faa UGANDA BUSINESS AND TECHNICAL EXAMINATIONS BOARD . [ PAPER CODE PROGRAMME | | | | DATE | DTME 213 NATIONAL DIPLOMA IN MECHANICAL MONDAY, TDME 213 ENGINEERING | 241 JANUARY 2022 | NATIONAL DIPLOMA IN REFRIGERATION | AND AIR CONDITIONING YEAR I SEMESTER 1 : ___| SERIES PAPER NAME | TIME ALLOWED | NOV/DEC 2020 | FLUID MECHANICS 3 HOURS YOU SHOULD HAVE THE FOLLOWING FOR THIS EXAMINATION | Answer Booklet | Non-Programmable Electronic Calculator | | Drawing Instruments INSTRUCTIONS TO CANDIDATES J. This paper consists of eight questions. 2. Answer only five questions. 3. All questions carry equal marks, 4. All answers to each question should begin on a fresh page. 5. Do not write anywhere on this question paper. 6. All rough work should be done in the official answer booklet provided. 7. Read other instructions on the official answer booklet.question One (a) Consider a bo intensity of pressure piston Is in equilibrium. ea A of 100cm’, Calculate fe of the piston if the (04 marks) dy of mass SOkg acting on 4 piston of aré on the water in contact with the undersid piston of hydraulic jack are 3cm and 10cm (b) The diameters of a small piston and a large all piston, find the load which will be respectively. If a force of 80N Is applied on the sm: lifted by the large piston in the following situations. (i) The pistons are at the same level. (06 marks) (il) Small piston is 40cm above the large piston. Take the density of the liquid in the jack to be 1000kq/m?. (10 marks) Question Two (a) State the principle of Archimedes, (01 mark) (b) Explain the application of Archimedes to a floating body. (05 marks) () There is a project of laying a steel pipeline across the bed of river Nile to convey gas. The bed is completely immersed in water and is anchored at intervals of 3m along its length. If the density of steel is 7900kg/m3, density of water is 1000kg/m? and Buoyancy force on each anchorage is 12150N/m, determine the intemal and extemal diameters of the pipe to be used in the project in order to with stand the stated forces. (14 marks) Question Three (a) Differentiate between a notch and weir. (02 marks) (b) Show from first principles that the theoretical rate of flow through a rectangular notch is given by Q = 2B |(2g). where, B=width of notch and H=height of the water. Explain why this expression requires modification in practice (09 marks) (c) Agriculture is the backbone of Uganda being practiced both on small and large scales, However, the activity is being affected by climate changes. You are a farmer who has knowledge of engineering and you have come up with an idea of Constructing a water reservoir to serve as a back up during dry seasons. On the reservoir you want to put a rectangular notch to discharge water at 0. 14m}/s when the level is 23cm above the sill, Taking 0.6 as coefficient of discharge, determine the width of the notch. (09 marks)® «, Question Four 4, (2) Explain three applications of Bernoull’s equations in engineering field other than (b) (c) venturimeter. Differentiate between the following flows of fluids. ae (i) Steady and unsteady flows (02 marks) (ii) Laminar and turbulent flows (02 marks) (iii) Shooting and Tranquil flows (02 marks) (iv) Steady uniform and steady non-uniform (02 marks) A rectangular plate 1.2m by 1.8m is submerged in water and makes an angle of 30° with the horizontal. With the 1.2m side being horizontal, calculate the magnitude of the net force on one face and the position of centre of pressure when the top edge of the plate is; (i) at the water surface (04 marks) (ii), 20m below the water surface. (05 marks) Question Five (a) Define the term venturimeter. (0% mark) (b) Uganda has oil in Bunyoro sub-region which is yet to be drilled. The drilling of oil starts ina particular region of Bunyoro which has specific gravity 0.85 and the intended volume flow rate of 60 litres/second through a vertical venturimeter with an inlet diameter of 200cm and a throat diameter of 100mm. If two pressure gauges are connected at the tapings such that they are positioned at the levels of their corresponding taping points, prove that the difference of the two pressure gauge readings is 2.675N/cm*. (19 marks) Question Six (a) Define the following terms as used in fluid mechanics. (i) Mass density (01 mark) (ii) Specific weight (01 mark) (ili) Specific volume (01 mark) (iv) Viscosity (01 mark) (v) Surface tension (01 mark) (vi) Specific gravity (01 mark) (b) Fluids are divided into liquids and gases which can be used as power transmission agents in mechanical engineering. 