Registration No: Total Number of Pages: 02 B.Tech BSCM1205 3rd Semester Back Examination 2017-18 MATHEMATICS-III BRANCH: AEIE, AERO, AUTO, BIOTECH, CHEM, CIVIL, CSE, ECE, EEE, EIE, ELECTRICAL, ETC, FASHION, FAT, IEE, IT, MANUTECH, MECH, METTA, MINING, MME, PE, PLASTIC, TEXTILE Time: 3 Hours Max Marks: 70 Q.CODE: B790 Answer Question No.1 which is compulsory and any five from the rest. Th e figures in the right hand margin indicate marks. Q1 a) b) c) d) e) f) g) h) Answer the following questions: Define Harmonic function and conjugate harmonic ? Define analytic function? Find the residue of f(z) = = 0? (2 x 10) Find the order of the pole of f(z) = =0? Let f(z) has zeros of order m and g(z) has zeros of order n , then what is the zeros of the fg(z) ? Find the partial differential equations by eliminating arbitrary function of = + ( + )? Write down Laplace equation in two dimension. ′ Find the complementary function of ( − ) =0;= , ′= ? i) Write only the complete integral of the partial differential equations − sin ; = , = ? j) Find the Radius of convergence of ∑∞ ( Q2 a) b) Find an analytic function whose real part is u(x, y) = 2xy + 2x ? Evaluate ∫ ( ) ; : ⃓ ⃓ =1 Q3 a) Solve the partial differential equations x( ) = , = ? b) Consider the wave equation = −∞ < ( , 0) = 1 then find the value of u( , ) ? a) ′ Solve + ′ −2 =0? Find the solutions of the one dimensional heat equation with u(0, t) = u(2 , )=0 , t>0 and u(x, 0) = ? The distribution function of Q4 b) Q5 a) b) Q6 a) b) ( − )p− ( + (5) (5) )q= ( < ∞ with u(x , 0) = sin ( ) ( ) + (5) , (5) (5) = ; 0<x<2 : ⃓ ⃓ =2? (5) (5) (5) ) ) ( + ? )! Using Cauchy integral formula find the value of ∫ Write down the singular point of f(z) = ( = ) What is Cauchy-Riemann equation and check whether f(z) = ( − ) + (2 ) satisfy Cauchy Riemann equation or not? Write down the Maclaurian series of f(z) = ? (5) (5)
Q7 Evaluate the real integral ∫ Q8 a) b) c) Write short answer on any TWO : Solve (D+ ′ − 1)(D+2 ′ − 2) z = 0 Solve the linear differential equation − Using Residue theorem find f(z)= ? d) Evaluate ∫ ; : ⃓ ⃓ =2 ? (10) ? (5 x 2) = ( − 2 )?