### Mechanical 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)**

Using Green's theorem evaluate $$ \int_c (xy+y^2)dx+x^2dy $$ where c is the closed curve of the region bounded by y=x and y=x^{2}

(5 marks)
**1 (b)** Use Cayley-Hamilton theorem to find A^{5}-4A^{4}-7A ^{ 3 }+11A^{ 2}-A-10 I in terms of A where $$ A= \begin{bmatrix} 1 & 4\\2 &3 \end{bmatrix} $$(5 marks)
**1 (c)** A continuous random variable has probability density function f(x)=6(x-x^{2}) 0≤x≤1. Find mean and variance.(5 marks)
**1 (d)** A random sample of 900 items is found to have a mean of 65.3cm. Can it be regarded as a sample from a large population whose mean is 66.2cm and standard deviation is 5cm at 5% level of significance.(5 marks)
**2 (a)** Calculate the value of rank correlation coefficient from the following data regarding marks of 6 students in statistics and accountancy in a test.

Marks in Statistics: | 40 | 42 | 45 | 35 | 36 | 39 |

Marks in Accountancy: |
46 | 43 | 44 | 39 | 40 | 43 |

(6 marks)
**2 (b)** If 10% of bolts produced by a machine are detective. Find the probability that out of 5 bolts selected at random at most one will be defective.(6 marks)
**2 (c)** Show that the matrix $$A=\begin{bmatrix} 8 & -6 & 2\\ -6& 7 &-4 \\ 2&-4 & 3 \end{bmatrix} $$ is diagonalisable. Find the transforming matrix and the diagonal matrix.(8 marks)
**3 (a)** In a laboratory experiment two samples gave the following results.

Sample | size | mean |
sum of square of deviations from the mean |

1 2 | 10 13 | 15 14 | 90 108 |

Test the equality of sample at 5% level of significance.(6 marks)
**3 (b)** Find the relative maximum or minimum of the function $$ z=x^2_1+x^2_2+x^2_3-6x_1 -10x_2-14x_3+130 $$(6 marks)
**3 (c)** Prove that $$ \bar{F}= (y^2 \cos x +z^3 )i + (2y\sin x -4)j+(3xz^2+2)k $$ is a conservative field. Find the scalar potential for F and the work done in moving an object and this field from (0, 1, -1) to (π/2, -1, 2).(8 marks)
**4 (a)** The weights of 4000 students are found to be normally distributed with mean 50kgs. And standard deviation 5kg. Find the probability that a student selected at random will have weight (i) less than 45kgs. (ii) between 45 and 60 kgs.(6 marks)
**4 (b)** Use Gauss's Divergence theorem to evaluate $$ \iint_s \bar{N}\cdot \bar{F}ds \ where \ \bar{F}= 4x\widehat{i} + 3y\widehat{j}-2z\widehat{k} $$ and s is the surface bounded by x=0, y=0, z=0 and 2x+2y+z=4(6 marks)
**4 (c)** Based on the following data, can you say that there is no relation between smoking and literacy.

| Smokers | Nonsmokers |

Literates Illiterates | 83 45 | 57 68 |

(8 marks)
**5 (a)** A random variable X follows a Poisson distribution with variance 3 calculate p(x=2) and p(x≥4).(6 marks)
**5 (b)** Use Stroke's theorem to evaluate $$ \int_c \bar{F}.d\bar{r} \ where \ \bar{F}=x^2i+xyj $$ and c is the boundary of the rectangle x=0, y=0, x=a, y=b(6 marks)
**5 (c)** Find the equations of the two lines of regression and hence find correlation coefficient from the following data.

x | 65 | 66 | 67 | 67 | 68 | 69 | 70 | 72 |

y | 67 | 68 | 65 | 68 | 72 | 72 | 69 | 71 |

.(8 marks)
**6 (a)** Two independent samples of sizes 8 and 7 gave the following results.

Sample 1: | 19 | 17 | 15 | 21 | 16 | 18 | 16 | 14 |

Sample 2: | 15 | 14 | 15 | 19 | 15 | 18 | 16 | |

Is the difference between sample means significant.(6 marks)
**6 (b)** $$ If \ A=\begin{bmatrix}2 &3 \\-3 &-7 \end{bmatrix} \ find \ A^{50} $$(6 marks)
**6 (c)** Use the Kuhn-Trucker Conditions to solve the following N.L.P.P $$ \begin {align*} Maximise \ z =&2x_1^2 -7x_2^2+12x_1x_2 \\ Subject \ to \ & 2x_1 +5x_2 \le 98 \\ & x_1x_2\ge 0 \end{align*} $$

(8 marks)