3 Turn Over(i) Suggest four reasons why oil as a fluid has gained more application in poy, (04 marks (10 marks) transmission. (ii) Explain five types of fluids. Question Seven (2) Define the term capillarity. (02 marks) (b) A glass tube of diameter, d and opened at both ends is inserted in a water trough. If the surface tension of the water iso, the angle of contact between the water and glass tube is @ and the density of water is p, Show that the rise (i) in the tube above the level of water is given by h = a (09 marks) (©) Calculate the capillary rise in a glass tube of 2.5mm diameter when immersed vertically in; (i) water. (04 marks) (il) mercury and comment on your answer. (05 marks) Take surface tensions ¢=0.072SN/m for water and o=0.52N/m for mercury in contact with air. The specific gravity for mercury is given as 13.6 and angle of contact 130° Question Eight (a) Show that the loss of head when a pipe under goes a sudden increase in diameter is 20 and V, is the velocity in the large pipe. (10 marks) given by where, Vis the velocity in the smaller pipe upstream of the enlargement (b) A pipe carrying 0.06m%/s of a fluid suddenly contracts from 200mm to 150mm diameter. ‘Assuming that vena contracta is formed in the smaller diameter, calculate the coefficient of contraction if the pressure head at a point upstream of the contraction is 0.655m greater than at a point just downstream of the vena contracta (10 marks) END—_——— UGANDA BUSINESS AND TECHNICAL EXAMINATIONS BOARD RSEABEXAMINATIONS BOAR PAPER CODE | PROGRAMME TOME 213 NATIONAL DIPLOMA IN MECHANICAL. MONDAY, ENGINEERING 10™ DECEMBER 2018 | NATIONAL DIPLOMA IN REFRIGERATION | AND AIR CONDITIONING YEAR II SEMESTER 1 SERIES PAPER NAME NOV/DEC 2018 FLUID MECHANICS TIME ALLOWED 3 HOURS YOU SHOULD HAVE THE FOLLOWING FOR THIS EXAMINATION Answer Booklet Drawing Instruments Graph Paper Non- Programmable Electronic Calculator NARWAWN HE INSTRUCTIONS TO CANDIDATES This paper consists of eight questions. Answer only five questions. All questions carry equal marks. “All answers to each question should begin on a fresh page. Do not write anywhere on this question paper. All rough work should be done in the official answer booklet provided. Read other instructions on the answer booklet provided. © 2018 Uganda Business and Technical Examinations BoardQuestion One (2) Consider a fluid jet impinging on a series of flat plates attached on the pe. of a wheel. Derive an expression for the efficiency of the system. (14 marks) (b) A jet of water 60mm in diameter, moving with a velocity of 20m/s strikes a fat vane which is normal to the axis of the stream. Find the force exerted by the jet; (i) When the vane moves with the velocity of 8m/s. (04 marks) (ii) TF instead of one flat vane, there is a series of vanes attached on the periphery of the wheel and always moving with a velocity of 8m/s. (02 marks) Question Two (2) _ State five limitations of Bernoulli's equation (05 marks) () Seoul's equation is for incompressible frictionless fluid. Explain the main parts of Bernoulli's equation. (10 marks) (©) Explain the following terms of fluid dynamics; (i) Stream tube. (03 marks) (it) Streak line (02 marks) Question Three (a) An inclined plane surface is immersed in water at angle @ to the free water Surface. From first principle, derive the expression for the depth vertical distance of the centre of pressure below the free water surface (14 marks) 5) A circular laminar 125cm in