UNIVERSITY OF ENGINEERING & MANAGEMENT, JAIPURQUESTION BANK

SUBJECT NAME: QUANTATIVE TECHNIQUES, SUBJECT CODE: MBA102

MBA, 1st YEAR, 1st SEMESTER

GROUP-A

(Objective/Multiple type question)

1. If Matrix and then find A + B.2. Define finite and infinite set with examples.3. Calculate (a) , (b) 4. Let A and B be two finite sets such that n(A) = 20, n(B) = 28 and n(A ∪ B) = 36, find n(A ∩ B). In tossing a fair die, the probability of getting on even number and greater than 2 is?5. Matrix and if then find the value of x, y, z, w.6. Determine whether the following functions are increasing or decreasing on given intervals: y = 2x - 57. Define Range.8. If P(A) = 5/8, P(B) = 3/7 then P(A ∩ B) = ? If A and B are mutually independent.9. If the wages and numbers of workers in a factory are as under, find the Arithmetic average wage.

Wages 100 125 150 175 200 500No. of workers 14 9 6 8 3 210. In tossing two coins, the probability of getting two heads.

11. In linear programming, the objective function and objective constraints are(i) Solved (ii) linear(iii) Quadratic (iv) Adjacent12. In Graphical solution the feasible solution is any solution to a LPP which satisfies ____________. (i)Only objective function. (ii) Non-negativity restriction. (iii)Only constraint. (iv) All the three. 13. The two forms of LPP are _________. (i) Standard form and canonical form. (ii) Standard form and general form. (iii) Matrix form and canonical form. (iv) Matrix form and standard form 14. In the simplex method, the slack, surplus and artificial variables are restricted to be (i)Multiplied (ii) Negative (iii)Non-negative (iv) Divided15. The probability of an event cannot be: (i) Equal to zero (ii) Greater than zero (iii) Less than zero (iv) Equal to one16. Define independent events.17. The scatter in a series of values about the average is called: (i) Central tendency (ii) Dispersion (iii) Skewness (iv) Symmetry18. If 35 is the upper limit of the class-interval of class-size 10, then the lower limit of the class-interval is: (i) 20 (ii) 25(iii) 30 (iv) None19. ) When the occurrence of one event has no effect on the probability of the occurrence of another event, the events are called: (i) Independent (ii) Dependent (iii)Mutually exclusive (iv) Equally likely20. Find the mode for the set of numbers 2, 2, 3, 6, 6, 6, 7, 8, 9. (i) 3 (ii) 7 (iii) 8 (iv) 6 21. In time series which is a mathematical expression of multiplicative model. (i) Y=T+S+C+I (ii) Y=TSC+I (iii) Y=TSCI (iv) Y=TS+CI22. Which model cannot be considered as a time series model. (i) Mathematical model (ii) Regression Model (iii) Additive Model (iv) Mixed Model23. While computing mean of grouped data, we assume that the frequencies are: (i) Evenly distributed over all the classes (ii)Centered at the class-marks of the classes

(iii)Centered at the upper limits of the classes (iv)Centered at the lower limits of the classes24. Find the median and mode for the set of numbers 2, 2, 3, 6, 6, 6, 7, 8, 9. (i) 3, 3   (ii) 7, 7 (iii) 8, 8 (iv) 6, 6  25. Differentiate 26. Determine whether the following functions are increasing or decreasing on given intervals: y = 2x - 527. If n(A) = 38, n(A ∪ B) = 70 and n(A ∩ B) = 25, then find n(B).28. What is ?29. What is ?30. Find f’’(1) and f’’’(-1) of the function f(x) = x3 + 2x2 – 6x + 4.31. In a group of 60 people, 27 like cold drinks and 42 like hot drinks and each person likes at least one of the two drinks. How many like both coffee and tea?32. In tossing a fair die, the probability of getting on even number and greater than 3 is?33. Define Probability with examples.34. If P(A) = 3/8, P(A ∪ B) = 7/8 and P(A ∩ B) = 2/5, then find P(B).35. In tossing two coins, the probability of getting two heads.36. A bag contains 7 red and 6 white balls find the probability that they will draw a red ball. 37. If P(A) = 3/8, P(B) = 7/8 then P(A ∩ B) = ?IfA and B are mutually independent.38. If the wages and numbers of workers in a factory are as under, find the Arithmetic average wage.

