Question Paper: Applied Mathematics 4 : Question Paper Dec 2015 - Computer Engineering (Semester 4) | Mumbai University (MU)

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## Applied Mathematics 4 - Dec 2015

### 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=x

_{1}+x

_{2}+3x

_{3}

Subject to

x

_{1}+2x

_{2}+3x

_{3}=4

2x

_{1}+3x

_{2}+5x

_{3}=7

and x

_{1}, x

_{2}, x

_{3}≥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 |

**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 |

**4 (b)**Solve the following LPP using Simplex method

Maximize z= 6x

_{1}-2x

_{2}+ 3x

_{3}

Subject to

2x

_{1}-x

_{2}+2x

_{3}≤2

x

_{1}+4x

_{3}≤4

x

_{1}, x

_{2}, x

_{3}≥ 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=2x

_{1}+2x

_{2}+4x

_{3}

Subject to

2x

_{1}+ 3x

_{2}+ 5x

_{3 }≥ 2

3x

_{1}+ x

_{2}+7x

_{3}≤3.

x

_{1}+4x

_{2}+6x

_{3}≤5

x

_{1}, x

_{2}, 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 2x

_{1}+x

_{2}≤5; and x

_{1}, x

_{2≥0.}(6 marks)

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

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 (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 μ

_{1}=μ

_{2}at 5% level of significance.(4 marks)