MU Applied Mathematics - 4 - December 2014 Exam Question Paper

MU Civil Engineering (Semester 4)
Applied Mathematics - 4
December 2014

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

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=x²

5 M

1 (b) Use Cayley-Hamilton theorem to find A⁵-4A⁴-7A ³+11A²-A-10 I in terms of A where

A = [\begin{matrix} 1 & 4 \\ 2 & 3 \end{matrix}]

$A= \begin{bmatrix} 1 & 4\\2 &3 \end{bmatrix}$

5 M

1 (c) A continuous random variable has probability density function f(x)=6(x-x²) 0≤x≤1. Find mean and variance.

5 M

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 M

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 M

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 M

2 (c) Show that the matrix

A = [\begin{matrix} 8 & - 6 & 2 \\ - 6 & 7 & - 4 \\ 2 & - 4 & 3 \end{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 M

3 (a) In a laboratory experiment two samples gave the following results.

Sample

size

mean

sum of square of deviations

from the mean

108

Test the equality of sample at 5% level of significance.

6 M

3 (b) Find the relative maximum or minimum of the function

z = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} - 6 x_{1} - 10 x_{2} - 14 x_{3} + 130

$z=x^2_1+x^2_2+x^2_3-6x_1 -10x_2-14x_3+130$

6 M

3 (c) Prove that

\bar{F} = (y^{2} \cos x + z^{3}) i + (2 y \sin x - 4) j + (3 x z^{2} + 2) k

$\bar{F}= (y^2 \cos x +z^3 )i + (2y\sin x -4)j+(3xz^2+2)k$ 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 M

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 M

4 (b) Use Gauss's Divergence theorem to evaluate

\iint_{s} \bar{N} \cdot \bar{F} d s w h e r e \bar{F} = 4 x \hat{i} + 3 y \hat{j} - 2 z \hat{k}

$\iint_s \bar{N}\cdot \bar{F}ds \ where \ \bar{F}= 4x\widehat{i} + 3y\widehat{j}-2z\widehat{k}$ and s is the surface bounded by x=0, y=0, z=0 and 2x+2y+z=4

6 M

4 (c) Based on the following data, can you say that there is no relation between smoking and literacy.

Smokers

Nonsmokers

Literates

Illiterates

8 M

5 (a) A random variable X follows a Poisson distribution with variance 3 calculate p(x=2) and p(x≥4).

6 M

5 (b) Use Stroke's theorem to evaluate

\int_{c} \bar{F} . d \bar{r} w h e r e \bar{F} = x^{2} i + x y j

$\int_c \bar{F}.d\bar{r} \ where \ \bar{F}=x^2i+xyj$ and c is the boundary of the rectangle x=0, y=0, x=a, y=b

6 M

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 M

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 M

6 (b)

I f A = [\begin{matrix} 2 & 3 \\ - 3 & - 7 \end{matrix}] f i n d A^{50}

$If \ A=\begin{bmatrix}2 &3 \\-3 &-7 \end{bmatrix} \ find \ A^{50}$

6 M

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 M

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