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

Paper Title: Engineering Mathematics–III (NAS401)
Course: B.Tech (Semester IV)
Max Marks: 100
Duration: 3 Hours

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

This section contains 10 compulsory questions testing fundamental concepts, such as:

Complex Analysis:
Evaluate ∫Cezz+1dz\int_C \frac{e^z}{z+1}dz∫C​z+1ez​dz, analyticity of sinh⁡z\sinh zsinhz, and Cauchy-Riemann conditions.

Fourier & Z-Transforms:
Proofs of modulation theorem and solving simple Z-transform difference equations.

Statistics:
Meaning of skewness and normal equations for linear regression.

Numerical Methods:
Newton-Raphson, Picard, and Euler’s methods for iterative and differential solutions.

Section B – Long Answer Questions (5 × 10 = 50 Marks)

Students must attempt any five.
Key topics include:

Analytic Function Proofs — showing continuity and failure of differentiability.

Cauchy Integral Formula Applications.

Fourier Transforms — sine and cosine forms.

Multiple Regression Problems — using given datasets.

Numerical Solutions — Regula-Falsi, Crout’s Method, Runge-Kutta of 4th order.

Example:
Solve

{2x+3y+z=9x+2y+3z=63x+y+2z=8\begin{cases} 2x + 3y + z = 9 \\ x + 2y + 3z = 6 \\ 3x + y + 2z = 8 \end{cases}⎩⎨⎧​2x+3y+z=9x+2y+3z=63x+y+2z=8​

using Crout’s method.

Section C – Advanced Problems (2 × 15 = 30 Marks)

Choose any two.
These questions require deep conceptual understanding and longer calculations.

Complex Functions: Testing regularity, evaluating integrals like ∫02πdθa+bsin⁡θ\int_0^{2\pi} \frac{d\theta}{a+b\sin\theta}∫02π​a+bsinθdθ​.

Z-Transform Applications: Solving difference equations such as
yk+2−4yk+1+3yk=5k.y_{k+2} - 4y_{k+1} + 3y_k = 5^k.yk+2​−4yk+1​+3yk​=5k.

Statistics:
Derivation involving regression line angles and performing a chi-square test for association between variables.

Interpolation & Approximation:
Missing data estimation, cubic polynomial fitting, and Simpson’s rule for numerical integration.

Key Topics Covered

In Summary

This exam thoroughly tests analytical, computational, and numerical skills vital for engineers. Students should focus on:

Derivation-based proofs (analyticity, transforms)

Problem-solving accuracy (Z-transform, Runge-Kutta)

Conceptual clarity in applied statistics and numerical computation.