Electronics Engineering (Semester 4)
Total marks: 80
Total time: 3 Hours
INSTRUCTIONS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Draw neat diagrams wherever necessary.
1.a.
Find the coefficient of correlation from the following data : N=10, $\sum \ X=225$
$\sum \ Y=189$, $\sum \ (X-22)^2=85$, $\sum \ (Y-19)^2=25$
(5 marks)
00
1.b.
Evaluate $\int_clogz \ dz$. Where c is |z|=1
(5 marks)
00
1.c.
Find the projection of u= (3, 1, 3) along and perpendicular to v=(4, -2, 2)
(5 marks)
00
1.d.
Find an eigen values of (i) Adj (A) (ii) 24$A^{-1}$+2A-I where,
$ \begin{equation}
\\A =
\begin{bmatrix}
1 & 2 & 3 & -2 \\
0 & 2 & 4 & 6 \\
0 & 0 & 4 & -5 \\
0 & 0 & 0 & 6 \\
\end{bmatrix}
\end{equation}$
(5 marks)
00
2.a.
Find the extremal of $\int_0^1(y^{"2}+x^2-y^2)dx$
(6 marks)
00
2.b.
Use Gram-schmidt process to transform the basis {$u_1$,$u_2$,$u_3$} in to orthonormal bases where
$u_1$= (1,1,1), $u_2$= (0,1,1), $u_3$= (0,0,1)
(6 marks)
00
2.c.
Show that the matrix
$ \begin{equation}
\\ A =
\begin{bmatrix}
-9 & 4 & 4 \\
-8 & 3 & 4 \\
-16 & 8 & 7 \\
\end{bmatrix}
\end{equation}$
Also find diagonal and transforming matrix.
(8 marks)
00
3.a.
If X is a normal variable with mean 10 and standard deviation 4, Find
(i) P[|X-4| $\lt$ 1]
(ii) P[5 $\lt$ X$\lt$ 18]
(iii) P[X$\lt$12]
(6 marks)
00
3.b.
Seven dice are thrown 729 times. How many times do you expect at least four dice to show 3 or 5.
(6 marks)
00
3.c.
Using Rayleigh-Ritz method find solution for the extremal of the functional
$\int_0^1(2xy-y^{'2}-y^2)dx$ given y(0)=0 and y(1)=0
(8 marks)
00
4.a.
For the 50 students in the class mean of X is 62.4 and 16Var(X)=9Var(Y).
Regression line of X on Y is 3Y-5X+180=0 Find
(i) Mean of Y
(ii) Correlation r between X and Y
(iii) Regression line of Y on X
(6 marks)
00
4.b.
Evaluate $\int_c\frac{(z+1)}{(z^3-2z^2)} \ dz$ where c is (i) |z|=1 (ii) |z-2-i|=2 (iii) |z-1-2i|=2
(6 marks)
00
4.c.
Check whether the set of all pairs of real number of the form (1,x) with operations (1,y)+(1,x)=(1,y+x) and k(1,y)=(1,ky) is a vector space.
(8 marks)
00
5.a
Using cauchy residue theorm evaluate $\int_0^{\infty}\frac{1}{(x^2+1)(x^2+9)} \ dx$
(6 marks)
00
5.b.
If
$ \begin{equation}
\\ A =
\begin{bmatrix}
1 & 0 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0 \\
\end{bmatrix}
\end{equation}$
find $A^{50}$
(6 marks)
00
5.c.
Find M.G.F. of poisson distribution. Hence find its mean and variance.
(8 marks)
00
6.a
Is the matrix derogatory? Justify your answer Where
$ \begin{equation}
\\ A =
\begin{bmatrix}
-2 & 0 & 1 \\
1 & 1 & 0 \\
0 & 0 & -2 \\
\end{bmatrix}
\end{equation}$
(6 marks)
00
6.b.
A random variable X has the following p.d.f.
f(x)=$kx^2e^{-x}$ for x$\gt$0. and f(x)=0 otherwise. Find
(i) k (ii) mean (iii) variance
(iv) M.G.F. (v) c.d.f. of X (vi) P[0$\lt$X$\lt$1]
(6 marks)
00
6.c.
Find all possible Laurent series of f(z)= $\frac{(z^2-1)}{(z^2+5z+6)}$
(8 marks)
00