# Deemed University 2009 A.M.I.E.T.E Electronics

Tuesday, 30 April 2013 02:40Web

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**(**0

**C)****(**¼

**D)****i.**Eliminating function from we find the partial differential formula

**(**

**A)****(**

**B)****(**

**C)****(**

**D)**j. If , then value of div F is

**(**

**A)****(**

**B)****(**

**C)****(**

**D)**ans any 5 ques. out of 8 ques..

every ques. carries 16 marks.

**Q.2**

**a.**A tightly stretched string of length l with fixed ends is initially in an equilibrium position. It is set vibrating by giving every point a velocity . obtain the Displacement (8)

**b.**An infinitely long plane uniform plate is bounded by 2 parallel edges and an end at right angles to them. The breadth of the plate is This end is maintained at a temperature at all points and other edges are at zero temperatur

**e.**Determine the temperature at any point of the plate in the steady stat

**e.**(8)

**Q.3**

**a.**A random variable X have the density function . If and zero otherwise obtain

**(**its distribution function.

**i)****(**probabilities and . (8)

**i****i)****b.**In a production of iron rods, let the diameter X be normally distributed

**(**What percentage of defectives can we expect? If the tolerance limits are set at in.

**i)****(**How should we set the tolerance limits to allow for 4% defectives?

**i****i)**presume

(8)

**Q.4**

**a.**A continuous random variable X has a pdf obtain K. Also obtain mean and variance of this random variabl

**e.**

**(**

**8)****b.**Prove that where and (8)

**Q.5**

**a.**obtain the directional derivative of in the direction of a unit vector which makes an angle of with x-axis. (8)

**b.**Use the Divergence theorem to evaluate where and S is the boundary of the region bounded by the paraboloid and the plane z = 4y. (8)

**Q.6**

**a.**Show that is independent of the path of integration from (1,1,

**2)**to (2,3,

**4)**and hence evaluate it. (8)

**b.**Verify the Greens theorem for and C is the square with vertices at (0,0), (8)

**Q.7**

**a.**Show that the function

(8)

satisfies Cauchy-Riemann equations at z = 0, but does not exist.

**b.**obtain the image of the region under the mapping

**(**

**8)****Q.8**

**a.**find the Taylor series expansion of about z = 0. Also obtain its radius of convergenc

**e.**(8)

**b.**Evaluate the integral a > 0 by using contour integration. (8)

**Q.9**

**a.**Using the method of separation of variables, solve where u(x,0)=6e–3

**x.**(8)

**b.**Evaluate by using Stokess theorem where c is the boundary the rectangle . (8)

Earning: Approval pending. |