## Applied Mathematics 4 - May 2014

### 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)**

Find a,b,c if $$ \bar{F}=(axy+bz^{3})\bar{i}+(3x^{2}-cz)\bar{j}+(3xz^{2}-y)\bar{k}\ $$is irrational.

(5 marks)**1(b)**

Find $$A^{5}-4A^{4}-7A^{3}+11A^{2}-A=10 I $$ in terms of A using Cayley-Hamilton theorem for $$A=\begin{bmatrix} 1 &4 \\ 2& 3 \end{bmatrix}\ $$

(5 marks)**1(c)**

A continuous random variable x has the p.d.f defined by f(x)=A+Bx, 0≤ x ≤1 if the mean of the distribution is $\dfrac{1}{3}$,Find A and B.

(5 marks)**1(d)**A sample of 50 pieces of certain type of string was tested.he mean breaking strength turned out to be 14.5 pounds.Test whether the sample is from a batch of a string having a mean breaking strength of 15.6 pounds and S.D of 2.2 pounds.(5 marks)

**2(a)**Obtain the rank correlation coefficient from the following data:

X | 10 | 12 | 18 | 18 | 15 | 40 |

Y | 12 | 18 | 25 | 25 | 50 | 25 |

**2(b)**The marks of 1000students of university are found to be normally distributed with mean 70 & SD 5.Estimate the number of student whose marks will be

between 60 & 75 (ii)more than 75.(6 marks)

**2(c)**

Show that the matrix $$A=\begin{bmatrix} 8 & -6 & 2\\ -6& 7 &-4 \\ 2&-4 & 3 \end{bmatrix}\ $$Find the diagonal form and transforming matrix.

(8 marks)**3(a)**A certain injection administered to 12 patients resultant in the following changes of 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)**

Optimize $Z=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-6x_{1}-8x_{2}-10x_{3}$

(6 marks)**3(c)**Verify Green's theorem in the plane for

$$ \displaystyle \oint (x^{2}-y)dx+(2y^{2}+x)dy$$ around the boundary of the region defined by $$ y =x^{2}$$ and y=4. (8 marks)

**4(a)**A car hire firm has two cars which it hire out day by day.The number of demands for a car each day is distributed as Poission variates with mean 1.5. Calculate the proportion of days on which (i)neither car is used (ii)some demand is refused.(6 marks)

**4(b)**Evaluate $$ \bar{F}=(2x-y+z))i+(x+y-z^{2})j+(3x-2y+4z)k$$ and S is the surface of the cylinder $$ x^{2}+y^{2}=4$$ bounded by the plane z=9 and open at the other end. (6 marks)

**4(c)**Table below shows the performances of students in Mathematics and physics.test the hypothesis that the performance in mathematics is independent of performance in physics.

Grades in Physics |
Grade in Maths | ||

High | Medium | Low | |

High | 56 | 71 | 12 |

Medium | 47 | 163 | 38 |

Low | 14 | 42 | 81 |

**5(a)**The ratio of the probability of 3 successes in 5 independent trials to the probability of 2 successes In 5 independent trials is 1/4.What is the probability of 4 successes in 6 independent trials?(6 marks)

**5(b)**Evaluate$$ \displaystyle \iint \bar{F}.\bar{ds} \ where \ \bar{F}=4xi-2y^{2}j+z^{2}k$$ and S is the region bounded by $$ y^{2}=4,x=1,z=0,z=3.$$(6 marks)

**5(c)**Find (i)the lines of regression (ii)coefficient of correlation for the following data.

X | 65,66,67,67,68,69,70,72 |

Y | 67.68,65,66,72,72,69,71 |

**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 | 5,6,8,1,12,4,3,9,6,10gms |

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

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

**6(b)**

If $A=\begin{bmatrix} -1 & 4\\\\ 2& 1 \end{bmatrix}$then prove that 3tan A=Atan3.

(6 marks)**6(c)**

Using the Kuhn-Tucker condition solve the following N.L.P.P.

Maximize$ Z=x_{1}^{2}-x_{2}^{2}-x_{3}^{2}+4x_{1}+6x_{2}$

Subject to $x_{1}+x_{2}\leq 2$

$2x_{1}+3x_{2} \leq12$

$x_{1},x_{2} \geq 0.$