diameter is immersed in water so that the distance of its perimeter measured vertically below the water surface varies between 60cm and 150cm. Find the; (Total force due to the water acting on one side of the laminar. (03 marks) (i)) Vertical distance of the centre of pressure below the surface. (03 marks)question Four (a) _ State the methods used for measuring the pressure difference between two ints in a fluid. e uid, (02 marks) (b) A differential U-tube manometer containing mercury is connected to two pipes A and B. Pipe A contains a liquid of specific gravity s:=1.6, and pipe 8, oil of specific gravity s2=0.8, Pipe A is at height of h above pipe B. (i) With the aid of a sketch, describe the process of determining the pressure difference of the two pipes. (14 marks) (ii) If the mercury in left limb is 50cm below A and 30cm in the right hand limb above the left limb, pipe B is 10cm below pipe A; determine the pressure difference of the two pipes. (04 marks) * Question Five One of the most important practical applications of Bernoull’s theorem is the venturi meter (a) With the aid of a simple sketch, derive the expression for discharge of @ fluid (12 marks) (b) With the aid of a sketch, explain how a pitot tube is used to determine the velocity of flow of the fluid. (08 marks) Question Six (a) Consider a small spherical droplet of a liquid with diameter « and identify the forces that are exerted on one half of the bubble. (04 marks) (b) The phenomenon of liquid rise or fall in the capillary tube is called capillary or meniscus effect. With the aid of sketches, explain this phenomenon. (11 marks) (©) Airis introduced through a nozzle into a tank of water to form a stream of bubbles. If the bubbles are intended to have a diameter of 2mm, calculate the; (i) Pressure at the nozzle that exceeds that of the surrounding. (03 marks) (li) Absolute pressure inside the bubble if the surrounding water is at 1O0kN/m?, Take the surface tension of water to be 0.073N/m. (02 marks) 3 Turn overQuestion Seven (a) With the aid of a sketch, illustrate the operation of a hydraulic jack when the effort cylinder is raised at a height, h. (05 marks) (b) With the aid of a sketch, derive an expression which shows that the intensity of pressure at a point in a fluid at rest is the same in all directions. (15 marks) Question Eight Fig. 1 Fig. 1 is a fluid element flowing from state 2 to state 2. Identify the forces that (a) are acting on the fluid element. (04 marks) (b) _Assiphon has a uniform circular bore of 75mm and consists of a bent pipe with its crest 1.8m above water level discharging into the atmosphere at a level of 3.6m below water level. Find the; (i) _ Velocity of flow. (06 marks) (ji) Discharge through the pipe (04 marks) (ii) Absolute pressure at the crest level ifthe atmospheric pressure is equivalent to 10m of water. Neglect losses due to friction (06 marks) ENDUGANDA BUSINESS AND TECHNICAL EXAMINATIONS BOARD | PAPERCODE | PROGRAMME "DATE | i NDME 212 | NATIONAL DIPLOMA IN THURSDAY, | | MECHANICAL | NOME 214 | | 19™ DEC. 2013 ENGINEERING | j SERIES “TIME ALLOWED PAPER NAME | NOV/DEC 2013 FLUID MECHANICS 3 HOURS | | "YOU SHOULD HAVE THE FOLLOWING FOR THIS EXAMINATION Answer booklet. Non-programmable electronic calculator. | Drawing instruments INSTRUCTIONS TO CANDIDATES This paper consists of eight questions. Answer five questions All questions carry equal marks. Begin each question on a fresh page All rough work should be done in the official answer booklet. Do not write on the question paper. ©2013 Uganda Business and Technical Examinations Board Turn OverQuestion One ‘ expression for th, 3) With the aid of a sketch, derive from Bernoulli's theorem an exp : theoretical discharge through a small orifice. / Cee b) An orifice meter consists of a 100mm diameter orifice in a (06 marisy and has a coefficient of discharge of 0.7. The pipe conveys oil of specific gravity 0.9 and the pressure difference between the ‘two sides of the orifice plate is measured by a