THEORY EXAMINATION (SEM-IV) 2016-17 ENGINEERING MATHEMATICS-III

Subject: Engineering Mathematics – III (EAS401)

Exam Type: Theory
Semester: IV (4th Semester)
Session: 2016–17
Time: 3 Hours
Maximum Marks: 100

Section A – Short Answer Questions (10 × 2 = 20 Marks)

This section tests quick conceptual and computational skills from complex analysis, numerical methods, and statistical mathematics.

Topics Covered:

Complex Integration: Evaluate ∫Cezz+1dz\int_C \frac{e^z}{z + 1} dz∫C​z+1ez​dz, where CCC is the circle ∣z∣=2|z| = 2∣z∣=2.

Analytic Function: Proof that f(z)=sinh⁡zf(z) = \sinh zf(z)=sinhz is analytic.

Modulation Theorem: Proof of F{f(x)cos⁡(ax)}=12[F(s+a)+F(s−a)]F\{f(x)\cos(ax)\} = \frac{1}{2}[F(s+a) + F(s-a)]F{f(x)cos(ax)}=21​[F(s+a)+F(s−a)].

Z-transform Equation: Solve recurrence yk+2+yk+1−2yk=0y_{k+2} + y_{k+1} - 2y_k = 0yk+2​+yk+1​−2yk​=0, given y0=4,y1=0y_0 = 4, y_1 = 0y0​=4,y1​=0.

Skewness Definition: Measure of symmetry/asymmetry in data distribution.

Regression Line Equation: Write normal equation of y=a+bxy = a + \frac{b}{x}y=a+xb​.

Finite Difference Relation: Prove Δ+∇=Δ∇−∇ΔΔ+∇\Delta + \nabla = \frac{\Delta\nabla - \nabla\Delta}{\Delta + \nabla}Δ+∇=Δ+∇Δ∇−∇Δ​.

Newton-Raphson Method: Find first approximation of 171/317^{1/3}171/3.

Picard’s Method: Find 3rd approximation of solution for dydx=1+xy\frac{dy}{dx} = 1 + xydxdy​=1+xy, y(0)=0y(0) = 0y(0)=0.

Euler’s Method: Find y(0.1)y(0.1)y(0.1) given dydx=log⁡(x+y)\frac{dy}{dx} = \log(x + y)dxdy​=log(x+y), y(0)=1.0y(0) = 1.0y(0)=1.0.

Section B – Descriptive Questions (Any 5 × 10 = 50 Marks)

This section evaluates understanding of advanced mathematical techniques applied to engineering problems.

Key Questions:

Continuity & Cauchy-Riemann Equations:
Show that f(z)=x3(1+i)−y3(1−i)x2+y2f(z) = \frac{x^3(1+i) - y^3(1-i)}{x^2 + y^2}f(z)=x2+y2x3(1+i)−y3(1−i)​ is continuous and satisfies C-R equations, but f′(0)f'(0)f′(0) does not exist.

Cauchy Integral Formula Application:
Evaluate ∫Cez(z+1)4dz\int_C \frac{e^z}{(z+1)^4}dz∫C​(z+1)4ez​dz where C:∣z∣=3C: |z| = 3C:∣z∣=3.

Fourier Transform:
Find Fourier cosine transform of 11+x2\frac{1}{1+x^2}1+x21​ and sine transform of x1+x2\frac{x}{1+x^2}1+x2x​.

Multiple Regression Analysis:
Calculate regression of X1X_1X1​ on X2,X3X_2, X_3X2​,X3​ for:

Regula-Falsi Method:
Find root of xex=cos⁡xxe^x = \cos xxex=cosx correct to four decimals.

Crout’s Method:
Solve linear equations:

Runge-Kutta Method:
Find y(1.1)y(1.1)y(1.1) using 4th order R-K method for dydx=y2+xy\frac{dy}{dx} = y^2 + xydxdy​=y2+xy, y(1)=1.0y(1) = 1.0y(1)=1.0, h=0.05h = 0.05h=0.05.

Section C – Analytical / Long Questions (Any 2 × 15 = 30 Marks)

Question 3

Show f(z)=∣xy∣f(z) = \sqrt{|xy|}f(z)=∣xy∣​ satisfies C-R equations but not regular at origin.

Evaluate ∫02πdθa+bsin⁡θ\int_0^{2\pi} \frac{d\theta}{a + b\sin\theta}∫02π​a+bsinθdθ​, for a>∣b∣a > |b|a>∣b∣.

Solve yk+2−4yk+1+3yk=5ky_{k+2} - 4y_{k+1} + 3y_k = 5^kyk+2​−4yk+1​+3yk​=5k by Z-transform.

Question 4

Using Convolution theorem, find Z−1[z2(z−1)(z−3)]Z^{-1}\left[\frac{z^2}{(z-1)(z-3)}\right]Z−1[(z−1)(z−3)z2​].

Show angle θ\thetaθ between regression lines satisfies

1. tan⁡θ=1−r2rσxσyσx2+σy2\tan \theta = \frac{1 - r^2}{r} \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}tanθ=r1−r2​σx2​+σy2​σx​σy​​

and interpret for r=0,±1r = 0, \pm1r=0,±1.

Apply Chi-square (χ²) test to determine association between income level and schooling type:

Question 5

Find missing value in:
| x | 2 | 3 | 4 | 5 | 6 |
| f(x) | 45 | 49.2 | 54.1 | ? | 67.4 |

Fit a cubic polynomial to:
| x | -2 | -1 | 2 | 3 |
| y(x) | -12 | -8 | 3 | 5 |

Find approximate value of log⁡e(5)\log_e(5)loge​(5) by Simpson’s 1/3 rule:

1. ∫05dx4x+5\int_0^5 \frac{dx}{4x + 5}∫05​4x+5dx​

correct to four decimal places.

Major Topics Covered

Complex analysis (analytic functions, Cauchy’s theorem, Z-transforms)

Fourier and Laplace transforms

Regression, correlation, and χ² test

Numerical methods: Newton-Raphson, Runge-Kutta, Crout’s method, Simpson’s rule

Curve fitting and interpolation

Purpose of the Paper

This exam tests both conceptual clarity and computational skill in applied mathematics.
It bridges theory and engineering applications — preparing students to solve real-world problems using mathematical models, transforms, and numerical methods.

2x + 3y + z = 9   x + 2y + 3z = 6   3x + y + 2z = 8