 Applied Mathematics-IV-Engineering-Mumbai University-Dec2019 - Grad Plus

# Applied Mathematics-IV-Engineering-Mumbai University-Dec2019

## Semester: 4

[Total Time: 3 Hours]
[Total Marks: 80 M]
N.B.
1) Question No. 1 is Compulsory.
2) Answer any THREE questions from Q.2 to Q.6.
3) Figures to the right indicate full marks.

1) a) If A=\begin{bmatrix}2&4\\0&3\end{bmatrix} then find the eigen values of 6A-1 + A3 + 2I [05M]

b) Determine whether the given vectors u= (-4,6,-10,1),v= (2,1,-2,9) are orthogonal with respect to the Euclidean inner product [05M]

c) The probability density function of a random variable x is zero except at x =0, I, 2 and
p(0) = 3α3 , p(1) =4 α – 10α3
p(2) =5 α – 1. Find α [05M]

d) Evaluate \phi\frac{z+6}{z^2-4}dz\;where\;c\;is\;(i)\;\left|z\right|-1\;(ii)\;\left|z-2\right|-1 [05M]

2) a) Using Rayleigh-Ritz method, find an appropriate solution for the extremal of the functional. [06M]
I=\int\left[2xy-y^2-y^{'2}\right]dx given y(0)=y(1)=0</p> <p>b) Using Cauchy’s Residue theorem evaluate [latex]\int_0^{2x}\frac{d\theta}{5+4\cos\theta} [06M]

c)A random variable X has the probability distribution given below:

3 a) Compute A9 – 6A8 + 10A7 – 3A6 + A + I where A=\begin{bmatrix}1&2&3\\-1&3&1\\1&0&2\end{bmatrix} [06M]

b) Verify Cauchy –Schwartz inequality for the vectors u=(-4,2,1) &v=(8,-4,-2) [06M]

c) Obtain Taylor’s or Laurent’s series expansion of the function f(z)=\frac1{z^2-3z+2} when
(i) |z| < 1 (ii)1 < |z |< 2 [08M]

4) a)Obtain the equation of the line of regression of Y on X for the following data and estimate Y when X = 73 [06M]

b)Show that the functional \int_{x_1}^{x_2}\left[y^2+x^2y^'\right]dx assumes extreme values on the straight line y = x [06M]

c)Let R3 have the Euclidean inner product. Use the Gram-Schmidt process to transform the basis vectors u1=(1,0,0),u2=(3,7,-2),u3=(0,4,1) in to an orthonormal basis [08M]

5) a)Evaluate \int_c\frac1z\cos zdz where c is the ellipse 9x2+4y2-1 [06M]

b) Seven dice are thrown 729 times. How many times do you expectants least four 10 dice to show three or five? [06M]

c) Show that the matrix A=\begin{bmatrix}-9&4&4\-8&3&4\-16&8&7\end{bmatrix} is diagnosable. Find the diagonal form D and the diagonalizing matrix M. [08M]

6) a) A continuous random variable X has the p.d.f. defined by f (x )=A +B x ,0 <x <1 . If the mean of the distribution is
1/3 find A and B [06M]

b) Find eA, IF A=\begin{bmatrix}\frac32&\frac12\\frac12&\frac32\end{bmatrix} [06M]

c) Evaluate \int_{-\infty}^\infty\frac{x^2dx}{\left(x^2+a^2\right)\left(x^2+b^2\right)}\;\left(a>0,\;b>0\right) [08M]

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