Q.1(a)(i)Prove that
(ii)Let
= k if x = 2
If f (x) is continuous for all x, then find out the value of ‘k’.
(b)(i)If , then prove that .
(ii)If , then prove that
(c)(i)If show that .
(ii)If , prove that
(d) (i)Evaluate
(ii)Evaluate, where
(e)(i)Solve
(ii)Solve.
(5 5 = 25)
Q.2(a)Find a unit vector parallel to the resultant of vectors
(i)
(ii)If find
(1)
(2)The unit vector parallel to but opposite in sense.
(b)(i)A particle is acted on by constant forces and is displaced from the point to point . Find the work done.
(ii)Using the line integral, compute the work done by the forces when it moves a particle from the point (0, 0, 0) to the point (2, 1, 1) along the curve .
(c)(i)Find all the values of such that (x, y, z) (0, 0, 0) and .
(ii)If show that F. curl F = 0.
(d)(i)Evaluate the integral
, where C is given by x = t, y = t2, z = 3 t, 1 t 2.
(ii)Verify Green’s theorem for
where C is bounded by y = x, and y = x2.
(e)(i)Show that the following vector is irrotational and find the scalar potential
(ii)If
find a, b, c such that curl F = 0, then find such that F = .
(5 5 = 25)
Q.3(a)Show that the adjoint of is 3A and the adjoint of is A itself.
(b)Find the rank of the matrix
(c)Find the characteristic equation of the matrix
Show that the equation is satisfied by A and hence obtain the inverse of the given matrix.
(d)Define Hermitian matrix, Unitary matrix and Orthogonal matrix.
Find k, l and m to make A a Hermitian matrix,
(10 + 5 + 5 + 5 = 25)
Q.4(a)(i)A box A contains 5 white and 2 black balls. Another box B contains 4 white and 5 black balls. A ball is transferred from box A to box B. Then a ball is drawn from the box B. Find the probability that it is white.
(ii)A coin is tossed 4 times. Let p be the event that the total number of heads are 2 and q be the event that the second toss results in tail. Find P (p) and P (q).
(b)(i)If events A and B are independent and , then find p (B)
(ii)A box contains 10 components, 3 of which are defective. A sample of size 5 is taken without replacement from this box. Obtain the probability that there are
2 defective components in the sample.
(c)(i)The mean life time of a sample of 100 fluorescent light bulbs produced by a company is computed to be 1570 hours with a standard deviation of 120 hours. The company claims that the average life of the bulbs produced by it is 1600 hours. Using the level of significance of 0.05, is the claim acceptable?
(ii)20% of the tools produced in a certain manufacturing process turns out to be defective. Find the probability that in a sample of 20 tools chosen at random,
(a)exactly 5 will be defective, and
(b)more than one will be defective.
by using Poisson’s approximation.
(d)(i)Assume that the probability of an individual coal miner being killed in a mine accident during a year is . Use appropriate distribution to calculate probability that in a mine employing 200 miners, there will be at least one such accident in a year.
(ii)Given the distribution function
Compute
(i),(ii)
(iii),(iv)
(e)(i)Two independent samples of sizes 6 and 5 have the following values :
Box A / 20 / 45 / 29 / 40 / 38 / 32Box B / 19 / 39 / 29 / 50 / 33
Examine whether the samples have been drawn from normal population having the same variance (5% level of significance).
(ii)Two hundreds digits were chosen at random from a set of tables. The frequencies of the digits are as below:
Digit / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9Frequency / 18 / 19 / 23 / 21 / 16 / 25 / 22 / 20 / 21 / 15
Use 2 test to assess the correctness of the hypothesis that the digits were distributed in an equal manner in the tables from which they are chosen.
(5 5 = 25)
Q.1(a)Explain the technological factors that necessitated the development of Numerical Control of machine tools.
(b)What are the applications in which Numerical Control can be found most suitable?
Q.2(a)Briefly explain the basis of designing the coordinate axes in CNC machine tools.
(b)How is a CNC control system organized? Briefly explain the functions of any three elements in the control.
Q.3(a)Explain the advantages of re-circulating ball screws compared to the conventional Acme screws.
(b)Describe five applications where the tough trigger probes can be used on a shop floor.
Q.4(a)In order to write CNC part programmes, what are the various types of information that needs to be considered by a part programmer.
