## Applied Mathematics 4 - May 2016

### 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)** Prove that F = (Z^{2} + 3y + 2x)i+(3x+2y+z)j+(y+2zx)k is irrational and find it's Scalar potential.(5 marks)
**1(b)** If $A=\begin{bmatrix} -2 & 2 & -3\\\\ 2 & 1 & -6\\\\ -1 & -2 & 0 \end{bmatrix} $

Find the characteristics roots of A and A^{2} + I.(5 marks)
**1(c)** The probability of a man hitting the target is 1/4. How many times must he fire so that the probability of his hitting the target atleast once is greater than 2/3?(5 marks)
**1(d)** A random sample of 400 members is found to have mean of 4.45 cms. Can it be reasonably regarded as a sample from large population whose mean is 5 cms and whose is 4 cms.(5 marks)
**2(a)** From the following data calculate the coefficient of rank correlation between X and Y.

X: | 32 | 55 | 49 | 60 | 43 | 37 | 43 | 49 | 10 | 20 |

Y: | 40 | 30 | 70 | 20 | 30 | 50 | 72 | 60 | 45 | 25 |

**2(b)**The daily consumption of electric power (in million kwh) is a random variable x with p.d.f $$\begin {align*} f(x)&=kx\ e^{-x/5}\ \text{for} \ x>0\\ &=0\ \text{for}\ x\le 0\\ \end{align*}$$

Find the value of k, the mathematical expectation and the probability that on a given day, the electric consumption is more than the expected value.(6 marks)

**2(c)**Show that the given matrix $ \begin{bmatrix} 2 & 2 & 1\\\\ 1 & 3 & 1\\\\ 1 & 2 & 2 \end{bmatrix}$ is diagonalizable. Find the transforming matrix and diagonal form.(8 marks)

**3(a)**A certain injection administered to 12 patients resulted in the following Changes in blood pressure.

5, 2, 8, -1, 3, 0, 6, -2, 1, 5, 0, 4

Can it be concluded that the injection will be in general accompanied by an increase in blood pressure.(6 marks)

**3(b)**Using the Langrangian multiplier method solve the following N.L.P.P $$\text{Optimize}\ z=2x^2_1+2x^2_2+2x^2_3-24x_1-8x_2-12x_3+196\\\text{subject to}\ x_1+x_2+x_3=11$$(6 marks)

**3(c)**Verify Green's theorem in the plane for $\oint \dfrac{1}{y}dx+\dfrac{1}{x}dy $ where C is the boundary of the region defined by y=1, x=4, $ y=\sqrt{x}$(8 marks)

**4(a)**A women with m keys with her wants to open the door of her house by trying the keys independently and randomly one by one. Find the mean And the variance of the no of trials required to open the door if unsuccessful Keys are kept aside.(6 marks)

**4(b)**Use Gauss theorem to evaluate $ \displaystyle \iint_s \overline{F}.d\overline{s}\ \text{where}\ \overline{F}=xi-3y^2j+zk$ over the surface of the cylinder x

^{2}+ y

^{2}= 16 between z=0 and z=5(6 marks)

**4(c)**Twele dice were thrown 4096 times and the number of appearance of 6 each time was noted.

NO. of Success: | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

Frequency: | 447 | 1145 | 1181 | 796 | 380 | 115 | 32 |

Test whether the dice are unbiased.(8 marks)

**5(a)**Marks obtained by students in an examination follow a normal distribution if 30% of students got below 35 marks and 10% got above 60 marks. Find the Mean and the standard deviation.(6 marks)

**5(b)**Using Stokes theorem find the work done in moving a particle once around the perimeter og the triangle ABC cut off by the plane 3x+2y+z=6 on the co-ordinate axes under the force F=(x+y)i+(2x-z)j+(y+z)k(6 marks)

**5(c)**The equations of two regression lines are 3x+2y=26 and 6x+y=31

Find (i) the means of x and y (ii) co efficient of correlation between x and y (iii) σ

_{y}if σ

_{x}=3(8 marks)

**6(a)**A group of 10 rats fed on diet A and another group of 8 rats fed on different diet B, recorded the following increase in weight.

Diet A: | 5 | 6 | 8 | 1 | 12 | 4 | 3 | 9 | 6 | 10 gms |

Diet B: | 2 | 3 | 6 | 8 | 1 | 10 | 2 | 8 gms |

Find if the variances are significantly different?(6 marks)

**6(b)**If $A=\begin{bmatrix} 1 & 4\\\\ 2 & 3 \end{bmatrix} $. Prove that $A^{50}-5A^{49}=\begin{bmatrix} 4 & -4\\\\ -2 & 2 \end{bmatrix} $(6 marks)

**6(c)**Using Kuhn-Tucker conditions solve following N.L.P.P

Maximize z=2x

_{1}+3x

_{2}-x

_{1}

^{2}-x

_{2}

^{2}

Subject to x

_{1}+x

_{2}≤ 1

2x

_{1}+3x

_{2}≤ 6 x

_{1}, x

_{2}≥ 0(8 marks)