Wages 100 125 150 175 200 500No. of workers 11 8 7 4 3 139. The arithmetic mean of 10 items is 4 and the arithmetic mean of 5 items is 10. Find combined arithmetic mean .

40. In a moderately symmetrical series, what is relation between arithmetic mean, median and mode .41. In a moderately skewed distribution, Mean = 45 and Median = 30, then calculate the value of mode .42. The arithmetic mean and geometric mean of two observations are 4 and 8 respectively, then find the harmonic mean .43. What is standard form of LPP. Explain? 44. What is difference between feasible and basic solution. 45. A cyclist pedals from his house to his collage at speed of 10km/h and back from the college to his house at 15km/h find average speed. 46. The first for raw moment is -1.5, 17, -30 and 108 find kurtosis coefficient.47. Define the Morden definition of probability.48. Let (1, 2, 3...50) one number is selected at random. What is the probability that selected number is multiple of 6 or 8.49. A, B and C are three arbitrary events find the expression for at least two events occur.50. A, B and C are three mutually exclusive and exhaustive events associated with a random experiment. Find P(A) given that :P(B)=3/2P(A) and P(C)=1/2P(B)51. Define (a) Disjoint Event (b) Independent Event.52. A urn contain 5red,3 green,2black balls. Two balls are selected at random, what is the probability that ball is red and green if first ball is replaced before second ball drawn.53. Total number of arrangements of the letters of the word STATISTICS is a) 3360 b) 504c) 16800d) 5040054. The sum of deviations taken from their A.M. is always equal to a) one b) zero c)depends on values d) none of these.

GROUP-B

(Short answer type questions)

1. Two unbiased dice are thrown. Find the probability that:(i) Both the dice show the same number,(ii) The total of the numbers on the dice is 13.

2. In how many ways 6 dashes and 7 dots can be arranged? And how many ways arrange 10 of them.3. An Urn contains 11 black and 19 white balls. Find the probability of drawn two balls of the same colour.4. Find the median wages of the following distributionWages (in Rs.) 2000-3000 3000-4000 4000-5000 5000-6000 6000-7000No. of workers 3 5 20 10 5

5. Find the mode of the following distributionClass-interval 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80Frequency 5 8 7 12 28 20 10 10

6. Determine whether the following functions are increasing or decreasing on given intervals: y = x2 - 57. In a group of 100 persons, 72 people can speak English and 43 can speak French. How many can speak English only? How many can speak French only and how many can speak both English and French?8. Use the graphical method to solve the following LP problemMax Z=10x1+10x2s.t. x1+x2 ≤ 10, x1+2x2 ≤ 8

2x1 - x2 ≤ 2, x1 - 2x2 ≤ 1 and x1,x2 ≥09. Explain (a) Basic and basic feasible solution.(b) Principle of duality in linear programming. 10. Write the dual of the following LP problemMin Z= x+2ys.t. x+y ≥1, x+2y≥1 2x+y≥1 and x , y≥0 11. Find the median wage of the following distributionWages 0-1000 1000-2000 2000-3000 3000-4000 4000-5000 5000-6000 6000-7000No. of employees

4 16 20 30 25 35 4012. An Urn contains 10 white, 4 red and 3black balls. If 3 balls are drawn at random. Find the probability that