(b)Explain the importance of part programme verification.
Q.5(a)Explain the difference between turning centre programming and machining centre programming.
(b)How the different lengths of the multiple tools used in milling is compensated? Explain the procedure.
Q.6(a)Discuss the different ways that computers can be used in the production of part programmes, detailing the advantages and limitations.
(b)What is a part programming language? Describe the three types of information, which are normally required for a part programming language.
Q.7(a)Can DNC be used for older NC machine tools? Explain your answer with a method showing how it can be applied.
(b)Explain cell control function as an advancement of DNC with the help of a block diagram.
Q.8(a)What are the factors that need to be considered while selecting a robot as a material handling equipment?
(b)What are the various tool monitoring systems available in an FMS? Explain any one system in detail.
Q.9(a)What are the different types of system layouts used in FMS? Explain their application.
(b)What are the steps involved in developing a cell layout? Briefly explain the functions.
Q.10(a)Explain how NC has helped to reduce the manufacturing time of a component.
(b)Describe the different methods of inputting information into the control unit of a CNC machine, detailing the advantages and limitations of each method.
Q.1(a)Juran defined quality as “fitness for use or purpose”. Critically evaluate the definition.
(b)An expert on quality has proposed that quality is the only issue for organizational survival. Discuss facing contemporary organizations.
Q.2(a)Explain Quality Function Development (QFD) with suitable examples. What are the benefits of its application?
(b)What is the significance of Risk Priority Number (RPN)? What is the follow up action after RPN is determined?
Q.3(a)Briefly explain the difference between SPC (statistical process control) and SQC (statistical quality control)?
(b)Name the statistical method used in checking the acceptability of parts supplied in lots by a vendor.
Q.4(a)What is the justification for Philip Crosby’s claim that “Quality is free”? Give reasons in support of your answer.
(b)Discuss and examine different views on costs and benefits of quality. Which of the views, do you think, you can support and why?
Q.5(a)“Quality is consistently achieving customer delight”. Comment.
(b)Can high quality and low costs go together? Illustrate your answer with examples.
Q.6(a)You are appointed as the coordinator of the steering committee in your organization to introduce TQM. What are the factors you should look for succeeding in this new position?
(b)“Production is an art and quality is the measure of this art”. Discuss.
Q.7(a)A factory manufacturing televisions has four units A, B, C, and D. The units A, B, C, D manufacture 15%, 20%, 30% and 35% of the total output respectively. It was found that out of their output 1%, 2%, 2%, and 3% are defective. A television is chosen at random from the total output, and found to be defective. What is the probability that it came from unit D?
(b)In a certain factory producing cycle tyres there is a small chance of 1 in 500 tyres to be defective. The tyres are supplied in lots of 10. Using Poisson distribution calculate the approximate number of lots containing the defective, one defective, and two defective tyres, respectively, in a consignments of 10,000 lots?
Q.8.(a)A firm producing automobile exhaust filters was required to provide the Environmental Protection Agency with failure rate data based upon a 10-hour test. If 200 units were tested and 8 failed, what are
(i)the failure rate in failures per unit per year and
(ii)the mean time between failures?
(b)A firm with a processing system using machines X and Y in sequence has now installed another machine, Z, which performs an equivalent job. If the respective reliabilities of X, Y, and Z are 0.9, 0.8, and 0.7 respectively, what is the total reliability of the system?
Q.9(a)A product design engineer must decide if a redundant component is cost-justified in a certain system. The system in question has a critical component with a probability of 0.98 of operating. System failure would involve a cost Rs. 20,000. For a cost of Rs. 100, a switch could be added that would automatically transfer the system to the backup component in the event of a failure. Should the backup be added if the backup probability is also 0.98?
(b)What is meant by the term availability? Why would this be a consideration in system design?
Q.10(a)Determine availability for each of these cases:
(i)MTBF = 40 days, average repair time = 3 days
(ii)MTBF = 300 hours, average repair time = 6 hours.
(b)A designer has estimated that she can increase the average time between failures of a part by 5 percent at a cost of Rs. 4500, or she can reduce repair time by 10 percent, at a cost of Rs. 2000. If only one option is possible, which one would be more cost effective? Currently, the average time between failures is 100 hours and the average repair time is 4 hours.
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