ASSIGNMENT COVER SHEET omicron

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1 ASSIGNMENT COVER SHEET omicron Name Question Done Backpack Ready for test Drill A differentiation Drill B sketches Drill C Partial fractions Drill D integration Drill E differentiation Section A integration Implicit differentiation differentiation 4 Implicit differentiation 5 Mechanics equilibrium 6 Mechanics equilibrium 7 Mechanics equilibrium with friction 8 kinematics Section B 9 Trig identities 0 differentiation differentiation differentiation logs 4 differentiation 5 R,alpha form 6 Trig equations 7 differentiation 8 differentiation 9 parabola 0 matrices Topic teacher comment student comment target X:\Maths\TEAM - Doubles & Furthers\A doubles - Assignments 05-6\05-6 assignments\dmy(7)omicron 5-6.doc

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 Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different. J.W.Goethe Y Double Maths Assignment ο (omicron) Remember - you need to complete this assignment, so start early and use the help available! Your tracking test is w/b rd February in lessons Drill and 0 questions plus Solomon C eam practice due in w/b 8//5 Drill Part A Use the quotient rule to differentiate the following functions with respect to : Part B Sketch the following functions: = = + ln( + ) y e y y = 0e Part C Epress the following as partial fractions: + ( )( + ) + 4 Part D Integrate the following with respect to : sec () d ( ) d Part E For each of these functions, find d y d : y = sin y = cos 6 ( ) d Section A Core, Mechanics and Core 4 X:\Maths\TEAM - Doubles & Furthers\A doubles - Assignments 05-6\05-6 assignments\dmy(7)omicron 5-6.doc y = cosec Eam practice C Solomon E (give in completed and marked with your assignment) C4

3 ( + ) 4 d ( ) 5 d e d (d) sin( + ) d (e) d (f) sec d Given that e + e y = y, find d y d in terms of and y. Find the coordinates of the stationary points of the curve + y y = 48 and determine their nature. 4 Find the gradient of the curve with equation 5 + 5y 6y = at the point (, ). Mechanics 5 A particle P of mass kg is held in equilibrium under gravity by two light inetensible strings. One string is horizontal and the other is inclined at an angle of 0º to the horizontal. The tension in the horizontal string is H N. The tension in the other string is T N. T N 0 P H N Find the value of T. Find the value of H. 6 A particle of mass 0. kg lies on a smooth plane inclined at an angle α to the horizontal, where tana = 4. The particle is held in equilibrium by a horizontal force of magnitude Q newtons. The line of action of this force is in the same vertical plane as a line of greatest slope of the inclined plane. Calculate the value of Q, to one decimal place. 7 A body of mass kg is held in limiting equilibrium on a rough plane inclined at 0 to the horizontal by a horizontal force X. The coefficient between the body and the plane is 0.. Modelling the body as a particle find X when the body is on the point of slipping Up the plane Down the plane X:\Maths\TEAM - Doubles & Furthers\A doubles - Assignments 05-6\05-6 assignments\dmy(7)omicron 5-6.doc

4 8 A car is moving along a straight road with uniform acceleration. The car passes a checkpoint A with a speed of m s - and another check-point C with a speed of m s -. The distance between A and C is 00 m. Find the time, in seconds, taken by the car to move from A to C. Given that B is the mid-point of AC, find, in m s - to decimal place, the speed with which the car passes B. C Section B Core, Core & Further Pure 9 Prove the following identities: sec + tan tan A sec tan + cot A cosec A cos A + sin A + tan Acosec A cos A sin A 0 Differentiate ln8 e ln( ) sin (d) e ln (e) ln ln (f) ln ln( / ) d y Given that y = cos + sin, find and show that d y d d + 4y = 0 On the curve with equation y = ( +) 6, the point P has coordinates of 0. Find the equation of the tangent to the curve at P. Solve the following equations giving your answers to sf: ln 6ln = ln( ) ln( + ) ln = 4 The normal to the curve y = sec at the point P π 4, meets the line y = at the point Q. Find the eact coordinates of Q. 5 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: cosθ + sinθ [use Rcos(θ α)] 5cosθ sinθ [use Rcos(θ + α)] X:\Maths\TEAM - Doubles & Furthers\A doubles - Assignments 05-6\05-6 assignments\dmy(7)omicron 5-6.doc

5 (d) sinθ + cosθ [use Rsin(θ + α)] sinθ 7cosθ [use Rsin(θ α)] 6 Solve the following equations in the range o 0 60 cos = sin + sec = tan cosec = cot + Complete the table below from memory function Sin Cos Tan Cosec Sec Cot derivative Now differentiate the following with respect to 7 sin 4 cos( π ) sin ( + π ) (d) sin (e) cos e (f) sin 8 sin sin cos sin e (d) ln ( cos ) (e) e cos (f) sin Further Pure 9 Prove that the equation of the tangent to the parabola y = 4a at the point P( at,at) is yt = + at. If this tangent meets the aes at X and Y, prove that XY = at + t 0 The triangle R has vertices at (0, 0), (, 0), and (0, ). Find the coordinates of the 4 vertices of the image of R under the transformation. Use the determinant of this matri to show that the area of the image is 4. X:\Maths\TEAM - Doubles & Furthers\A doubles - Assignments 05-6\05-6 assignments\dmy(7)omicron 5-6.doc

6 Answers to Drill Part A: ( ) 6 ( 5 ) Part B: Check using your graphic calculator or autograph Part C: ( ) ( ) Part D: tan( ) + c ln + c Part E: sin 6sin 6 Answers to Section A 4 ( + ) + + ( ) ( + ) + c -cosec cot ( ) 6 6 c e) ln + c f) tan + c a) ( + ) + b) ( ) + c c) e + c d) cos( + ) + c () y e y e 6 () ma (,4), min(,6 ) (6) 7 (5a) 9N (5b) 4 N (6). (7a) N (sf) (7b).0N (sf) (8a) 50 s (8b) 4. m s - Answers to Section B (9) Proof (0a) (0d) e ( ln ) (ln ) (0b) e 4 (0c) (0e) sin + cos ( sin ) (0f) () Proof () 8 y + = 0 4 or (6) y = 4 π + e (5a) R =, α = π 4 min, θ = 5π (5b) R =, α =. 8 min, θ = (5c) R =, α = π min 7π 7, θ = (5d) R = 58, a = arctan min 6 58, θ = 5.88 (6a) 0, 0, 70 (6b) 45, 5 (6c) 90, 70, 0, 0 5 (9) (0) proof X:\Maths\TEAM - Doubles & Furthers\A doubles - Assignments 05-6\05-6 assignments\dmy(7)omicron 5-6.doc