mercury manometer, the leads to the gauge being filled with oil. If the difference in mercury levels in the gauge is 760mm. Calculate from first principles the rate of flow of oil in the pipe line (14 marks) Question Two 2) Explain the following terms as applied to properties of fluids; i) Mass density (02 marks) 1) Coefficient of dynamic viscosity. (02 marks) iil) Kinematic viscosity. (02 marks) iv) Surface tension. (02 marks) 5) The ram and plunger of a hydraulic jack are 150mm and 25mm in diameter respectively, With a plunger stroke of 300mm, the jack is able to lift a load of 20kN in 30 seconds, through 1.6m Calculate; ; ') The effort on the plunger, (06 marks) ii) Power required to drive the plunger. (03 marks) ii) Number of strokes done by the pump, (03 marks)Question Three a) State Archimedes principle. gs ' (02 marks) ) Explain, with the aid of a simple diagram, the application of Archimede: a floating body, ' principle to Considering the forces involved, their lines of action, the centre of buoyancy and the metacentre (05 marks) An empty tank is rectangular in plan, side elevation and end elevation and is made Of mild steel sheet. The top is closed by a sheet of the same metal and the dimensions of the tank are 120cm by 66cm by 60cm. If the steel sheet weighs 369N/m’and the tank is allowed to float in fresh water with the 60cm edges vertical; i) Prove that the equilibrium is unstable. (10 marks) ii) Find the inclination of these edges to the vertical after the tank has heeled over to a stable position. (03 marks) Question Four a) Derive an expression for the total pressure on one side of a plane surface of area, A immersed vertically in a liquid of specific weight win terms of 4,1 and tie vertical depth F to the centroid of the immersed surface below the free surface of the liquid, (08 marks) b) A closed vertical cylindrical tank 60cm diameter and 1.8m deep, contains water to 8 depth of 1.2m Air at a pressure of 3SKNV/m? above the atmosphere is pumped into the cylinder Determine; i) The total normal pressure on the vertical wall of the tank, (10 marks) ii) The distance of the centre of pressure from the base. (02 marks) Question Five a) Outline three conditions to be observed while using & piezometer. (03 marks) b) With the aid of a sketch, describe the construction of the piezometer. (05 marks) 3 Turn Overpipe at two points Aand Bin, <° ©) Fig, 115 a differential manometer connected to 2 liquid + contains mercury with specite fen the two points in the pipe (05 marks) of density 900kg/m*, The U-tube manometer 13.6. Calculate the pressure difference betwe gravity d) State three characteristics of manometric liquids. (03 marks) ) Give four reasons for using mercury as a manometric liquid. (04 marks) Question Six a) With the aid of a sketch, explain how a pitot tube is used to determine the velocity of the flowing stream at a point. (09 marks) b) A 300mm x 150mm venturimeter is provided in a vertical pipe line carrying oil of specific gravity 0.9, the flow being upwards, 4 ~ne difference in ele n ence in elevations of the throat Section and entrance section of the venturimeter is 300mm. The differential U ~ tube mercury manometer shows 2 e deflection of 250mm, ws a gauge Calculate; i) The discharge of the oil, (07 marks) ii) The pressure difference between the entrance and the throat. (04 marks) Take cd=0.98 and the specific gravity of mercury as 13.6. Question Seven @) Explain the following terms with reference to the flow of fluids:~ i) Steady flow. (02 marks) ii) Unsteady flow. (02 marks) ili) Rotational Flow. (02 marks) iv) Compressible flow. (02 marks) b) Oil flows through a pipe line shown in Fig. 2, which tapers from 450mm diameter at Ato 300mm diameter at 8. The pipe forks, one branch being 150mm diameter and discharging at Cand the other branch being 225mm diameter and discharging at O. If the velocity at A is 1.8m/s and the velocity at Dis 3.6m/s. Determine: i) The discharge through Cand 0. (07 marks) ii) The velocities at Band C. (05 