isical.ac.in : JRF Quality Reliability & Operations Research Sample Question for ISI Admission Test
Name of the University : Indian Statistical Institute
Exam : ISI Admission Test
Document Type : Sample/Previous Year Question Paper
Name of the Subject : Quality Reliability And Operations Research
Year : 2016
Website : http://www.isical.ac.in/~admission/IsiAdmission2017/PreviousQuestion/Questions-Jrf-QROR.html
Download Sample/Previous Years’ Questions :
MMA 2016 : https://www.pdfquestion.in/uploads/11370-MMAphy-2016.pdf
QRB 2016 : https://www.pdfquestion.in/uploads/11370-JRF-QRB-2016.pdf
MMA 2015 : https://www.pdfquestion.in/uploads/11370-MMAphy-2015.pdf
QRB 2015 : https://www.pdfquestion.in/uploads/11370-JRF-QRB-2015.pdf
MMA 2014 : https://www.pdfquestion.in/uploads/11370-MMAphy-2014.pdf
QRB 2014 : https://www.pdfquestion.in/uploads/11370-JRF-QRB-2014.pdf
JRF Operations Research Question for Admission Test
Test Code : QRB
Session : Afternoon
Time : 2 Hours
Related : JRF Computer Science Sample Questions for ISI Admission Test 2017 : www.pdfquestion.in/11366.html
Instructions
Read the following carefully before answering the test :
** Write your Registration Number, Test Code, Booklet No. etc., in the appropriate places on the answer booklet.
** The question paper is divided into two groups: Group A and Group B.
** Group A has one question on Mathematics, two on Probability and one on Statistics.
** Group B has totally eight questions – two on each of Operations Research, Reliability, Statistical Quality Control and Quality Management.
** Each question carries 24 marks. You have to answer two questions from Group
** A and three questions from Group B. Marks are specfied against the questions.
** Partial credit may be given to partial answers.
** Full credit may be given to complete and rigorous arguments.
** All Rough Work Must Be Done On This Booklet And/Or On The Answer-Booklet. You Are Not Allowed To Use Calculators In Any Form.
Model Questions
Mathematics :
1. (a) Let A = CCt and B = DDt, where C and D are real matrices so that AB is well dened. Prove that if trace of AB is equal to 0, then CtD = 0. Here Xt stands for the transpose of the matrix X. (8)
(b) Define f(x; y) = 2×2+3y+4 for any point (x; y) on the line segment joining the points (0; 0) and (-4; 3). Transform this function into a uni variate function of non negative variable and sketch the graph of the transformed function. (8)
(c) Let A be a xed real matrix with 3 rows. For any 3 1 real vector b, dene S(b) = fx : Ax = bg. Assume that S(bi) 6= ; for i = 1; 2; 3, where b1 = (1; 2; 3)t, and b2 = (3; 2; 1)t and b3 = (4; 4;
4)t. Further assume that for any b, S(b) 6= ; implies that b is a linear combination of b1; b2 and b3. Find the rank of the matrix A and justify your answer. (8)
Probability :
2. Each human has two chromosomes. While all males have one X chromosome inherited from mother and the Y chromosome inherited from father,all females have two X chromosomes inherited one from mother and one from father. Call an X chromosome bad if it leads to a disease called haemophilia, else call it good. It is known that ladies having children have at most one bad chromosome. Consider a lady whose brother has a bad chromosome but her father has a good chromosome. Find the probability that the lady is a carrier, that is, the lady has a bad chromosome if
(a) the lady has at most one bad chromosome
(b) the lady has only one child, a son with a bad X chromosome
(c) the lady has only one child, a son with a good X chromosome
(d) the lady has only two children, sons, both with good X chromosomes
(e) the lady has only two children, sons, both with bad X chromosomes
(f) the lady has only one child, a daughter. (4 6 = 24)
3. (a) A basket contains 50 items of which 20 are defective. Two items are drawn, one after the other, without replacement. Dene Yi = 1 if ith item drawn is defective, Yi = 0 if it is good, i = 1; 2.
i. Show that Y1 and Y2 are identically distributed. (4)
ii. Are Y1 and Y2 independent? (2)
iii. What is the expected value of Y1 + Y2? (2)
(b) An urn contains n strings. Two ends are picked up randomly, the ends are tied and the resulting string/loop is dropped back into the urn. This process is repeated n times (because there are a total of 2n ends to begin with). The number of loops formed at the end of such a process, X, is a random variable. Find the probability that the fifth knot results into a loop. Hence or otherwise, find out the expectation of X. (8)
(c) A software code has N lines and each line is either defective or good. The problem is to estimate the number of defective lines. The code s independently tested by two engineers – one of them experienced and the other is fresher. Each engineer marks each line as good or defective, separately. Assume that none of them marks a good line as defective. Let x be the number of lines marked as defective by the first engineer, y be the number of lines marked as defective by the second engineer, and z be the number of lines marked as defective by both engineers. Estimate the number of defective lines and explain your method. (8)