1 + 5sin (b) f ( ) = 7 sin( + π ) (c) f ( ) = hint: think about ma and min values of sin 10 4sin. Epress the following in the form Rsin(θ ± α) or Rcos(θ ± α) as appropriate (with α in radians) and hence find the minimum value of the function, and the value of θ for which it occurs: (a) cosθ + sinθ [use Rcos(θ α)]

3 (b) (c) (d) 5cosθ 1sinθ [use Rcos(θ + α)] 3sinθ + 3cosθ [use Rsin(θ + α)] 3sinθ 7cosθ [use Rsin(θ α)] π 3. Epress 1sin 5cos in the form R sin( α) where R > 0 and 0 < α <. Hence solve the equation 1sin 5cos = 7 for 0 < < π giving correct to 3 decimal places. Consolidation: 4. Prove the following identities: 1 (a) cosec sec tan cosec +1 (b) sin A sin B + cos A sin(a + B) cosb sin B 5. The function f is given by f: ln (3 6), R, >. (a) Find f 1 (). (b) Write down the domain of f 1 and the range of f 1. (c) Find, to 3 significant figures, the value of for which f() = Find the equation of the normal to the curve y = e (cos + sin ) at the point (0,1) 7 A curve has equation y = e. Find the coordinates of the turning point in terms of natural logarithms, and show that it is a minimum point. 8. c Solve the following equations in the interval 0 π giving in terms of π or to 1dp as appropriate: (a) π 1 cos = sin 4 (b) sin + sin = 0 M1 Practice (Preparation for M) 9 A particle of mass 5 kg is moving with velocity (3i + 4j) ms 1 when it is given an impulse of ( i + 6j) Ns. Find the speed of the particle after the impact If = 0, what is the value of +

4 Preparation: Start of C4, Read* about Partial Fractions C4 new tetbook pages1-9, and old C4 tetbook pages 1-8 * you are not epected to work through questions in this preparation section but read the tetbook, making notes if you wish, to help you to understand the topic. Complete this past paper (C3 June 005) in 1 hour 30mins. If it takes longer draw a line under where you got to in the time and continue Do it under eam conditions. Do not use the mark scheme until you have done the whole paper Mark it yourself using the mark scheme on the VLE and write down your % on the paper. Do your corrections in another colour. Hand it to your teacher with your assignment

5 C3 June 005 (7 questions 75 marks) 1. (a) Given that sin θ + cos θ 1, show that 1 + tan θ sec θ. (b) Solve, for 0 θ < 360, the equation () tan θ + sec θ = 1, giving your answers to 1 decimal place. (6). (a) Differentiate with respect to (i) 3 sin + sec, (ii) { + ln ()} Given that y =, 1, ( 1) (b) show that d y 8 = 3 d ( 1). (6) 3. The function f is defined by f: , > 1. + (a) Show that f() = (b) Find f 1 ()., > 1. 1 (4) The function g is defined by g: + 5, R. (c) Solve fg() = 41. X:\Maths\TEAM - A\A Assignments 15-16\A Mechanics\MPM(6)zeta doc Updated: 15/10/015

6 4. f() = 3e 1 ln, > 0. (a) Differentiate to find f (). The curve with equation y = f() has a turning point at P. The -coordinate of P is α. (b) Show that α = 61 e α. () The iterative formula n + 1 = 1 n 6 e, 0 = 1, is used to find an approimate value for α. (c) Calculate the values of 1,, 3 and 4, giving your answers to 4 decimal places. (d) By considering the change of sign of f () in a suitable interval, prove that α = correct to 4 decimal places. () () 5. (a) Using the identity cos (A + B) cos A cos B sin A sin B, prove that (b) Show that cos A 1 sin A. () sin θ 3 cos θ 3 sin θ + 3 sin θ (4 cos θ + 6 sin θ 3). (c) Epress 4 cos θ + 6 sin θ in the form R sin (θ + α ), where R > 0 and 0 < α < π 1. (4) (d) Hence, for 0 θ < π, solve sin θ = 3(cos θ + sin θ 1), (4) giving your answers in radians to 3 significant figures, where appropriate. (5) X:\Maths\TEAM - A\A Assignments 15-16\A Mechanics\MPM(6)zeta doc Updated: 15/10/015

7 6. Figure 1 y 1 O 3 b (1, a) Figure 1 shows part of the graph of y = f(), R. The graph consists of two line segments that meet at the point (1, a), a < 0. One line meets the -ais at (3, 0). The other line meets the -ais at ( 1, 0) and the y-ais at (0, b), b < 0. In separate diagrams, sketch the graph with equation (a) y = f( + 1), (b) y = f( ). () Indicate clearly on each sketch the coordinates of any points of intersection with the aes. Given that f() = 1, find (c) the value of a and the value of b, (d) the value of for which f() = 5. () (4) X:\Maths\TEAM - A\A Assignments 15-16\A Mechanics\MPM(6)zeta doc Updated: 15/10/015

8 7. A particular species of orchid is being studied. The population p at time t years after the study started is assumed to be p = 800ae 1+ ae 0.t 0.t, where a is a constant. Given that there were 300 orchids when the study started, (a) show that a = 0.1, (b) use the equation with a = 0.1 to predict the number of years before the population of orchids reaches (4) (c) Show that p = e 0.t. (1) (d) Hence show that the population cannot eceed 800. () Finished and done all the questions? Now mark it using the mark scheme which can be found on the VLE Answers to the assignment are on the net page X:\Maths\TEAM - A\A Assignments 15-16\A Mechanics\MPM(6)zeta doc Updated: 15/10/015

9 Drill Answers Part A 1 (a) ( 1 ) (b) 1 + ln ( 1 ) (c) ( 3) ( + 1) 3 e 4 (d) ln + (e) 1+ 4 ( ) 5 3 (f) 3 sec (g) sin e cos (h) 6e 3 e Part B Use your calculator or Autograph to check these Part C Use your graphical calculator or autograph to check your sketches. Ranges given below: f 5 f 0 < f < or R (a) ( ) (b) ( ) (c) ( ) Part D (a) 1 cosec + c (b) 1 ln 1 + c (c) e 3 + c Current work (1) check on graphic calculator ()a) R =, α = π 4 min, θ = 5π b) R = 13, α = min 13, θ = c) R = 3, α = π 3 min 7π 7 3, θ = d) R = 58, a = arctan min 58, θ = R = 13, α = = or.968 Consolidation: (4) proof ( ) (5b)sketch f() to find range of f equals domain of (5a) 1 3 e f (5c) 8.70 (6) y + = 0 (7) ( ln, ln 4) min by showing second derivative is positive (8a).7, 5.8 radians (8b) 0, π, π, π/3, 4π/3 radians M1 practice: (9) 5.81ms -1 Challenge Question: (10) 7 X:\Maths\TEAM - A\A Assignments 15-16\A Mechanics\MPM(6)zeta doc Updated: 15/10/015

10 Mα THS ASSIGNMENT COVER SHEET Name Current Maths Teacher Please tick honestly: Have you ticked/crossed your answers using the answers given? Have you corrected all the questions which were wrong? Yes No - eplain why. How did you find this assignment? Use this space to outline any problems you ve had and how you overcame them as well as the things which went well or which you enjoyed/learned from. X:\Maths\TEAM - A\A Assignments 15-16\A Mechanics\MPM(6)zeta doc Updated: 15/10/015