1 (a)
Evaluate the line integral \( \int^{1+i}_{0} (x^2 iy)dz \) along the path y=x.

5 M

1 (b)
State Cayley-Hamilton theorem & verify the same for \( A= \begin{bmatrix} 1 &3 \\2 &2 \end{bmatrix} \)

5 M

1 (c)
The probability density function of a random variable x is

Find i) k ii) mean iii) variance

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 M

1 (d)
Find all the basic solutions to the following problem.

Maximum

z=x

Subject to

x

2x

and x

Maximum

z=x

_{1}+x_{2}+3x_{3}Subject to

x

_{1}+2x_{2}+3x_{3}=42x

_{1}+3x_{2}+5x_{3}=7and x

_{1}, x_{2}, x_{3}≥0.
5 M

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 M

2 (b)
Evaluate \( \oint_c \dfrac {dz}{z^3 (z+4)} \) where c is the circle |z|=2.

6 M

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.

i) less than 62 inches, ii) between 65 and 71 inches.

8 M

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 M

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 M

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 M

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 M

4 (b)
Solve the following LPP using Simplex method

Maximize z= 6x

Subject to

2x

x

x

Maximize z= 6x

_{1}-2x_{2}+ 3x_{3}Subject to

2x

_{1}-x_{2}+2x_{3}≤2x

_{1}+4x_{3}≤4x

_{1}, x_{2}, x_{3}≥ 0.
6 M

4 (c)
Expand \( f(z) = \dfrac {2}{(z-2)(z-1)} \) in the regions

i) |z| <1, ii) 1<|z|<2, iii) |z|>2.

i) |z| <1, ii) 1<|z|<2, iii) |z|>2.

8 M

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 M

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 M

5 (c)
Solve the following LPP using the Dual Simplex method.

Minimize

z=2x

Subject to

2x

3x

x

x

Minimize

z=2x

_{1}+2x_{2}+4x_{3}Subject to

2x

_{1}+ 3x_{2}+ 5x_{3 }≥ 23x

_{1}+ x_{2}+7x_{3}≤3.x

_{1}+4x_{2}+6x_{3}≤5x

_{1}, x_{2}, x_{≥0}
8 M

6 (a)
Solve the following NLPP using Kuhn-Tucker conditions

Maximum \(z=10x_1+ 4x_2 - 2x^2_1 - x^2_1 \)

subjected to 2x

Maximum \(z=10x_1+ 4x_2 - 2x^2_1 - x^2_1 \)

subjected to 2x

_{1}+x_{2}≤5; and x_{1}, x_{2≥0.}
6 M

6 (b)
In an experiment on immunization of cattle from Tuberculosis the following results were obtained.

Use X

Use X

^{2}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 M

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 M

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 μ

Test the hypothesis that μ

_{1}=μ_{2}at 5% level of significance.
4 M

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