GAYAZA HIGH SCHOOL MATHS SEMINAR 2016 APPLIED MATHS

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1 GAYAZA HIGH SCHOOL MATHS SEMINAR 06 APPLIED MATHS STATISTICS AND PROBABILITY. (a) The probability that Moses wins a game is /. If he plays 6 games, what is (i) the epecte number of games won? (ii) the chance of winning at least two games? (b) A machine manufacturing nails makes approimately 5% that are outsie set tolerance limits. If a ranom sample of 00 nails is taken, fin the chance that (i) more than nails will be outsie the tolerance limits, (ii) between 0 an 0 nails inclusive, will be outsie the tolerance limits.. The table below shows marks obtaine by some stuents: Marks 5 -<9 -<5 -<40 -<50 -<55 -<60 -<70 -<75 -<80 Frequency (a) Estimate the (i) variance (ii) moe (b) Construct an Ogive an use it to etermine the; (i) 68 th percentile (ii) number of stuents who score above 47%. (a) Events A, B an C are such that P(A) =, P(B) = y an P(C) = + y. If P(AuB) = 0.6 an P(B/A) = 0., (i) Show that 4 + 5y =. (ii) Given that B an C are mutually eclusive an that P(BuC) = 0.9, etermine another equation in an y. (iii) Hence fin the values of an y. Deuce whether A an B are inepenent events (b) The events A an B are inepenent with P(A) = an P(AuB) (i) P(B) (ii) P(A/B) (iii) P(B I /A) Prepare by: Theoe Niyirina. GHS Tel / niyirina0@yahoo.com. Fin; 4. The weekly eman for petrol in thousans of units in a house is a continuous ranom variable with a probability ensity function of the form; a ( );0 f ( ) 0elsewehere a) Given that the average eman per week is 600 units, etermine the values of a an b) Fin P(0.9 < < ) 5. (a) A biase tetraheral ie has its faces numbere,,, 4. If this ie is tosse, the probability of the face that it lans on, is inversely proportional to the square of the number on the face. If is the ranom variable the number on the face the ie lans on, etermine; (i) the probability istribution for. (b) Var() (c) P( )

2 GAYAZA HIGH SCHOOL MATHS SEMINAR 06 APPLIED MATHS (b) The probability istribution of a iscrete ranom variable X is given by; P(X=) = k; =,,,,n. Where k is a constant Given that E() = 5, fin the; (i) value k an n (ii) Var(+) 6. The continuous ranom variable X has a probability function k( ); 0 k;0 f()= (5 ) k ; 0, elswhere where k is a constant. a) Sketch the p..f an hence fin the value of constant k b) Fin the cumulative istribution function, F(). c) Calculate P ( 0.5 ) 7. The weights of cabbages in the S. garen is normally istribute with a mean of 50g an a stanar eviation of 6g (a) If a cabbage is selecte at ranom, fin the probability that it weighs; (i) between 48 an 57.4g (ii) less than 46.g (b) If 0% of the cabbages weigh more than y g, etermine y 8. (a)in a certain country, on average one stuent in five has blue eyes. For a ranom selection of n stuents, the probability that none of the stuents has blue eyes is less than Fin the least possible value of n. (b) John plays two games of squash. The probability that he wins his first game is 0.. If he wins his first game, the probability that he wins his secon game is 0.6. If he loses his first game, the probability that he wins his secon game is 0.5. Given that he wins his secon game, fin the probability that he won his first game (c)dan has a pack of 5 cars. 0 cars have a picture of a robot on them an 5 cars have a picture of an aeroplane on them. Emma has a pack of cars. 7 cars have a picture of a robot on them an cars have a picture of an aeroplane on them. One car is taken at ranom from Dan s pack an one car is taken at ranom from Emma s pack. The probability that both cars have pictures of robots on them is 7/8. Write own an equation in terms of an hence fin the value of. Prepare by: Theoe Niyirina. GHS Tel / niyirina0@yahoo.com

3 GAYAZA HIGH SCHOOL MATHS SEMINAR 06 APPLIED MATHS MECHANICS. (a) Fin the centre of gravity of the lamina shown below. E cm D B 9cm 6cm C O 4cm A (b) If the lamina is suspene from O, fin the angle that OB makes with the vertical.. a). The impulse of a force of (6i+j+k)N acts for.5secons. If this force acts on a boy of mass kg with an initial velocity of (-4i+j-k)m/s, fin the velocity at the en of the perio. b). A vertical spring supports a boy which stretches it 50cm when in equilibrium. The boy is then pulle own an release. Show that the boy eecutes simple harmonic 5 motion with perio secons 5. (a) A car of mass one tone has an engine which eerts a constant tractive force of 900N. The car moves up a hill of inclination of in 0. There is a constant frictional resistance of 0.4 KN to its motion. If the initial spee of the car is 08km/hr, how far has it then travelle before it comes to rest? (g=0m/s ) (b) Three forces i+5 j-4 k, - j+7 k an i+ k, act on a boy from a point A(4,-,) to point B(8,4,5). Fin the work one on the boy. 4. A smooth bea of mass 0. kg is threae on a smooth circular wire of raius r metres which is hel in a vertical plane. If the bea is projecte from the lowest point on the circle with spee rg. Fin the; (a) spee of the bea when it has gone one sith of the way roun the circle. (b) force eerte on the bea by the wire at this point. Prepare by: Theoe Niyirina. GHS Tel / niyirina0@yahoo.com

