Bihar Engineering University, Patna
B.Tech 2nd Semester Examination, 2026
Engineering Mathematics II
Time: 03 HoursCode: 100202Full Marks: 70
Instructions:
- The marks are indicated in the right-hand margin.
- There are NINE questions in this paper.
- Attempt FIVE questions in all.
- Question No. 1 is compulsory.
Q.1 Choose the correct option/answer the following (Any seven question only):[2×7=14]
-
The harmonic conjugate of the function u(x, y) = x² − y² is
-
If a function is analytic inside and on a simple closed contour C, then ∮C f(z) dz
-
The residue of the function f(z) = ez/z at z = 0 is
-
The auxiliary equation of y'' − 5y' + 6y = 0 is
-
The standard form of a Cauchy–Euler differential equation is
-
The series Σn=1∞ 1/np converges if
-
Cauchy's Condensation Test can be applied to
-
The Laplace Transform of 2t is
-
The inverse Laplace Transform of 1/(s² + 9) is
-
The half-range cosine series is used when the function is extended as a
Q.2 (a) Prove that the function f(z) = z³ is analytic.[7]
(b) Show that the function u(x, y) = ex(x cos y − y sin y) is harmonic. Hence, find its harmonic conjugate v(x, y) and obtain the corresponding analytic function f(z).[7]
Q.3 (a) Evaluate the contour integral ∮C (2z+1)/[(z−1)(z−2)] dz, where C is the circle |z − 1| = 2, using the Cauchy Integral Formula.[7]
(b) Find the Laurent series expansion of the function f(z) = 1/[z(z−1)] valid in the region 0 < |z| < 1. Hence, determine the nature of the singularity at z = 0.[7]
Q.4 (a) Evaluate the definite integral ∫02π dθ/(5 + 4 cos θ), using the method of residues.[7]
(b) Solve the differential equation y'' − 4y' + 4y = xe2x.[7]
Q.5 (a) Solve the Cauchy–Euler equation x²y'' + 5xy' + 4y = x³.[7]
(b) Using the Method of Variation of Parameters, solve the differential equation y'' + y = sec x.[7]
Q.6 (a) Test the convergence of Σn=1∞ 2n/n! using D'Alembert's Ratio Test.[7]
(b) Apply Raabe's Test to determine the nature of Σn=1∞ (2n)!/[4n(n!)²].[7]
Q.7 (a) Find the inverse Laplace Transform of (s) = (2s+5)/[(s−1)(s²+4)].[7]
(b) Solve the initial value problem using the Laplace Transform y'' + 5y' + 6y = e−2t, where y(0) = 1, y'(0) = 1.[7]
Q.8 (a) Find the Fourier series of the function f(x) = x, where −π < x < π. Hence, deduce the value of 1 − 1/3 + 1/5 − 1/7 + − ······[7]
(b) Obtain the half-range sine series of f(x) = x(π − x), where 0 < x < π.[7]
Q.9 Write short note on any two of the following:[7×2=14]