(SEM III) THEORY EXAMINATION 2021-22 MATHEMATIVS-IV

B.Tech Mathematics 0 downloads

₹29.00

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: