About this previous year paper
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∫Cz+1ezdz, analyticity of sinhz\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+bsinθ\int_0^{2\pi} \frac{d\theta}{a+b\sin\theta}∫02πa+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
| Unit | Topic | Core Areas |
|---|---|---|
| I | Complex Variables | Analytic functions, Cauchy-Riemann equations, Cauchy integral theorem |
| II | Transforms | Fourier, Z-Transforms, Modulation Theorem |
| III | Statistics | Correlation, regression, chi-square test |
| IV | Numerical Techniques | Newton-Raphson, Runge-Kutta, Crout’s, Regula-Falsi |
| V | Interpolation & Integration | Newton’s and Simpson’s methods |
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.