This is the B.Tech Semester III examination paper for MATHEMATICS-IV (KAS302).
The paper carries 100 marks and comprehensively evaluates a student’s understanding of:
Partial Differential Equations (PDEs)
Heat & Wave equations
Regression, moments & distributions
Probability theory
Statistical tests (t-test, χ²-test, F-test)
Statistical Quality Control (SQC)
The question paper is divided into three sections – A, B, and C, moving from basic theory to analytical questions to full-length numerical/statistical problems.
SECTION A – Short Questions (20 Marks)
This section contains 10 brief questions (2 marks each) covering:
Solving basic linear PDEs
Forming PDEs by eliminating arbitrary constants
Radio wave equations
Classification of PDEs (elliptic, parabolic, hyperbolic)
Mode calculation from mean & median (using empirical relation)
Correlation coefficient from regression lines
Probability problem: selecting 2 children from 4 persons
PDF normalization & probability evaluation
t-test definition and purpose (small samples)
Meaning of Statistical Quality Control (SQC)
These questions test quick understanding of PDEs, probability, correlation, and statistical definitions.
SECTION B – Descriptive / Medium-Length Questions (30 Marks)
Students must attempt any three out of five questions (10 marks each).
Topics include:
(a) Solving non-homogeneous PDE using operators (D, D′)
— PDE of the form (D−D′−1)(D−D′−2)z=sin(2x+3y)(D − D′ − 1)(D − D′ − 2) z = \sin(2x + 3y)(D−D′−1)(D−D′−2)z=sin(2x+3y)
(b) Heat Conduction Problem
— Temperature distribution in a laterally insulated bar whose boundary temperatures change suddenly.
(c) Central Moments from given frequency distribution
— Computation of μ₁, μ₂, μ₃, μ₄ from symmetric data.
(d) Normal Distribution Applications
— Using mean 14, SD 2.5 to find:
Students scoring between 12–15
Students scoring above 18
Students scoring below 8
Using provided normal table values.
(e) Chi-square Test for Independence
— Testing effectiveness of cattle tuberculosis vaccine using 2×2 contingency table.
This section checks analytical solving, heat equation modeling, statistical tables, and hypothesis testing.
SECTION C – Long Answer / Advanced Questions (50 Marks)
This section contains five questions (Q3–Q7) with two alternatives each, and students must attempt one from each.
Q3 – Advanced PDE Solving
Solve first-order PDE using Lagrange’s method
OR
Solve linear PDE (x2D2−4xyDD′+4Dy2+6D′)z=x3y4(x² D² − 4xy D D′ + 4Dy² + 6D′)z = x³ y⁴(x2D2−4xyDD′+4Dy2+6D′)z=x3y4
Q4 – Heat & Wave Equation Solutions
(a) Solve Laplace’s equation zxx+zyy=0z_{xx} + z_{yy} = 0zxx+zyy=0 with boundary conditions given
(b) Derive displacement of a stretched vibrating string:
u(x,t)=Asin(πxl)cos(πctl)u(x,t) = A \sin\left(\frac{\pi x}{l}\right) \cos\left(\frac{\pi c t}{l}\right)u(x,t)=Asin(lπx)cos(lπct)
Q5 – Regression & Random Variables
(a) Fit a parabolic regression curve for given x,yx, yx,y data.
OR
(b) Find MGF, mean & variance of geometric distribution:
P(X=r)=qr−1pP(X=r)=q^{r-1} pP(X=r)=qr−1p
Q6 – Binomial & Poisson Distribution Problems
(a) Fit a binomial distribution to frequency data.
(b) Poisson distribution problem: number of taxi drivers with
0 accidents
More than 3 accidents
Q7 – F-Test & NP-Chart
(a) Use F-test to check difference of population variances (sample size 8 & 10).
(b) Construct np-chart from inspection data of 10 lots, each with sample size 400, and check process control.
OVERALL SUMMARY
The MATHEMATICS-IV (KAS302) question paper gives a complete assessment of:
PDE & Engineering Mathematics
PDE formation
PDE solutions (operator method, Lagrange method)
Heat & wave equation modeling
Probability & Statistics
Measures of central tendency
Regression
Central moments
Correlation
Geometric, binomial, Poisson distributions
Advanced Statistical Testing
t-test
χ²-test
F-test
Statistical Quality Control (np-chart)
It combines theory, computation, proofs, modeling, and data analysis, preparing students for higher-level engineering mathematics.
