Test

No

Date

Topic Covered

References

Linear Algebra

1

July 2

 2023

Matrix:

Find Rank of Matrices

Given Matrix, Find Inverse and Express High order Matrix Polynomial in terms of A

Find Inverse using ONLY ELEMENTART ROW OPERATION

Given Matrices, find Eigen Values and eigen Vectors

State and Prove Cayley Hamilton theorem

Given Matrix A, Find A(power of 300 ) using Cayley Hamilton

Given Homogeneous Equation, Find Dimension and Basis

Linear Eqn , Find Condition for ----No Solution , Unique Solution ,Infinite Solutions

Theorems and Problems from Krishna Series Book

Eigen Values, Vectors

Orthogonality

Similarity

Diagonalizable

Quadratic Form

Matrices-Krishna Series

2

July 9

 2023

Vector Spaces, Subspaces Basis, Dimensions, Nullity

Linear Transformation:

Given Linear Transformation, Find a) Rank, b) Nullity c) Range Space d) Null Space

Given Matrix Form of Linear Transformation, Find a) Rank, b) Nullity c) Range Space d) Null Space

Given Linear Transformation with Two Different Bases Sets, Find Matrix

Given Linear Transformation with Standard Basis. Find Matrix relative to New Basis Set

Any Graduation Text Book covers this Syllabus

Schaum Series Linear Algebra : Selected Problem Solving

SuccessClap Free Study material on Linear Algebra

Calculus

3

July 16

 2023

Limits: Continuity, Types of Discontinuity, Heine Continuity method, Bolzano Intermediate Value Theorem,

Uniform Continuity

Differentiability,

Indeterminants

Max/ Min Single Variable

Find Max/Min of function.

Problems to find Max/Min Area, Surface, Height

Max Min TWO Variables

Find Max/Min of function

Max/Min Multiple Variables

Lagrange Multiplier

Length of Arc

Areas: Cartesian, Polar, Loop/Asymptotes

Volumes: Cartesian, Polar , Spherical System

Surface Areas

Asymptote: General Method, Inspection method, Intersection of Curve and Asymptote

Curve Tracing

Krishna Series : 1)Differential Calculus,

 2)Integral Calculus

4

July 23

 2023

Mean Value theorem: MVT, Taylor Maclurin Expansion, Generalized MVT

Rolle Theorem

Lagrange Theorem (Also proof)

Cauchy MVT

Function Increasing/Decreasing

Jacobians

Partial Differentiation

Formation,

Eulers

Total Differentiation

Krishna Series : 1)Differential Calculus,

 2)Integral Calculus

5

July 30

 2023

Beta Gamma

Properties

Relation between Beta and Gamma

Legendre Duplication Formula and Properties

Beta Gamma Several Variables

Evaluation of Integrals

Definite Integral as Sum

Differentiation under Integral Sign

Multiple Integrals

Change of Order of Integration

Evaluation of Integrals

Krishna Series : 1)Differential Calculus,

 2)Integral Calculus

REAL ANALYSIS

6

Aug 6

 2023

Riemann Integrals

Summation of Series

Theorems :