(a) Two of the balls drawn are white (b) one is white(c) one is of each color (d) one is black13. For a distribution the mean is 10, variance is 16, ɣ1 (gamma) is +1 and β2 is 4. Obtain the first four moments about the origin i.e. zero. Commit upon the nature of distribution.14. Explain Simplex method.15.Use the graphical method to solve the following LP problemMax Z=12x1+10x2s.t. x1+2x2 ≤ 10, x1+x2 ≤ 82x1 - x2 ≤ 2, x1 - 2x2 ≤ 1 and x1,x2 ≥016.Write the dual of the following LP problemMin Z= 4x+2ys.t. 2x+2y ≥1, x+2y≥1 2x+y≥1 and x , y≥017.Find the median wage of the following distributionWages 0-1000 1000-2000 2000-3000 3000-4000 4000-5000 5000-6000 6000-7000No. of employees

4 16 Y3 Y4 Y5 6 418.An Urn contains 10 white, 4 red and 3black balls. If 3 balls are drawn at random. Find th probability that

(a) Two of the balls drawn are white (b) at least one is white(c) one is of each color (d) at least one is black(e) none is red19.Fit trend line in given data from the method of least square.1 2 3 4 5 6 718 20 16 21 26 23 2720.Determine whether the following functions are increasing or decreasing on given intervals: y = 3x3 + 6x2 + 4x - 521.In a group of 100 persons, 72 people can speak English and 43 can speak French. How many can speak English only? How many can speak French only and how many can speak both English and French?22.Determine whether the following functions are increasing or decreasing on given intervals: y = x2 - 523.Find the maxima and minima for y = 5x3 + 2x2 − 3x.24.Calculate the mean and standard deviation from the following data:

x (observations) 2.7 2.9 3.1 3.3 3.5 3.7f (frequency) 2 7 15 21 2 3

25.Calculate the Mean and Standard deviation from the following data:x (observations) 2.7 2.9 3.1 3.3 3.5 3.7f (frequency) 2 7 15 21 2 3

26.Calculate the Mode and Median from the following data:x (observations) 2.7 2.9 3.1 3.3 3.5 3.7f (frequency) 2 7 15 21 2 3

27.An Urn contains 10 black and 10 white balls. Find the probability of drawn two balls of the same colour.28.Use the graphical method to solve the following LP problem Maximize Z=2X1+3X2 S.T. X1+X2≤30 X2≥3 0≤X2≤12 X1≤20 X1-X2≥0 And X1,X2≥0

29.Find dual of given LP problem Min Z= 4x+2y s.t. 2x+2y ≥1 x+2y≥1 2x+y≥1 And x,y≥0 30.Find mode from following data

31.Find feasible region of given LPP Maximize Z=10X1+15X2 S.T. 2X1+X2≤26 2X1+4X2≤56 -X1+X2≤5 And X1,X2≥0 32. Construct simplex table 1 and find pivot element of given LPP Maximize Z=10X1+15X2 S.T. 2X1+X2≤26 2X1+4X2≤56 And X1,X2≥0 33.Data on the readership of a certain magazine show that the proportion of ‘male readers under35 is 0.40’ and ‘over 35 is 0.20’. if the proportion of readers under 35 is 0.70. find the probability that a randomly selected male subscriber is under 35 year of age .34.A and B are two weak students of mathematics and their chances of solving a problem in mathematics correctly are 1/6 and1/8 respectively. If the probability of their making a common error is1/525 and they obtain the same answer, find the probability that their answer is correct.35.In jaipur40% are male and 60% are female in these 5% are male drinker and 1% are female drinker. One person selected at random. What is the probability that selected person is drinker.36.A company produce three type of boly A,B and C. The total production 50% of type A, 30% of type B, 20% of type C. In these bolts 5% of A, 3% of B and 2% of c are defective. One bolt is selected at random, selected bolt found to be defective what is the probability that bolt is type B.37.Toss a coin trice find the probability that(a) at least one head occur (b) at most one head occur (c) none of head occur.

