Give a formula for the velocity as a function of the displacement given that when s = 1 metre, v = 2 m s 1. (7)

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1 . The acceleration of a bod is gien in terms of the displacement s metres as s a =. s (a) Gie a formula for the elocit as a function of the displacement gien that when s = metre, = m s. (7) (b) Hence find the elocit when the bod has traelled 5 metres. () (Total 9 marks). d Sole the differential equation =, gien that = 0 when =. Gie our answer in the form = f(). (Total 6 marks). (a) Show that the solution of the differential equation d = cos cos, gien that = 4 when =, is = arctan ( + sin ). (5) (b) Determine the alue of the constant a for which the following limit eists arctan( sin ) a lim and ealuate that limit. () (Total 7 marks) 4. The population of mosquitoes in a specific area around a lake is controlled b pesticide. The rate of decrease of the number of mosquitoes is proportional to the number of mosquitoes at an time t. Gien that the population decreases from to in a fie ear period, find the time it takes in ears for the population of mosquitoes to decrease b half. (Total 8 marks) IB Questionbank Mathematics Higher Leel rd edition

2 5. A particle moes in a straight line in a positie direction from a fied point O. The elocit m s, at time t seconds, where t 0, satisfies the differential equation 50. The particle starts from O with an initial elocit of 0 m s. (a) (i) Epress as a definite integral, the time taken for the particle s elocit to decrease from 0 m s to 5 m s. (ii) Hence calculate the time taken for the particle s elocit to decrease from 0 m s to 5 m s. (5) (b) (i) Show that, when 0, the motion of this particle can also be described b the differential equation where metres is the displacement from O. 50 (ii) Gien that = 0 when = 0, sole the differential equation epressing in terms of. 0 tan (iii) Hence show that = tan 50 (4) (Total 9 marks) 6. A certain population can be modelled b the differential equation the population at time t hours and k is a positie constant. d = k cos kt, where is (a) Gien that = 0 when t = 0, epress in terms of k, t and 0. (5) (b) Find the ratio of the minimum size of the population to the maimum size of the population. () (Total 7 marks) 7. The acceleration in m s of a particle moing in a straight line at time t seconds, t 0, is gien b the formula a =. When t = 0, the elocit is 40 m s. Find an epression for in terms of t. (Total 6 marks) IB Questionbank Mathematics Higher Leel rd edition

3 8. (a) Sole the differential equation cos e e d e = 0, gien that = 0 when =. (7) (b) Find the alue of when =. () (Total 8 marks) 9. Find in terms of, gien that ( + ) d tan and when = 0. (Total 7 marks) 0. A bod is moing through a liquid so that its acceleration can be epressed as 00 m s, where m s is the elocit of the bod at time t seconds. The initial elocit of the bod was known to be 40 m s. (a) Show that the time taken, T seconds, for the bod to slow to V m s is gien b 40 T = 00V 80. (4) (b) (i) Eplain wh acceleration can be epressed as metres, of the bod at time t seconds. d, where s is displacement, in d s (ii) Hence find a similar integral to that shown in part (a) for the distance, S metres, traelled as the bod slows to V m s. (7) (c) Hence, using parts (a) and (b), find the distance traelled and the time taken until the bod momentaril comes to rest. () (Total 4 marks) IB Questionbank Mathematics Higher Leel rd edition

4 . A skdier jumps from a stationar balloon at a height of 000 m aboe the ground. Her elocit, m s, t seconds after jumping, is gien b = 50( e 0.t ). (a) (b) Find her acceleration 0 seconds after jumping. How far aboe the ground is she 0 seconds after jumping? () () (Total 6 marks). A jet plane traels horizontall along a straight path for one minute, starting at time t = 0, where t is measured in seconds. The acceleration, a, measured in m s, of the jet plane is gien b the straight line graph below. (a) Find an epression for the acceleration of the jet plane during this time, in terms of t. () (b) Gien that when t = 0 the jet plane is traelling at 5 m s, find its maimum elocit in m s during the minute that follows. (4) (c) Gien that the jet plane breaks the sound barrier at 95 m s, find out for how long the jet plane is traelling greater than this speed. () (Total 8 marks) IB Questionbank Mathematics Higher Leel rd edition 4

5 . (a) s a = s a = d s s ds s s ds s = ln s + + k Note: Do not penalize if k is missing. When s =, = = ln + k k = ln s ln s ln ln (b) EITHER 6 ln = ln + 4 ln 4 OR = ln 6 + ln = ln ln = ln 6 4 ln [9] IB Questionbank Mathematics Higher Leel rd edition

