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 M

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 M

1 (c)
A continuous random variable has probability density function f(x)=6(x-x

^{2}) 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 \[A=\begin{bmatrix} 8 & -6 & 2\\ -6& 7 &-4 \\ 2&-4 & 3 \end{bmatrix} \] is diagonalisable. Find the transforming matrix and the diagonal matrix.

8 M

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

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

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 M

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 M

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 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 \[ \iint_s \bar{N}\cdot \bar{F}ds \ where \ \bar{F}= 4x\widehat{i} + 3y\widehat{j}-2z\widehat{k} \] 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 | 83 45 | 57 68 |

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 \[ \int_c \bar{F}.d\bar{r} \ where \ \bar{F}=x^2i+xyj \] 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.

Is the difference between sample means significant.

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

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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