class 0-10 10-20 20-30 30-40 40-50frequency 7 8 20 10 5

38.A manufacturer supplies quarter horsepower motors in lots of 25. A buyer, before taking a lot, tests at random a sample of 5 motors and accepts the lot if they are all good; otherwise he rejects the lot. Find the probability that :(i) he will accept a lot containing 5defective motors ;(ii) he will reject a lot containing only one defective motors.39.If 10% of the pens manufactured by the company are defective, find the probability that a box of 12 pens contain (1) Exactly two defective pens . (2) At least two defective pens. (3) No Defective pens.40. In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?41.In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there?42.Prove D’ Morgan’s laws for the following sets : U = { 2, 3, 4, 5, 6, 8, 9 }, A = { 3, 5, 9 }, B = { 4, 6, 8 }.43.Find the matrix is invertible. Find its inverse if possible and verify the result.44.If u=log (x3+ y3+z3−3xyz ) , then prove ∂u

∂x+ ∂u∂ y

+ ∂u∂z

= 3x+ y+ z .

45.Verify Euler’s Theorem For 46.Find the inverse of the matrix .47.Solve the following by matrix inversion method: x + y + 2z = 4 2x + 5y - 2z = 3 X + 7y - 7z = 548.Verify Euler’s theorem for the function 49.A sample of 100 observations had mean 64 and s.d. 7•5. Two observations whose values were 14 and 18 were wrongly recorded as 24 and 24. Make necessary correction of mean and s.d.50.Find the 14 Arithmetic means between 5 to 8 and show that their sum is 14 times the A.M. between 5 to 8.

GROUP-C

(Long type questions)

1. Solve the system of linear equation 2. Calculate the mean and standard deviation for the following table giving the age distribution of 542 members.Age (in years) 20-30 30-40 40-50 50-60 60-70 70-80 80-90Frequency 3 61 132 153 140 51 2

3. In a factory employee 3,000 persons, in a day 5% work less than 3 hrs, 580 works 3.01 to 4.50 hrs, 30% work from 4.51 to 6.00 hrs, 500 works from 6.01 to 7.50 hrs, 20% work from 7.51 to 9.00 hrs and the rest work 9.01 to more hrs. What is the median work hrs of work?4. An analysis of monthly wages paid to the workers of two firm A and B belonging to the same industry gives the following results:Firm A FirmB No. of workers 500 600Average daily wage Rs. 186.00 Rs. 175.00Variance of distribution of wages 81 100

(i) Which firm, A or B, has a larger wage bill?[8](ii) In which firm, A or B, is there greater variability in individual wages? [7]5. Determine whether the following functions are increasing or decreasing on given intervals:

y = 3x3 + 6x2 + 4x – 56. Each student in a class of 40 plays at least one indoor game chess, carom and scrabble. 18 play chess, 20 play scrabble and 27 play carom. 7 play chess and scrabble, 12 play scrabble and carom and 4 play chess, carom and scrabble. Find the number of students who play (i) chess and carom. (ii) Chess, carom but not scrabble.7. Define(a) Events(b) Morden axioms of probability(c ) total probability formula(d) baye’s theorem8. Find feasible region of given LPP Maximize Z=10x1+15x2s.t.. 2x1+x2≤2, 2x1+4x2≤56 -x1+x2≤5 and x1, x2≥0 9. Solve the following LPP by using simplex methodMin Z =x1-2x2-3x3s.t. -2x1+x2+3x3 =2, 2x1+3x2+4x3 =1and x1, x2, x3 ≥ 010.a) A and B are two weak students of mathematics and their chances of solving a problem in mathematics correctly are and respectively. If the probability of their making acommon error is and they obtain the same answer, find the probability that their answer is correct. b) Data on readership of a certain magazine show that the proportion of ‘male readers under is ’ and ‘over is ’. If the proportion of readers under is Find probability that a randomly selected male subscriber is under 35 year of age .11.The probability of X, Y and Z becoming managers are 4/9, 2/9 and 1/3 respectively. The probability that the Bonus Scheme will be introduced if X, Y and Z becomes managers are 3/10, 1/2 and 4/5 respectively.[Baye’s ]