4 GAYAZA HIGH SCHOOL MATHS SEMINAR 06 APPLIED MATHS 5. (a) A boy of mass kg is place on a rough plane incline at an angle of 0 0 to the horizontal. The coefficient of friction between the boy an the plane is 0.5. Fin the least force neee to prevent the boy from slipping own the plane if this force acts upwars at an angle of 0 0 to the line of greatest slope. (b) C 4N B N 8N 8N O N A OABC is a square in which OA =cm. Taking OA an OC as the an y aes respectively, fin the: (i) (ii) equation of the line of action of the resultant force. istance from C of the point where the line in (i) above crosses the sie C 6. Two uniform ros AB an BC each of length b but of weights W an 5W respectively are freely jointe at B an stan incline at 90 to each other in a vertical plane on a smooth floor with ens A an C connecte by a rope. Show that the; (i) tension in the rope is W (ii) W reaction at the hinge B is 7. A light inetensible string has one en attache to a ceiling. The string passes uner a smooth moveable pulley of mass kg an then over a smooth fie pulley. Particle of mass 5 kg is attache at the free en of the string. The sections of the string not in contact with the pulleys are vertical. If the system is release from rest an moves in a vertical plane, fin the: (i) (ii) acceleration of the system. tension in the string. (iii) istance move by the moveable pulley in.5 s. Prepare by: Theoe Niyirina. GHS Tel / niyirina0@yahoo.com 4

5 GAYAZA HIGH SCHOOL MATHS SEMINAR 06 APPLIED MATHS 8. a) A particle is projecte with spee u at an elevation θ from a point O on level groun. Show that the equation of trajectory is; y = tanθ g u ( + tan θ). b) A particle is projecte with spee 0 g ms - from a point O on the groun at an elevation θ. If the particle must clear a vertical tower of height 40m an at a horizontal istance of 40m from O, prove that tanθ NUMERICAL ANALYSIS 9. (a)locate graphically the smallest positive real root of sin ln = 0 (b) Use the Newton Raphson metho to approimate this root of the equation in (a) above correct to.p. 0. The mock eamination an average final eamination marks of a certain school are given in the following table. Mock Marks AV. Final Marks (a) (i) Plot the marks on the scatter iagram an comment on the relationship between the two marks. (ii) Draw a line of best fit an use it to preict the average final mark of a stuent whose mock mark is 50. (b) Calculate the rank correlation coefficient between the marks an comment on your result.. If the numbers an y are approimations with errors an y respectively. Show that the maimum absolute error in the approimations of y is given by y + y Hence fin the limits within which the true value of that =.8 ± 0.06 an y =.44 ± y lies given Prepare by: Theoe Niyirina. GHS Tel / niyirina0@yahoo.com 5

6 GAYAZA HIGH SCHOOL MATHS SEMINAR 06 APPLIED MATHS. (a) Two sies of a triangle PQR are p an q are such thatprq. (b) (i) Fin the maimum possible error in the area of this triangle. (ii) Hence fin the percentage error mae in the area if p = 4.5cm, q = 8.4cm an α = 0 o. Fin the range within which (a) Use the trapezium rule with si orinates to estimate ln. Give your answer correct to ecimal places. (b) Hence fin the percentage error mae in your estimate an suggest how it can be reuce 4. The quaratic formula metho for solving the equation lies. a b c 0 is escribe by the following parts of the flowchart: Print:, is 0? No Yes Start b 4ac Stop Rea: a, b, c b a b a b i a b i a (i) By rearranging the given parts, raw a flow chart that shows the algorithm for the escribe metho (ii) By performing a ry run for the flow chart using 4 0, copy an complete the table below: Prepare by: Theoe Niyirina. GHS Tel / niyirina0@yahoo.com 6

National Quali cations

National Quali cations AH06 X70/77/ Mathematics of Mechanics TUESDAY, 7 MAY :00 PM :00 PM Total marks 00 Attempt ALL questions. You may use a calculator. Full credit will be given only to solutions which