marks) Fig. 2 Turn OverQuestion Eight meets a vane having a | jet of water 75mm in diameter with 2 velocity of 12m/s y i, - velocity of 4.8m/s in the direction of the jet. If the water meets the vane tangentially and is deflected through 120°, Find:- ) The force in the di ection of the jet. (13 marks) ) The work clone per se he work done per second ona single vane moving in a straight line. (02 marks) ©) The work done per second on a service of vanes mounted on the circumference of a large wheel. Ae marks) ENDsy) 178 rm | UGANDA BUSINESS AND TECHNICAL EXAMINATIONS BOARD PAPER CODE "PROGRAMME ar ‘DATE 2) DTME 217 NDME, NDEE, NOCE, NDWE, NDAR WEDNESDAY, | | YEAR II SEMESTER I ee | "SERIES PAPER NAME TIME ALLOWED NOV/DEC 2020 FABRICATION AND WELDING 3 HOURS | TECHNOLOGY | | YOU SHOULD HAVE THE FOLLOWING FOR THIS EXAMINATION | Answer Booklet Non-Programmabie Electronic Calculator | Drawing Instruments INSTRUCTIONS TO CANDIDATES 1. This paper consists of eight questions. 2. Answer only five questions. 3. All questions carry equal marks. 4. All answers to each question should begin on a fresh page. 5. Do not write anywhere on this question paper. 6. All rough work should be done in the official answer booklet provided. 7. Read other instructions on the answer booklet.s & & Question One > »s (a) Define oxyfuel gas welding. Qn @ (b) Explain two types of torches used in oxyfuel gas welding. (04 map, © (©) You are tasked as a technician to join two hollow section pipes of diameter SO using oxyacetylene welding equipment. Explain the procedure for settin: '9 UD the equipment before executing the task. (14 marks) gg Question Two fa) (b) (c) Question Three Define polarity as applied to arc welding machines, With the aid of a neat sketch, explain the dire welding machine. (02 marks) ct Current straight polarity (DCSP) of a (10 marks) the body during a welding operation. As ieved, Its necessary to protect eyes and the skin of 2 trainee welder, discuss how this can be achi (08 marks) (2) A client took a mild steet sheet to the workshop to be used to repair a leaking cast ‘ron saucepan. If the only available process in the ‘workshop was braze welding; (0. Discuss four steps the welder would Undertake to weld the saucepan, (08 marks) (ii) Identify four Gesign considerations made when welding the saucepan with the process in 3(a). (04 marks) (©) Explain four methods of brazing operation, (08 marks) Question Four (2) Explain welding electrode as used in fusion welding (02 marks) (©) List eight functions of flux coatings on welding electrodes, (08 marks) (©) Electric arcs of electrodes having three flux Coatings like E7018 are initiated using the scratch method technique so as to improve contact between the electrode and the metal. With the aid of a neat sketch, discuss the procedures followed when initiating the electric arcs with such electrodes, (10 marks) Question Five (2) Explain welding as used in metal jolning processes, (02 marks) (b) A client from a certain Organization is inquiring whether pipeline welders need to be trained more in arc welding than gas welding, 2D 3, ae %, (i) _ Differentiate between the two welding processes. (04 marks) yy Y wi % %, (ii) With aid of neat sketches, describe the four types of joints that would be used oy when training the pipeline welders. (12 marks) %e (ili) State two appli i fe two applications of the welding processes in 5(b) used to train the pipeline welders. (02 marks) Question Six (a) State two advantages of oxyacetylene gas welding. (02 marks) (b) Flying hot debris from welding is one of the biggest hazards welders face wnen welding. Explain eight means of preventing these hazards. (08 marks) (©) You have been provided with a piece of mild steel plate 15cm long by 40cm wide by 12.5mm thick, a strip of angle iron and two G-clamps. As a welder, explain how you would cut the plate into two parts using oxyfuel gas cutting. (40 marks) Question Seven (a) (v). Inspection and testing provide assurance that applicable standards