(a) What is the probability that Bonus Scheme will be introduced. (b) If the bonus scheme has been introduced , what is the probability that the manager appointed was Y.12. The contents of urns I,II and III are as follows: 1 white,2 black and 3 red balls 2 white,1 black and 1 red balls and 4 white,5 black and 3 red balls. One urn is chosen at random and two balls drawn from it(a) They happen to be black and white. (b) They happen to be black and red.

13. Use the simplex method to solve the following LPPMax Z=3x1+5x2+4x3s.t. 2x1+3x2 ≤ 8, 2x2+5x3 ≤ 103x1+2x2+4x3 ≤ 15 and x1, x2, x3 ≥014. A company makes two kinds of leather belt. Belt A is high quality belt and B belt is of lower quality. The respective profits are 4 rs and 3 rs per belt. Each of type A requires twice as much time as a belt of type B, and if all belts were of type B, the company could make 1000 per day. The supply of leather is sufficient for only 800 per day (both A and B). Belt A requires a fancy buckle and only 400 per day are available. There are only 700 buckles a day available for belt B. Formulate the standard form of LPP.15.Solve the following LPP by using two-phase simplex methodMin Z =x1-2x2-3x3s.t. -2x1+x2+3x3 =2, 2x1+3x2+4x3 =1 and x1, x2, x3 ≥ 016.The contents of urns I,II and III are as follows: 1 white,2 black and 3 red balls 2 white,1 black and 1 red balls and4 white,5 black and 3 red balls. One urn is chosen at random and two balls drawn from it(a) They happen to be black and white. (b)What is the probability that they come from urns I and II.17.a)A can hit a target 4 times in 7 shots, B 3 times in 5 shots and C three times in 5 shots. All of them fire one shot each simultaneously at the target. What is the probability that (i) 2 shots hit (ii) At least two shots hit ?b) The probability that a student A solves a mathematics problem is 2/5 and the probability that a student B solves the problem is 2/3. What is the probability that (a) the problem is not solved (b) the problem is solved (c) both A and B solve the problem.18.a) In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?b) From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done?19.If (0,1)(0,-1)(0,0)(1,1)(-1,-1) are the critical points of the function , find the extrema and the saddle points.20.Examine the consistency and if consistent find whether they have unique solution or not and hence solve. x + 2y - z = 10 x - y - 2z = -2 2x + y - 3z = 821.The profits of 50 firms in thousand rupees is given below :

28 35 61 29 36 48 57 67 69 5048 40 47 42 41 37 51 62 63 33

31 32 35 40 51 54 56 38 37 6037 46 42 38 61 59 58 44 39 5738 44 45 45 47 38 44 47 47 64a) Arrange the above data into classes of interval 5 startingfrom 25.b) Find the relative frequency, frequency density and morethan and less than cumulative frequencies of each class.c) Draw the ogives and find the median profit

22.Test Score and Sales Data of Salesmen are given in the following table. Salesmen A B C D E F G H I J Test Score 50 80 60 70 90 60 80 50 70 90( X ) Sales 3•5 7•0 5•0 6•0 5•0 4•0 6•0 4•0 5•54•0 ( 000 Rs.)

a) Calculate the regression coefficient of Y on X.b) From above table calculate the regression line ofY on X.c) From above table calculate the regression line ofX on Y.23.Find the rank correlation from the following table :Candidates 1 2 3 4 5 6 7 8 9 10Marks by I 58 62 45 30 90 72 65 45 50 55Marks by II 65 60 55 45 65 80 50 52 62 70