Continuous is integrable

Bounded and Finite Set of Discontinuity is integrable

Bounded and Discontinuity point has fixed limit point is integrable

Monotonic is integrable

Integral Inequality Eqn

First Mean Value Theorem

Generalised Mean Value Theorem

Second Mean Value Theorem

Improper Integrals

Limit Test

Cauchy Test

Absolute Convergence and conditional convergence

Abel Test

Dirichlet Test

Real Analysis by Raisinghania S.Chand

7

Aug 13

 2023

Sequence

Bolzano-Weistress Theorem

Cauchy First Theorem on Limit

Cauchy second Theorem on Limit

Caseros Theorem

Cauchy sequence

Monotone Convergence Theorem

Series

Necessary Condition

Comparison Test

Comparison Test of Second Kind

D Alembert Ratio Test

Cauchy nTH Root Test

Raabe Test

Logarithmic Test

DE Morgan and Bertrand Test

Second Logarithmic Ratio Test

Kummer Test

Gauss Test

Cauchy Integral Test

Cauchy Condensation Test

Real Analysis by Raisinghania S.Chand

8

Aug 20

 2023

Uniform Convergence

Cauchy Principle

Mn Test

Weir strass Test

Abel Test

Dirichlet Test

Properties on a) Sum b) Differentiability c) Integrability

Functions of Several Variables

Limit

Continuity

Differentiability

Maxima and Minima

Real Analysis by Raisinghania S.Chand

VECTOR ANALYSIS

9

Aug 27

 2023

Directional Derivative : Max/Min ,Angle made,

Vector Identities: Proof -Curl (A XB), Div (A X ,Grad (A.B) ,Curl(curl A)

Invariance under Transformation

Green Theorem

Stoke Theorem

Divergence Theorem

Work done.

Differential Geometry: Derive Serret Frennet formula, Show curve lie in Plane, Find Curvature Vector, Problems on finding Radius of Curvature, Torsion

Krishna Series Vector Analysis

Differential Geometry by Paramanda Gupta

ODE

10

Sep 3

 2023

Formation of DE

Eqn of 1^st Order, 1^st Degree

Integrating Factors

Linear Differential Eqns

Eqn reducible to Linear form

Bernauli Eqn

List of Important Results like Subnornal,subtangent,

length of normal and tangent,

Eqn of Tangent and Normal

Tangent and Normal X and Y intercept

Polar subtangent, subnormal, length of tangent, normal

Orthogonal Trajectory/Oblique Trajectory Eqn

Eqn of First Order but not of first degree

Solvable for p, x, y

Lagrange form

Eqn in Claurait Form

Eqn reducible to Claurait form

Singular Solution

Ordinary and Partial Differential Equations by Raisinghania

11

Sep 10

 2023

Linear Differential Equations with Constant Coefficients

Method of variation of parameter

Linear Equation of Second Order

Euler Cauchy

Reduction with One known integral

Reduction to Normal

Changing independent variable

Ordinary and Partial Differential Equations by Raisinghania

12

Sep 17

 2023

Laplace,

Inverse Laplace

Application to initial value problem for 2^nd order linear equations with constant coefficients

Ordinary and Partial Differential Equations by Raisinghania

PDE

13

Sep 24

 2023

Formation of PDE of a given function

Lagrange Equations: Various Methods

Surface Orthogonal to a given Surface

Charpit method

Clairaut equations

Jacobi method

Ordinary and Partial Differential Equations by Raisinghania

Advance PDE by Raisinghania

14

Oct 1

 2023

Homogeneous Linear PDE with Constant Coefficient

Linear PDE of Second Order with Constant Coefficient

Cauchy Method of Characteristic Strip (Total 10 Qns )

Canonical Form

Ordinary and Partial Differential Equations by Raisinghania

Advance PDE by Raisinghania

15

Oct 8

 2023

Wave Eqn

Initial Velocity is zero, Initial Displacement is Given

Initial Velocity is Given, Initial Displacement is Zero

Heat Eqn

a) Initial Temperature f(x) .Both Ends suddenly changed to  Zero Temperature

b) Initial Tempearature (T1 , T2 ) , suddenly change to ( T3 , T4)

c)Initial Temperature f(x). Both Ends Insulated suddenly

d)At t=0 distribution is f(x). Suddenly One end is kept at T1 and other end Insulated

Laplace Eqns

a) Three sides Temperature is 0, Other side f(x)

b) Two sides Temperature is 0, One side f(x), Other side at Infinite Long

c)One side Insulated (X-Axis)

d)One Side Insulated (Opposite side of X-Axis)

e) Two sides Insulated , X-axis side f(x), Other side 0

f) Two sides Insulated, X-axis side 0, Other side f(x)

g) Three sides Insulated

Laplace in Polar Coordinate Sysytem

a) Semicircular Plate

b) Circular Arc

c) Circular Place

d) Circular Annulus

Advance PDE by Raisinghania

Complex Analysis

16

Oct 15

 2023

Analytic Function:

