STUPIDSID
1(a)
Find the Eigenvalues and eigenvectors of the matrix.
A=\( \begin{bmatrix} 2 & 2& 0\\ 0 & 2& 1\\ 0& 0& 2 \end{bmatrix} \)/
A=\( \begin{bmatrix} 2 & 2& 0\\ 0 & 2& 1\\ 0& 0& 2 \end{bmatrix} \)/
5 M
1(b)
Evaluate the line integral \[\int_{0}^{l+i}\left ( x^2+iy \right )\] dz along the path y=x
5 M
1(c)
Find k and then E (x) for the p.d.f.
\( f(x)=\left\{\begin{matrix} k(x-x^2),0\leq x\leq 1,k> 0& \\ 0, & otherwise \end{matrix}\right. \)/
\( f(x)=\left\{\begin{matrix} k(x-x^2),0\leq x\leq 1,k> 0& \\ 0, & otherwise \end{matrix}\right. \)/
5 M
1(d)
Calculate Karl person's coefficient of correlation from the following data.
| x | 100 | 200 | 300 | 400 | 500 |
| y | 30 | 40 | 50 | 60 | 70 |
5 M
2(a)
Show that the matrix \( A=\begin{bmatrix}
2 & -2& 3\\
1& 1& 1\\
1& 3& -1
\end{bmatrix} \)/ is non-derogatory.
6 M
2(b)
Evaluate \[\int \frac{e^2^z}{\left ( z+1 \right )^4}\] dz where C is the circle |z-1|=3
6 M
2(c)
If x is a normal variate with mean 10 and standard deviation 4 find
i) P(|x-14|<1)
ii) P(5≤x≤18)
iii) P(x≤12)
i) P(|x-14|<1)
ii) P(5≤x≤18)
iii) P(x≤12)
8 M
3(a)
Find the relative maximum of minimum (if any) of the \[Z=X_{1}^{2}+X_{2}^{2}+X_{3}^{2}-4X_1-8X_2-12X_3+100\]
6 M
3(b)
If x is Binomial distributed with E(x)=2 and V(x)=4/3,find the probability distribution of x.
6 M
3(c)
If \( A=\begin{bmatrix}
2& 1\\
1 & 2
\end{bmatrix} \)/,
find A50.
find A50.
8 M
4(a)
Solve the following L.P.P by simplex method Minimize
z=3x1+2x2 Subject to 3x1+2x2≤18
0≤x1≤4
0≤x2≤6
x1,x2≥0.
z=3x1+2x2 Subject to 3x1+2x2≤18
0≤x1≤4
0≤x2≤6
x1,x2≥0.
6 M
4(b)
The average of marks scored by 32 boys is 72 with statndard deviation 8 while that of 36 girls is 70 with standard deviation 6. Test 1% level significance whether the boys perform better than the girls.
6 M
4(c)
Find Laurent's series which represents the function
\[f(z)=\frac{2}{\left ( Z-1 \right )\left ( z-2 \right )}\] When
i) |z| <1,
ii) 1<|z|<2
iii) |z|>2
\[f(z)=\frac{2}{\left ( Z-1 \right )\left ( z-2 \right )}\] When
i) |z| <1,
ii) 1<|z|<2
iii) |z|>2
8 M
5(a)
Evaluate \[\int \frac{Z^2}c_{\left ( z-1 \right )^2\left (z+1 \right )}\] dz where C is|z| =2 using residue theorem
6 M
5(b)
The regression lines of a sample are x+6y=6 and 3x+2y=10 Find
i) Sample means
\[\bar{x} \ \text{and}\ \bar{y}\]
ii) Correlation coefficient between x ad y. Also estimate y When x=12
i) Sample means
\[\bar{x} \ \text{and}\ \bar{y}\]
ii) Correlation coefficient between x ad y. Also estimate y When x=12
6 M
5(c)
A die was thrown 132 times and the following frequencies were observed
Using χ2-test examine the hypothesis that the die is unbiased.
| No.obtained | 1 | 2 | 3 | 4 | 5 | 6 | Total |
| Frequency | 15 | 20 | 25 | 15 | 29 | 28 | 132 |
8 M
6(a)
Evaluate \[\int ^\infty _\\-\infty\frac{x^2+x+2}{x^4+10x^2+9}\] dx using contour integration.
6 M
6(b)
If a random variable x follows Poisson distribution such that P(x-1)=2(x=2) Find the mean the variance of the distribution Also find P(x=3).
6 M
6(c)
Use Penalty method to solve the following L.P.P. Minimize
z=2x,sub>1+3x2
x1+x2≥5
x1+2x2≥6 x1, x2≥0.
z=2x,sub>1+3x2
x1+x2≥5
x1+2x2≥6 x1, x2≥0.
8 M
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