Engineering Mathematics 4 - Dec 2013

Electronics & Communication (Semester 4)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks. 1 (a) Employ Taylor's series method to obtain the values of y at x=0.1 and 0.2 for the differential equation dy/dx=2y+3e'.y(0)=0 considering upto fourth degree term.(6 marks) 1 (b) Determine the value of y when x=0.1, given that y(0)=1 and y=x²+y^{2 using modified Euler's formula. Take h=0.05."}(7 marks) 1 (c) Apply Adams-Bashforth method to solve the equation dy/dx=x²(1+y), given y(1)=1, y(1,1)=1.233, y(1,2)=1.548, y(1,3)=1.979. Evaluate y(1,4).(7 marks) 2 (a) $$ \dfrac {dy}{dx}=1+zx, \ \dfrac {dz}{dx}=-xy, \ y(0)=0, \ z(0)=1 \ at \ x=0.3 $$ by taking h=0.3. Applying Runge-Kutta method of fourth order.(6 marks) 2 (b) Applying Picard's method to compute y(1,1) from the second approximation to the slolution of the differential equation y+y² y'=x³. Given that y(1)=1(7 marks) 2 (c) Using the Mitni's method obtain an approximate solution at the point x=0.8 of the problem $$ \dfrac {d^2y}{dx^2}=1-2y \dfrac {dy}{dx} $$ give that y(0)=0, y'(0)=0, y(0.2)=0.02, y'(0.2)=0.1996, y(0.4)=0.0795, y'(0.4)=0.3937, y(0.6)=0.1762, y'(0.6)=0.5689.(7 marks) 3 (a) Derive Cauchy-Riemann equations in Cartesian form.(6 marks) 3 (b) Give u-v(x-y)(x²+4xy+y²) find the analytic function f(z)=u+iv.(7 marks) 3 (c) If f(z)=u+iv is an analytic function then prove that $$ \left [] \dfrac {\partial }{\partial x} |f(z)| \right ]^2 + \left [ \dfrac {\partial }{\partial y} |f(z)| \right ]^2 =|f'(z)|^2 $$(7 marks) 4 (a) Find the image of the straight lines parallel to coordinate axes in z-plane under the transformation w=z².(6 marks) 4 (b) Find the bilinear transformation which maps the points z=1, i, -1 onto the points w=0, 1, ?.(7 marks) 4 (c) Evaluate $$ \int_c \dfrac {e^{2z}} {(z+1)(z+2)} $$ where c is the circle |z|=3.(7 marks) 5 (a) Find the solution of the Laplace equation in cylindrical system leading to Besseis differential equation.(6 marks) 5 (b) If ? and ? are two distinct roots of J_n(x)=0, then prove that $$ \int^1_0 x J_n (\alpha x)J_m (\beta x)dx=0 , \ m\ne n $$(7 marks) 5 (c) Express f(x)=x⁴- 2x¹+ 3x²- 4x+5 in terms of legendre polynomial.(7 marks) 6 (a) A committee consists of 9 students, 2 from first year, 3 from second year and 4 from third year, 3 students are to be removed at random. What is the probability that (i) 3 students belongs to different class (ii) 2 belongs to the same class and third belongs to different class. (iii) All the 3 belongs to the same class.(6 marks) 6 (b) State and prove Baye's theorem.(7 marks) 6 (c) The chance that a doctor will diagnose a disease correctly is 60%. The chance that a patient will die after correct diagnose is 40% and the chance of death after wrong diagnose is 70%. If a patient dies, what is the chance that discase was correctly diagnosed.(7 marks) 7 (a) The probability distribution of finite random variable x is given by the following table: