Computer Engineering (Semester 4)
TOTAL MARKS: 80
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks.
1 (a) Evaluate the line integral $ \int^{1+i}_{0} (x^2 iy)dz $ along the path y=x.(5 marks)
1 (b) State Cayley-Hamilton theorem & verify the same for $ A= \begin{bmatrix} 1 &3 \\\\2 &2 \end{bmatrix} $(5 marks)
1 (c) The probability density function of a random variable x is
|
x |
-2 |
-1 |
0 |
1 |
2 |
3 |
|
p(x) |
0.1 |
k |
0.2 |
2k |
0.3 |
K |
Find i) k ii) mean iii) variance(5 marks)
1 (d) Find all the basic solutions to the following problem.
Maximum
z=x1+x2+3x3
Subject to
x1+2x2+3x3=4
2x1+3x2+5x3=7
and x1, x2, x3 ≥0.(5 marks)
2 (a) Find the Eigen values and the Eigen vectors of the matrix $ \begin{bmatrix}
4 &6 &6 \\\\1
&3 &2 \\\\-1
&-5 &-2
\end{bmatrix} $(6 marks)
2 (b) Evaluate $ \oint_c \dfrac {dz}{z^3 (z+4)} $ where c is the circle |z|=2.(6 marks)
2 (c) If the heights of 500 students is normally distributed with mean 68 inches and standard deviation of 4 inches, estimate the number of students having heights
i) less than 62 inches, ii) between 65 and 71 inches.(8 marks)
3 (a) Calculate the coefficient of correlation from the following data:
|
x |
30 |
33 |
25 |
10 |
33 |
75 |
40 |
85 |
90 |
95 |
65 |
55 |
|
y |
68 |
65 |
80 |
85 |
70 |
30 |
55 |
18 |
15 |
10 |
35 |
45 |
(6 marks)
3 (b) In sampling a large number of parts manufactured by a machine, the mean number of defectives in a sample of 20 is 2. Out of 100 such samples, how many would you expect to contain 3 defectives i) using the Binomial distribution, ii) Poisson distribution.(6 marks)
3 (c) Show that the matrix $ \begin{bmatrix}
-9 &4 &4 \\\\-8
&3 &4 \\\\-16
&8 &7
\end{bmatrix} $ is diagonalizable. Find the transforming matrix and the diagonal matrix.(8 marks)
4 (a) Fit a Poisson distribution to the following data:
|
x |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
f |
56 |
156 |
132 |
92 |
37 |
22 |
4 |
0 |
1 |
(6 marks)
4 (b) Solve the following LPP using Simplex method
Maximize z= 6x1-2x2 + 3x3
Subject to
2x1-x2+2x3 ≤2
x1+4x3 ≤4
x1, x2, x3 ≥ 0.(6 marks)
4 (c) Expand $ f(z) = \dfrac {2}{(z-2)(z-1)} $ in the regions
i) |z| <1, ii) 1<|z|<2, iii) |z|>2.(8 marks)
5 (a) Evaluate using Cauchy's Residue theorem $ \oint_c \dfrac {1-2z}{z(z-1)(z-2)}dz $ where is |z|=1.5.(8 marks)
5 (b) The average of marks scored by 32 boys is 72 with standard deviation 8 while that of 36 girls is 70 with standard 6. Test at 1% level of significance whether the boys perform better than the girls.(6 marks)
5 (c) Solve the following LPP using the Dual Simplex method.
Minimize
z=2x1+2x2+4x3
Subject to
2x1 + 3x2 + 5x3 ≥ 2
3x1+ x2+7x3≤3.
x1+4x2+6x3≤5
x1, x2, x≥0(8 marks)
6 (a) Solve the following NLPP using Kuhn-Tucker conditions
Maximum $z=10x_1+ 4x_2 - 2x^2_1 - x^2_1 $
subjected to 2x1+x2≤5; and x1, x2≥0.(6 marks)
6 (b) In an experiment on immunization of cattle from Tuberculosis the following results were obtained.
Use X2 Test to determine the efficacy and vaccine in preventing tuberculosis.
|
|
Affected |
Not Affected |
Total |
|
Inoculated |
267 |
27 |
294 |
|
Not Inoculated |
757 |
155 |
912 |
|
Total |
1024 |
182 |
1206 |
(6 marks)
6 (c) (i) The regression lines of a sample are x+6y=6 and 3x+2y=10 find (a) sample means $ \overline x $ and $ \overline y $ (b) coefficient of correlation between x and y.(4 marks)
6 (c) (ii) If two independent random samples of sizes 15 & have respectively the mean and population standard deviations as $ \overline x_1 = 980, \ \overline x_2=1012: \ \sigma_1 = 75, \ \sigma_2 = 80 $
Test the hypothesis that μ1=μ2 at 5% level of significance.(4 marks)
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