are being met Weld testing services must be performed in accordance with standards such 2s European Norm (EN), American welding Society (AWS), the American Petroleum Institute (API) along with other applicable standards. As a welder, state six weld testing types that should be performed in accordance with these standards. (06 marks) Explain seven importance of qualification of welding procedures, welder performance and welding consumable: (14 marks) Question Eight (a) (b) Josntify six inert gases used in metal inert gas welding (06 marks) Weders from UMA have come to your institution to benchmark the applicability of GMAW process in all and gas sector. (i) Explain the working principles of the welding process with the aid of a well labelled diagram. (10 marks) (il) Enumerate four advantages of the welding process in 8(b) in the oil and gas sector. (04 marks) END_UGANDA BUSINESS AND TECHNICAL EXAMINATIONS BOARD P, APER CODE PROGRAMME | DATE OT™E 217 NATIONAL DIPLOMA IN MECHANICAL | TUESDAY, ENGINEERING | 177 DECEMBER 2019 | NATIONAL DIPLOMA IN ELECTRICAL | ENGINEERING | YEAR II SEMESTER | | jo t _ TIME ALLOWED SERIES PAPER NAME NOV/DEC 2019 FABRICATION AND WELDING 3 HOURS, TECHNOLOGY E THE FOLLOWING FOR THIS EXAMINATION YOU SHOULD HAV! Answer Booklet Non-Programmable Electronic Calculator Drawing Instruments INSTRUCTIONS TO CANDIDATES This paper consists of eight questions. 2 Answer only five questions. equal marks. All questions cary uestion should begin on a fresh page. All answers (0 each 9 Do not write anywhere on this question Paper. be done in the official answer booklet provided. wae w All rough work should 2 ead other instructions on (he answer booklet. © 2019 Uganda Business and Technical Examinations Board Turn Over2 ? (YY ~ 2 lates Using > tal pl in two me! sked to jor ° ane id of a neat diagram, show , wy ASU 5, With the 2 oe ra oxy-tuel welding proces aratus. truction of the oxy-fuel gas welding app ipe. a e to light the torch/blow pip: nt would tak ) (i) Explain the steps the studer (04 marks) ibe the formation of the various types of flames (ii) With the help of sketches, descr (10 marks) in oxy-acetylene gas welding processes. Question Two (a) Define oxy-acetylene cutting as applied in fabrication. (03 marks) (b) With the help of neat sketches, illustrate the following welding joints and positions ()) Single bevel joint in a horizontal position (02 marks) i) Vee bi n ( Jutt joint in a flat position (02 marks) (i) Square butt joint in a horizontal position (02 marks) ‘ marks () Square butt joint in a vertical position (c) Assuming you are the workshop manager, (02 marks) (a) ion in refer i (>) Outline three advantages ade, Tee enc 9eS and two disadvantage of metal ee ‘oe, inert with -: the ai i '€ aid of 3 sketch, Gescribe the : i von Principle of OPerati, . a 2 ‘ON OF th, Pain the principle OF Self-adjust srry etal oe AT inert gas weidin ue toon . ~ ‘dare j elation te (06 Marks)(a) (b) >, Question Four % A student was tasked to weld a pipe line during industrial training however, when te weld was tested, it was found to be having defects. (i) Define what is meant by 2 weld defect. (02 marks) (ii) With the aid of simple sketches, explain six possible causes of weld defects. (12 marks) (i) Define the term flux in relation to welding. (02 marks) (04 marks) (ji) Identify four functions or uses of flux on SMAW electrodes Question Five (a) (b) () g is one of the widely applied modern welding Tungsten inert gas (TIG) weldin (02 marks) processes. State two of its applications. When welding with tungsten inert gas (TIG), there is high chances of danger due to the following; (i) Electric shock and burns, (ii) High frequency and Question Six (a) (iii), Fumes ‘As a trained technician, explain how the dangers from each of the above can be avoided (06 marks) With the aid of a sketch, describe the principle of operation of the tungsten inert gas (TIG) welding process. (12 marks) Define the following as applied to welding technology (i) Arc