24.A survey among hotels in the city reveals the figures below :No. of customers 0 — 20 20 — 40 40 — 60 60 — 80 80-100 No. of days 5 15 40 30 10Calculatei) the upper and lower quartile ii) quartile deviation

iii) co-efficient of skewness.25.a) Calculate the mean, median and mode from the following data : . Age : 20 – 25 25 – 3030 – 35 35 – 40 40 – 45 45 – 50 50 – 55Frequency : 50 70 100 180 150 120 70

b) Calculate the missing frequencies f 1 and f 2 : Class : 0 — 20 20 — 40 40 — 60 60 — 80 80 — 100 Frequency : 19 f 1 32 f 2 19 Mean = 50 N = 120.26.a) If A = { x : 1 ≤ x ≤ 3 }, B = { x : 2 ≤ x ≤ 4 }, find the sets A ≈ B, A ↔ B and A – B.b) Find the number of ways in which a mixed doubles game can be arranged from among 9 married couples if no husband and wife play in the same game.c) Find the area bounded by the curves and y = x.27. a) The equations of two sides of a triangle PQR are x + 2y + 4 = 0 and 3x – 4y + 37 = 0. Given thatPQR = 90˚ and that P is the point ( 6, – 5 ), find the coordinates of QB) Find the equation of the circle passing through the intersection of the circles – 8x – 2y + 7 = 0 and – 4x + 10y + 8 = 0 and also passing through the point ( 3, – 3 ).C) A man is known to speak truth 3 out of 4 times. He throws a dice and reports that it is a six. What is the probability that it is actually a six?28.A departmental store gives in service training to its salesmen.  Which is followed by a test.  The store management is considering whether it should terminate the services of any salesmen who does not do well in the test.  The following data gives the test scores and sales made by the nine salesmen during a certain period. You as a manager are required to analyze the given data, and determine whether there is any associations between the test scores and sales.  Subsequently discuss as to whether, the termination of services of low test scoring employees is justified? Also determine if the store management wants a minimum sales volume of Rs. 30,000. What should be the minimum test scores the shall ensure continuation of service? Further estimate

the most probable sales volume of a salesman who has scored a test score of 28.Test scores 14 19 24 21 26 22 15 20 19Sales (in 000Rs.) ‘

31 36 48 37 50 45 33 41 39

29.A) The results of 21 footballs (win, loser, draw ) are to be predicted. How many different forecasts can contain exactly 18 correct results?B) A fair coin and a fair die are thrown. Find the probabilities of (i) head on the coin and the number 6 on the die, (ii) head on the coin and even number on the die.30.Draw the histogram and cumulative frequency polygons (more than as well as less than type ) from the following table : Marks 1-10 11-20 21-30 31-40 41-50Frequency 5 13 12 10 8Also find the median from the graph.31.A) A box contains 20 tickets of identical appearance, the tickets being numbered 1, 2, 3, ............. 20. If 3 tickets are chosen at random, what is the probability that the numbers on the tickets drawn are in arithmetic progression?B) The line y = mx and the curve y = – 2x intersect at the origin O and meet again at a point A. If P is the mid-point of OA, find the equation of the locus of P as m varies.32.A) Find the equation of the straight line that passes through the interesection of the lines 3x + 4y = 17 and 4x – 2y = 8 and which is perpendicular to the line 7x + 5y = 12.B) Find the equation of the circle which.passes through the points ( 3, 4 ) and ( 3 – 6 ) and which has its centre on the straight line 2x + 3y = 3.33. A) The letters of the word "THEORY" are permuted andthen the words are arranged as in a dictionary. What is the rank of the word "THEORY" in that dictionary ?B) A motorist plans a journey and event X is arrival at his destination in less than 3 hrs. He estimates the probability of dry weather, rain or snow to be 1/3, 1/2 and 1/6 respectively. The probabilities of event X in these conditions are 3/4, 2/5 and 1/10 respectively. What is the probability that the motorist completes his journey in 3 hours ? What is the probability that if he fails to arrive in less than 3 hours, there was a fall of snow ?

