[Total Time: 3 hours]
[Total Marks: 80]
N.B.: 1) Question Nal is compulsory
2) Answer any three questions from 0.2 to Q.6.
3) Use of Statistical Tables permitted
4) Figures to the right indicate full marks
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Q. 1) a) Find all the basic solutions to the following problem. (5M)
Maximise z =x_{1}+3x_{2} +3x_{3}
subject to x_{1}+ 2x_{2} + 3x_{4} =4
2x_{1} +3x_{2}+5x_{3}= 7
x_{1},x_{2},x_{3}≥0
b) Evaluate \int_c\begin{pmatrix}z-&z^2\end{pmatrix}dz where is upper half of the circle |z| = 1 (5M)
c) Ten individuals are chosen at random from a population & heights are found to be 63. 63. 64, 65, 66, 69, 69, 70, 70, 71. inches Discuss the suggestion that the height of the universe is 65 inches. (5M)
d) If A=\begin{bmatrix}2&3\\-3&-4\end{bmatrix}, find A^{100 }(5M)
Q. 2) a) Evaluate \int_c\frac{z+2}{\left(z-3\right)\left(z-4\right)}dz ,where c is the circle |z|= 1 (6M)
b) An I.Q. test was administered to_5 persons and after they were trained. The results are given below.
I | II | III | IV | V | |
I.Q. Before training | 110 | 120 | 130 | 132 | 125 |
I.Q. after training | 120 | 118 | 125 | 136 | l121 |
Test whether there is any change in I.Q. after the training program, use 1% LOS. (6M)
c) Solve the following LPP using Simplex Method (8M)
Maximise z = 4x_{1}+10x_{2}
subject to 2x_{1} +x_{2}≤ 10
2x_{1}+5x_{2} ≤20
2x_{1} +3x_{2} ≤ 18
x_{1},x_{2}≥0
Q. 3) a) Find the Eigenvalues and Eigenvectors of the following matrix. (6M)
A=\begin{bmatrix}2&2&1\\1&3&1\\1&2&2\end{bmatrix}b) If the height of 500 students is normally distributed with mean 68 inches and standard deviation 4 inches. Find the expected number of students having heights between 65 & 71 inches. (6M)
c) Obtain Taylor’s and Laurent’s expansions of f(z)=\frac{z^2-1}{z^2+5z+6} around z=0 (8M)
Q. 4) a) A machine is claimed to produce mails of men length 5 cms & standard of 0.45 cm. A random sample of 100 nails gave 5.1 as their average length. Does the performance of the machine justly the claim? Mention the level of significance you apply. (6M)
b) Using the Residue theorem, Evaluate \int_0^{2\pi}\frac{d\theta}{5+3\;\sin\;\theta} (6M)
c) i) a certain manufacturing process 5% of the tools produced turn out to be defective. Find the probability that in a sample of 40 tools at most 2 will be defective.
ii) A random variable x has the probability distribution (4+4M)
p\left(X=x_3\right)=\frac{1_3}8C_x, X= 0,1,2,3 Find the moment generating function of x
Q. 5) a) Check whether the following matrix is Derogatory of Non-Derogatory: (6M)
A=\begin{bmatrix}6&-2&2\\-2&3&-1\\2&-1&3\end{bmatrix}
b) In an industry 200 workers employed for a specific job were classified according to their performance & training received to test independence of training received & performance. The data are summarized as follows. (6M)
Performance | Good | Not good | Total |
Trained | 100 | 50 | 150 |
Untrained | 20 | 30 | 50 |
Total | 120 | 80 | 200 |
Use x^{2}-test for independence at 5% level of significance & write your conclusion
c) Use the dual simplex method to solve the following L.P.P. (8M)
Minimise z =2x_{1} +x_{2}
subject to 3x_{1} + x_{2} ≥ 3
4x_{1} +3x_{2} ≥ 6
x_{1}+2x_{2} ≤ 3
x_{1, }x_{2} ≥ 0
Q. 6) a) Show that the matrix A satisfies the Cayley-Hamilton theorem and hence find A^{-1 } (6M)
Where A=\begin{bmatrix}1&3&7\\4&2&3\\1&2&1\end{bmatrix}
b) A discrete random variable has the probability density function given below. (6M)
X = x_{1} | -2 | -1 | 0 | 1 | 2 | 3 |
P(x_{1}) | 0.2 | K | 0.1 | 2K | 0.1 | 2K |
Find K. Mean, Variance
c) Using Kuhn-Tucker conditions, solve the following NLPP (8M)
Maximise z = 2x^{2}_{1} -7x^{2}_{2}+12x_{1} x_{2}
subject to 2x_{1}+5x_{2} ≤ 98
x_{1},x_{2} ≥ 0
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