BTECH (SEM III) THEORY EXAMINATION 2023-24
TIME: 3 Hours | MAX MARKS: 70
This examination evaluates a student’s understanding of Partial Differential Equations, Fourier Methods, Probability Theory, Curve Fitting, Statistical Moments, Hypothesis Testing, Quality Control Charts, and Wave/Heat Equations.
The paper is designed to assess both theoretical knowledge and analytical problem-solving ability required in engineering mathematics.
The question paper is divided into three major sections, progressing from basic conceptual recall to high-level analytical and numerical computations.
SECTION A – Short Conceptual Questions (14 Marks)
7 Questions × 2 Marks
This section checks foundational understanding of PDE formation, classification, probability computations, hypothesis concepts, and control chart limits.
Key Concepts Covered:
1. Formation of PDE
Deriving the partial differential equation from the relation:
z=f(2x−y)z = f(2x - y)z=f(2x−y)
Tests understanding of eliminating arbitrary functions.
2. Classification of PDE
Students classify the given second-order PDE as elliptic, parabolic, or hyperbolic using discriminant analysis.
3. Normal Equations for Curve Fitting
Writing normal equations for fitting
y=c0+c1xy = c_0 + \frac{c_1}{x}y=c0+xc1
Checks understanding of least squares method.
4. Expected Mean of a Probability Distribution
Based on given discrete values of xxx and p(x)p(x)p(x).
5. Probability Density Function Condition
Finding value of parameter ppp using
∫04px dx=1\int_0^4 p x \, dx = 1∫04pxdx=1
6. Null Hypothesis
Basic theoretical definition used in hypothesis testing.
7. Control Limits of R-Chart
Understanding statistical quality control (SQC) charts for range.
This section checks clear conceptual recall and precise formula usage.
SECTION B – Analytical / Problem-Solving Questions (21 Marks)
Attempt any three (7 Marks each)
This section includes solving PDEs, computing moments, handling Poisson relations, and testing significance through t-tests.
Topics Covered:
1. Solving First-Order PDE
Solving
y2p+(x2+y)q=x+yzy^2 p + (x^2 + y)q = x + yzy2p+(x2+y)q=x+yz
using standard first-order PDE methods.
2. Laplace PDE via Separation of Variables
Solving
∂2u∂x2+∂2u∂y2=0\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0∂x2∂2u+∂y2∂2u=0
with boundary conditions involving sine series — classical Laplace equation solution.
3. Moments → Skewness and Kurtosis
Using four moments about 5 to classify the distribution as symmetric, skewed, leptokurtic, or platykurtic.
4. Poisson Distribution with Given Relationship
Solving
P(X=2)=9P(X=4)+90P(X=6)P(X=2)=9P(X=4)+90P(X=6)P(X=2)=9P(X=4)+90P(X=6)
to determine the standard deviation (√λ).
5. t-Test for Difference in Means
Comparing lifetimes of motors from two batches using sample means and standard deviations (given t₀.₀₅ = 2.13).
This section evaluates mathematical reasoning, application of theorems, and computation accuracy.
SECTION C – Long Descriptive / Application Questions (35 Marks)
Students must attempt one question from each of the five sub-sections.
These questions require in-depth understanding and detailed steps.
C1 – Advanced PDE Solutions (7 Marks)
Option A – First-Order PDE
Solve
px+qy=pqpx + qy = pqpx+qy=pq
using Charpit or Lagrange method.
Option B – Operator Method for PDE
Solve
(D2−6+D′2)z=cos(2x+y)(D^2 - 6 + D'^2)z = \cos(2x + y)(D2−6+D′2)z=cos(2x+y)
using differential operator techniques.
C2 – Wave Equation / Heat Equation (7 Marks)
Option A – Vibrating String (Wave Equation)
A stretched string of length 2 with fixed ends is at initial position
y=3sin(πx2)y = 3\sin\left(\frac{\pi x}{2}\right)y=3sin(2πx)
Find displacement y(x,t)y(x,t)y(x,t) after release from rest.
Option B – One-Dimensional Heat Equation
Solve
ut=uxx, x>0, t>0u_t = u_{xx},\; x>0,\; t>0ut=uxx,x>0,t>0
subject to piecewise initial temperature distribution and boundary condition u(0,t)=0u(0,t)=0u(0,t)=0.
C3 – Regression & Curve Fitting (7 Marks)
Option A – Exponential Curve Fitting
Fit a curve of form
y=aebxy = ae^{bx}y=aebx
using least squares method.
Option B – Regression Lines & Correlation
Given two regression equations:
2x+2y−5=0,2x+3y−8=0,σx=122x + 2y - 5 = 0,\quad 2x + 3y - 8 = 0,\quad \sigma_x =122x+2y−5=0,2x+3y−8=0,σx=12
Find
(a) Means of x and y
(b) Variance of y
(c) Correlation coefficient rrr
C4 – Probability & Normal Distribution (7 Marks)
Option A – Binomial Probability
Out of 320 families with 5 children, compute number of families with:
(i) 2 boys & 3 girls
(ii) At least one boy
Option B – Normal Distribution Applications
Given mean = 140 and SD = 10 for 1000 workers, estimate number of workers with wages:
(i) 140–144
(ii) <126
(iii) >160
using area under normal curve.
C5 – Chi-Square Test / Control Charts (7 Marks)
Option A – Chi-Square Test for Independence
Using classification of workers (male/female × skilled/unskilled), test whether nature of work is independent of gender.
Option B – np-Chart Construction
Given defectives in 10 samples of size 100 each, construct an np-chart and interpret whether the process is in control.
