Mahatma Gandhi University (MGU) 2005 B.Tech Computer Science and Engineering Mathematics - Question Paper

2005 Mahatma Gandhi University B.Tech Computer Science and Engineering Mathematics november 2005 ques. paper

F 3612    (Pages : 2)

Reg. No.. Name......

B.TECH. DEGREE EXAMINATION, NOVEMBER 2005

' Fifth Semester Branches : Computer Science and Engineering/Information Technology ENGINEERING MATHEMATICSIV (RT)

(New Scheme2002 admission onwards)

Maximum : 100 Marks

Answer one full question from each module.

All questions carry equal marks.

Module I    - -

1.    (a) A repair shop attended by a single machine has an average of 4 customers an hour who bring

small appliances. The mechanic inspects them for defects and quite often can fix them right away or otherwise render a diagnosis. This takes him 6 minutes on the average. Arrivals are Poisson and service time has the exponential distribution.

(i)    Find the probability that the shop is empty.

(ii)    Find the probability of finding at least one customer in the shop.    '

(iii)    What is the average number of customers in the system ?

.    (iv) Find the average time spent, including service.

(b) A postal clerk can service a customer in 5 minutes, the service time being exponentially distributed with an average of 10 minutes during the early morning slack period and an average of 6 minutes during the afternoon peak period. Assess the (i) average que length ; and (ii) the expected waiting time in the que during the two periods.

2.    (a) Explain what are queueing problems. How does the queueing theory apply to these problems,

(b) Arrivals at a telephone booth are considered to be Poisson with an average time of 10 minutes between one arrival and the next. The length of a phone call is assumed to be exponential with mean 3 minutes :

(i)    What is the probability that a person arriving at the booth will have to wait ?

(ii)    Find the average number of unite ia the system.

Module II

3.    (a) Find the roots of xe? - cos x 0 correct to four places of decimals using Newton-Raphson

method.

(b) Using Jacobi method, solve correct to three decimal places, the system of equations : x + YJy-2z = 48, 30* - 2y + 3z = 75, 2* + 2y + I8z = 30.

4.    (a) Find the root of 6.r -13 = 0 correct to these decimal places using Homere method.

(b) Using Gauss Seidel iterative method solve correct to four decimal places the system of equations 10i - 2y + z - 12 a 0, x + 9y - z - 10 = 0, 2x -y + 11s- 20 = 0.

v* v-^,j. #*/. r"C^f ' **'

S*?-'¹'     '

^rf vt'sT'¹i

v (_a; compute y (42) and y (.8$) from the following data:

lllkyv-* - * ^; 40 50 60 70 80 t' ^:    .v : 184 204 226 250 276 304

(b) Find and atx = 2 and x = 9 5 from the ibilowing data :

x : 2 4 6 8 10    ,

y : 105 42-7 25'3 167 13 .

6.    (a) Derive Newtons divided difference interpolation formula.

(b) Using Trapezoidal rule and Simpsons | th rule find the value of f I f x - ~] dx by dividun

* (1,2) into 10^J equal intervals.    '

v .    '    Module IV

7.    (a) Solve graphically the following linear programming problem : .

Maximize Z = 60*j + 40k₂ such that 2x_r + x₂ 60, x₁ 25, *_a 35, atj, x₂ > 0.

(b) Solve the following linear programming problem by Big M method :_

Minimize Z = 4x₁ + &e₂ + 5x₃ subject to

x₁ + 2x₃ +3x12, x₁ + x₂~2x₃ >8, 2x_L + x_s + x₃ 2. 14, *₂, ig > 0.

8.    (a) Using simplex method solve the following linear programming problem :_

Maximize Z = 7.1 + x₂ + 2x₃ subject to

x₁+x₂-2x₃< 10, 4x_r+ jc₂ +1₃ <, 20, x_ltx₃, x₃Z0.

(b) Using the principle of duality, solve the linear programming problem:

Minimize Z = 4x₁ + I2x₂ + 1S*₃ such that , x_t + 3*₃ > 3, 2x₃ + 2x_s k 5, x_v x₂, a:_a 2 0.

Module V

9.    Solve the following transportation problem for mininrniTn cost :

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