34. Find the standard deviation and coefficient of variationfrom the following table given the marks of 150 students :

35. A student obtained the mean and the standard Deviation of 100 observations. as 40•1 and 5•0 respectively. It was later found that he had occupies 50 wrongly instead of the correct value 40. Find the correct mean and the correct standard deviation.36.A) If z is a function of x & y, where show that .

B) If prove that 37. Group the following data set in eight equal intervals using tally mark and then calculate frequency density and cumulative frequency. With the frequency distribution draw pie-chart, histogram, frequency polygon and ogives. Show.the required calculations in the table. Then find the median using ogives. 119 37 48 78 56 79 90 108 39 5675 48 56 89 83 96 100 50 60 7077 85 63 90 8145 49 67 65 87

102 123 38 84 55 44 67 86 101 11296 63 73 116 59 40 82 73 59 6838. A) If prove that:

b) Verify Euler’s Theorem For .

MarksNo. of Marks No. of

studentsstudents.1-10 5 51-60 2211-20 12 61-70 15 21-30 20 71-80 631-40 W t25 81-90 441-50 40 91-100 1

39.A) If then show that b) Solve by matrix Inversion method, the following system of equations:

40. A) Define trace of a matrix. If is a square matrix, then find the trace of 5A.

b) Find the adjoint of .c) Find X from the matrix equation AX=B where

& .

d) Find the matrix A if .41. A) Define homogenous function of the two variables x & y.B) If find c) Verify Euler’s Theorem for the following function D) Find where 42.A) Define singular and non-singular matrices.

B) Find the adjoint of the matrix A= . C) Find X from the equation AX=B given the system of equations:

D) If find matrix A.43. For the function show that a) . b) .

44.If show that 45. If and P.T.A) B) c) 46. Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?47. a) A multiple-choice test consists of 8 questions with 3 answer to each question (of which only one is correct). A student answers each question by rolling a balanced die and checking the first answer if he gets 1 or 2, the second answer if he gets 3 or 4 and the third answer if he gets 5 and 6. To get distinction, the student must secure at least 75% correct answer, if there is no negative marking, what is the probability that the student secure a distinction?b) The probability of a man hitting a target is .¼ (i) If he fires 7 times what is the probability of his hitting the target at least twice? (ii) How many times must he fire so that the probability of his hitting the target at least once is greater than 2/3?48.(a) Define the normal distribution function. Define mean and variance for normal distribution.(b) In a sample of 1000 cases, the mean of a certain test is 14 and standard deviation is 2.5. Assuming the distribution to be normal, find

(i) How many students score between 12 to 15? (ii) How many scores above 18? (iii) How many score below 8?[P(0≤Z≤0.8)= 0.2881, P(0≤Z≤0.4)= 0.1554, P(0≤Z≤1.6)=0.4452, P(0≤Z≤2.4)=0.4918].(c) The local authorities in a certain city install 10,000 electric lamps in the streets of the city. If these lamps have an average life of 1000 burnings hours with a standard deviation of 200 hours, assuming normality, what number of lamps might be expected to fail (i) In the first 800 burning hours? (ii) between 800 and 1200 burning hours? After what period of burning hours would you expect that (a) 10% of the lamps would fail? (b) 10% of the lamps would be still burning?[P(0<Z<1) = 0.3413, P(0<Z<1.28) = 0.40].49. If 10% of the pens manufactured by the company are defective, find the probability that a box of 12 pens contain.i) Exactly two defective pens.ii) At least two defective pens.iii) No Defective pens.50.A department in a works has 10 machine which may need adjustment from time to time during the day. Three of these machine are old, each having a probability of 1/11 of needing adjustment during the day, and 7 are new, having corresponding probabilities of 1/21. Assuming that no machine needs adjustment twice on the same day, determine the probabilities the probabilities that on a particular day. i) Just 2 old and no new machines need adjustment.ii) If just 2 machines need adjustment, they are of the same type.