6 d d., Separating ariables d arctan = ln + c = 0, = arctan 0 = ln + c ln = c arctan = ln ln = ln () () = tanln N [6]. (a) this separable equation has general solution sec d = cos tan = sin + c the condition gies tan = sin + c c = 4 the solution is tan = + sin = arctan ( + sin ) ()() AG IB Questionbank Mathematics Higher Leel rd edition

7 (b) the limit cannot eist unless a = arctan sin = arctan R in that case the limit can be ealuated using l Hopital s rule (twice) limit is (arctan( sin )) lim lim where is the solution of the differential equation the numerator has zero limit (from the factor cos in the differential equation) so required limit is lim finall, = sin cos cos cos sin () since cos 5 = at 5 the required limit is 0 R [7] IB Questionbank Mathematics Higher Leel rd edition

8 4. Let the number of mosquitoes be. d k d k ln = kt + c = e kt+c = Ae kt when t = 0, = A = = e kt when t = 5, = = e 5k 4 = e kt 5 4 5k = ln 5 4 k = ln 5 5 (= ) = e kt = e kt ln kt 5 t = ln = 5.5 ears 4 ln 5 [8] IB Questionbank Mathematics Higher Leel rd edition 4

9 5. (a) (i) EITHER Attempting to separate the ariables 50 OR Inerting to obtain 50 THEN () () () () 5 0 t 50 N t A N 0 (ii) 0.7sec 5ln sec (b) (i) () Must see diision b ( > 0) AG N0 50 (ii) Either attempting to separate ariables or inerting to obtain () (or equialent) 50 Attempting to integrate both sides arctan = C 50 Note: Award for a correct LHS and for a correct RHS that must include C. When = 0, = 0 and so C = arctan0 = 50(arctan0 arctan ) N IB Questionbank Mathematics Higher Leel rd edition 5

10 (iii) Attempting to make arctan the subject. arctan = arctan0 50 = tanarctan0 50 Using tan (A B) formula to obtain the desired form. 0 tan 50 AG N0 0 tan 50 [9] 6. (a) d = k cos (kt) d = k cos(kt) () d k cos( kt) ln = sin(kt) + c = Ae sin(kt) t = 0 0 = A = 0 e sin kt () (b) l sin kt () 0 e 0 e so the ratio is e : e or : e [7] IB Questionbank Mathematics Higher Leel rd edition 6

11 7. ln = t c tc t = e Ae t = 0, = 40, so A = 40 () () () = t 40e (or equialent) [6] 8. (a) cos e d e rearrange e 0 to obtain cos e e d e cos() as cos sin() C 4 e e and e e d e C Note: The aboe two integrations are independent and should not be penalized for missing Cs. () a general solution of cos e e e d 0 is sin( 4 ) e e C gien that = 0 when =, C = e sin() e e 4 (or.5)() so, the required solution is defined b the equation sin( ) e e or lnln sin() e 4 4 N0 (or equialent) (b) for =, ln ln e (or 0.47) 4 [8] IB Questionbank Mathematics Higher Leel rd edition 7

12 9. ( + d d ) tan d tan cos sin ln sin d ln C Notes: Do not penalize omission of modulus signs. Do not penalize omission of constant at this stage. EITHER ln sin OR ln C C 0 ()() sin =, A = e C THEN sin 0 A = arcsin ( ) Note: Award M0A0 if constant omitted earlier. [7] 0. (a) T V T = 00 V 80 () AG (b) (i) ds a = ds = d s R AG IB Questionbank Mathematics Higher Leel rd edition 8

13 (ii) 80 ds 00 S V 00 ds S ds 0 V 80 S = V 80 () (c) letting V = 0 () distance = =. metres 0 80 time = =.6 seconds 0 80 [4]. (a) a = 0e 0.t ()() at t = 0, a =.5 (m s ) (accept 0e ) (b) METHOD 0 0.t d = 50( e ) 0 () = so distance aboe ground = 70 (m) ( s.f.) (accept 76 (m)) METHOD s = 50( e 0.t ) = 50t + 50e 0.t (+ c) Taking s = 0 when t = 0 gies c = 50 So when t = 0, s = 8... so distance aboe ground = 70 (m) ( s.f.) (accept 76 (m)) [6]. (a) 5 equation of line in graph a = t a t 5 IB Questionbank Mathematics Higher Leel rd edition 9

14 5 (b) t 5 () 5 = t + 5t + c () 4 when t = 0, = 5 m s 5 = t + 5t from graph or b finding time when a = 0 maimum = 95 m s (c) EITHER graph drawn and intersection with = 95 m s t = = 4.8 OR ()() 5 95 = t + 5t + 5 t = ; ()() 4 t = = 4.8 (8 0 ) [8] IB Questionbank Mathematics Higher Leel rd edition 0