(SEM III) THEORY EXAMINATION 2023-24 MATHEMATICS-IV

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∫04​pxdx=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.