welding (02 marks) (ii) Weldability (02 marks) ‘A student on industrial training was tasked to weld two copper plates together using spot welding. As a technician for welding, (i) With the help of a sketch, illustrate spot welding to the student. (03 marks) (ii) Describe the procedure the student will follow to accomplish the task. (07 marks) 3 Turn Over(oO fety is one of os ‘hat shoule jified welder, explain six safety precautions tl (06 marks In a welding workshop, catequard against. As 2 qua erved in a welding workshop, obs Question Seven (a) _ In reference to oxy-fuel gas welding, enumerate four precautions taken when storing and using compressed gas cylinders (04 marks) (b) Explain the following abnormalities observed in oxy-fuel gas welding. (i) Back fire (02 marks) (ii) Flash back (02 marks) (©) ()__ Explain five causes of back fire and how they can be prevented, (10 marks) (ii) Distinguish between the oxygen and acetylene cylinders. (02 marks) Question Eight 2) Explain the following divisions of weldiaag (i) Forge or pressure welding (i) Fusion or non-pressure welding a (02 marks) (b) (c) With the aid of oo — illustrate the following weld joint features or parts; (li) Bevel angle, (Wi) Groove angle, (™) Root edge and (¥) Groove face, An electrode f¢ ‘or welding a mild steel rod is coded as ER4: 211. Explai in the each charact er in the electr rode nomeni clature aby ove, ENDDATE PAPER CODE PROGRAMME | OTEM 211 | NOCE, NDWE, NDEE, NOME, NDCR, | THURSDAY OEM 211 NORA, NDAR, NOTE, NDAE 27" JANUARY 2022 YEAR II SEMESTER | SERIES PAPER NAME TIME ALLOWED | i NOV/DEC 2020 ENGINEERING MATHEMATICS I rie 3 HOURS YOU SHOULD HAVE THE FOLLOWING FOR THIS EXAMINATION i | Answer Booklet | Non-Programmable Electronic Calculator | Graph Paper | mathematical Tables _ Drawing Instruments ' INSTRUCTIONS TO CANDIDATES J. This paper consists of eight questions. 2. Answer only five questions, All questions carry equal marks. All answers to each question should begin on a fresh page. Do not write anywhere on this question paper. All rough work should be done in the official answer booklet provided. NOW ay Read other instructions on the answer booklet. © 2022 Uganda Business and Technical Examinations Board Turn OverQuestion One g (a) Table.1 shows the length of 250 electronic components which were accurately .o 4 x Length (cm) i < 7.00 7.00 - 7.05 Given that 10% of the components are scrapped because they are too short and 8% are scrapped because they are too long, Use a cumulative frequency graph to estimate limits for the length of an acceptable component. (08 marks) (b) The mean and standard deviation of the heights of 12 boys in a class are 148.8cm and 5.4cm respectively. A boy of height 153.4cm joins the class. Find the mean and standard deviation of the heights of 13 boys. (08 marks) (c) The depth of water in a lake was measured at SO different points on the surface of the lake. The depths, x metres are summarized by Ex = 934.5 and Ex? = 19275.81. Find the mean and variance of the depth. (04 marks) Question Two (a) Given that cos 60° = 0.5, determine the value of cos 70° using Taylor's series correct to four decimal places. (09 marks) (b) Evaluate.’ 28" de using Maclaurin’s theorem. (11 marks) Question Three (04 marks) (a) Find the middle term of (2p ~ (b) Determine the value of (3.039)* correct to 6 significant figures using the binomial series. (06 marks) (c) Express as power series as far as the term in x*. State the range of values for which the series is convergent. (10 marks)2 : %, Question Four \ The data shows ; the frequency distribution for the valu 48 resistors of a similar value. . 21.0 : 22.4 228 215 22.6 211 216 22.9 20.5 21.8 22.2 21.0 21.7 22.5 23.2 229 21.7 214 22.1 22.2 22.3 22.1 21.8 22.0 22.7 217 219 211 21.4 22.4 22.3 20.9 22.8 21.2 22.7 22.2 216 2163) 22 215) 22.0). 2314) O23. 