Code: math2Information Technology2nd Semester5 Examination Papers

Engineering Mathematics - II

Official Bihar Engineering University examination question papers, marks distribution, and questions transcript. Available exam sessions: 2026, 2025, 2021, 2019.

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Official Exam Paper TranscriptB.Tech 2nd Semester Examination, 2026
Bihar Engineering University, Patna
B.Tech 2nd Semester Examination, 2026

Engineering Mathematics II

Time: 03 HoursCode: 100202Full Marks: 70

Instructions:

  1. The marks are indicated in the right-hand margin.
  2. There are NINE questions in this paper.
  3. Attempt FIVE questions in all.
  4. Question No. 1 is compulsory.
Q.1 Choose the correct option/answer the following (Any seven question only):[2×7=14]

  1. The harmonic conjugate of the function u(x, y) = x² − y² is

    1. 2xy
    2. −2xy
    3. x² + y²
    4. xy

  2. If a function is analytic inside and on a simple closed contour C, then ∮C f(z) dz

    1. 2πi
    2. 0
    3. ∞
    4. Depends on the contour

  3. The residue of the function f(z) = ez/z at z = 0 is

    1. 0
    2. e
    3. 1
    4. 2πi

  4. The auxiliary equation of y'' − 5y' + 6y = 0 is

    1. m² + 5m + 6 = 0
    2. m² − 5m + 6 = 0
    3. m² − 6m + 5 = 0
    4. m² + 6m − 5 = 0

  5. The standard form of a Cauchy–Euler differential equation is

    1. y'' + py' + qy = r(x)
    2. x²y'' + axy' + by = 0
    3. y'' + x²y = 0
    4. y' + xy = 0

  6. The series Σn=1∞ 1/np converges if

    1. p < 1
    2. p = 1
    3. p > 1
    4. p ≤ 1

  7. Cauchy's Condensation Test can be applied to

    1. Any alternating series
    2. Any finite series
    3. Positive, decreasing series
    4. Oscillating series

  8. The Laplace Transform of 2t is

    1. ln 2
    2. 1/(s − 2)
    3. 1/(s − ln 2)
    4. 1/s

  9. The inverse Laplace Transform of 1/(s² + 9) is

    1. sin 3t
    2. (1/3) sin 3t
    3. 3 sin 3t
    4. cos 3t

  10. The half-range cosine series is used when the function is extended as a

    1. Periodic function
    2. Odd function
    3. Even function
    4. Constant function
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]
  1. Cauchy–Riemann Equation
  2. Laurent's Series
  3. Convolution Theorem
  4. Parseval's theorem

Related Official Syllabus

Engineering Mathematics-II

100202

Official BEU curriculum units & topics for this examination subject.

Unit 1.0: Complex Analysis - I

Functions of complex variable • Limit • Continuity • Differentiability • Analytic function • Cauchy-Riemann Equations in Cartesian and polar form • Harmonic function and harmonic conjugate

Unit 2.0: Complex Analysis - II

Line Integral • Contour integrals • Cauchy theorem • Cauchy's Integral formula (without proof) • Taylors series • Zero of analytic functions • Singularities • Laurent's series • Residue • Cauchy residue theorem (without Proof) and its applications

Unit 3.0: Ordinary Differential Equations

Linear differential equations of nth Order with constant coefficients • Solution of Homogeneous and Non-Homogeneous Equations • Equations with variable coefficients • Cauchy-Euler Equations • Method of Variation of Parameters

Unit 4.0: Sequence and Series

Introduction of Sequence and Series • Nature of series Tests of convergence of Series • Comparison test • D'Alembert ratio test • Cauchy's Root test • Raabe's test • Logarithmic test • Cauchy's condensation test

Unit 5.0: Laplace Transform

Laplace Transform • Existence theorem • Properties of Laplace Transform • Laplace Transform of Periodic functions • Inverse Laplace Transform • Convolution theorem • Application of Laplace Transform to solve Ordinary differential equations

View Complete 2nd Semester Syllabus

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