207, 2113, 226 2166) 21 2) Of resistance in ohms of a batch of (a) Form a frequency distribution table for the data having 6 classes. (06 marks) (b) Draw a histogram to represent the data and use it to estimate the mode. (06 marks) (c) Find the; (i) Mean. (02 marks) (ii) Median. (03 marks) (jii)Standard deviation of the resistance. (03 marks) Question Five (a) The diameter of a round bar is measured at 25 points selected at random along its length and the mean of these values is 0.425mm. If the standard deviation of the diameter of the bar given by the manufacturer is 0.030mm, determine the, (i) 80% confidence interval of the estimated mean diameter of the bar (2¥2 marks) (ii) Degree of confidence for the mean diameter of 0.425 + U.012mm (4¥2 marks) (b) The probability of a student getting a job during the industrial training period is 0.0003. For a population of 7500 students, draw a histogram showing the probabilities of 0, 1, 2, 3, 5 and 6 students getting a job during this period (13 marks) Question Six (a) Two tetrahedral dice are thrown and the sum of their scores noted. Find; ‘unction of the random variable \ representing the sum (i) The probability distribution f (05 marks) of the scores. a (i), Pa > 5) ie cae) (ii), Pa) (06 marks) Question Two (2) The mean of four numbers is 25. Their median is 23. If the largest number exceeds the smallest number by 30 and largest number is one and a half times the second largest number, determine the four numbers. (11 marks) (b) An urn contains red and green beads in the ratio 3:5. Several beads are selected at random from the urn. Obtain the number of beads that must be picked so that the probabillity is greater than 0.90 and there is at least one green bead amongst them. (09 marks) Question Three (a) Expand (1 ~ 16x)" in ascending powers using binomial theorem, Use your expansion and the substitution of x = —— to evaluate (39)% to 6 significant figures, (08 marks) (b) Find the first 4 terms in the expansion log, Sin(x + h) in ascending powers of h, hence find the value log, Sin 31° to four places of decimals given that log. 2 0.69315, (12 marks) Question Four (a) (i) Determine the first S terms for the series fm(1+x) and én(1-x) respectively, (10 marks) (ii) Use the series for ¢n(1 + x) and for ¢n(1 ~ x) to determine the series for Ly fn (03 marks) J}Fine ind the Fange of values of x for which th 7 e es x followin series , 25 3s? es 19 Series is absolutely convergent (07 marks) Question Five (a) (b) (c) (d) A deep well drill in 1g Company charges a flat fee of 500 dollars set up charge plus 5 dollars for the fi ee irst metre of the well drilled, 6 dollars for the second metre drilled, 7 . for the third metre drilled and so on up to the 70" metre. Find; i (ii) How much the company will charge for the 70" metre. (02 marks) The total charge for the 70m well drilled (use series of A.P or G.P). (03 marks) A stack of electric poles has 30 in the bottom row, 29 in the next and so on with one pole in the top row. Use series (AP or G.P) to find the total number of poles in the stack. (05 marks) ‘A parachutist falls 10m during the first second, then falls 20m during the next second, 30m during the third second and so on. Using series (A.P or GP), find the number of metres the parachutist will fall during the; (i) 10 second. (03 marks) (ii) First ten seconds. (02 marks) The mean weight of the whole class of 30 boys is «kg. When two boys with a total weight of 150kg are absent, the mean weight of those present is 2kg less than that of the whole class. Find the value of x. (05 marks) Question Six (a) In a certain country, 60% of the cars are privately owned, Of the privately owned cars, 70% are small, whereas of the cars which are not privately owned, 40% are small. Independent of ownership, 20% of the small cars and 30% of the large cars are less than two years old. If a car is selected at random, calculate the probability that the car is; (i) Small (06 marks) (ii) Privately owned given that it is large. (02 marks) ‘ore than two years old. (02 marks) (iii) Privately owned large and m (iv) Large given that it is privately owned and more than two years old (02 marks) Turn over