Cauchy Riemann Equation

Given Function and Given Point

a) Show it satisfy Cauchy Riemann eqn

b) Show Analytic or Non-Analytic

c) Show Existence of Derivative or Not

Given U or V as Harmonic Function, find its Conjugate and also Function

Complex Integration: Cauchy Integrals, Zeroes, Singularity, Poles, Rouche theorem

Krishna Series Complex Analysis

17

Oct 22

 2023

Series Expansion

Expand in Taylor Series

Expand in Laurent Series

Power Series representation

Use Cauchy Residue Theorem, to Evaluate the Integral

Contour Integrations

Krishna Series Complex Analysis

Analytic Geometry

18

Oct 29

 2023

Straight Lines

Find coordinate the foot of perpendicular from given point to a plane. And also find distance

Plane and St line: Parallel/ Perpendicular / Coincide

Projection of Line on Plane

Condition for Coplanar Lines

Determine Eqn of St Line intersecting two given Lines

Find Perpendicular distance of a point from line and its foot coordinate.

Intersection of 3 Planes

Shortest Distance

Find SD between two lines and obtain eqn of SD (Very Imp)

Projection Method

Two Lines in Symmetric Form

One-line General, Other Symmetric Form

Two lines in General Form

Skew Lines

Find locus of line, which intersects set of lines or function.

Planes

Eqn of plan through 3 points

Angle between planes

Variable Plane Problems

Bisecting Planes

Combined Eqn of Planes

Projection of planes

Krishna Series 3D Geometry

19

Nov 5 2023

Spheres

Sphere touching plane.

Sphere through Circle

Eqn of tangent plane to sphere and its condition

Polar plane eqn and find pole.

Intersection of Spheres and orthogonality condition

Cylinder

Eqn parallel to line and guiding curve

Right Circular Cylinder

Tangent Plane

Enveloping Cylinder

Cone

Find eqn of cone with Vertex origin cone

Find cone, with non origin vertex

Condition for second degree to represent cone and find coordinates

Tangent Line and Tangent Plane of Cone

Condition for Tangency

Reciprocal cone

Angle between lines in which plane cuts Cone

Condition for 3 mutually perpendicular generators

Condition for 3 mutually perpendicular tangent planes

Right Circular Cone

Enveloping Cone

Krishna Series 3D Geometry

20

Nov 12 2023

Central Conicoids

Find Eqn of Tangent plane

Condition for Tangency

Director Sphere

Polar Planes, Pole

Locus of Chord Bisected at a given point

Normal to Conicoid

Prove 6 normal to ellipsoid

Find cubic curve through feet of normal

Diametral Plane

Problems on Semi conjugate diameter properties

Paraboloid

Generating Lines

Krishna Series Analytic Geometry (Slim Book)

LPP

21

Nov 19 2023

Formulation of LPP

Graphical Method of Solution

Simplex Method

Big M Method

Construct Dual and Solve

Operation Research by Taha

ANY BOOK

22

Nov 26 2023

Transportation Problem

Assignment Problem

Operation Research by Taha

Numerical Analysis

23

Dec 3 2023

Obtain derivation

(a)Quadrature Formula,

(b)Trapezoid Rule,

(c)Simpson 1/3,

(d) Simpson3/8. Rule    and also

(e)derive their Error Formula for ALL RULES.

Newton Raphson Method

a)Derivation,

b) Find Condition for its convergence

c) Show rate of convergence is quadratic

d) Explain its merits and demerits

Bisection Method

Regula Falsi Method

Secant Method

Iteration Method and its Convergence

Derive Newton Gregory Forward interpolation formula, and its Error.

Derive Newton Gregory’s Backward Interpolation formula and its Error

Lagrange Interpolation

Derivation of  formula and derive its Error formula

Prove that Lagrange’s formula can be put in the form of

P n x = r =1 n ∅ x f x r x - x r ∅ ' x r

w h ere ∅ x = r =0 n x - x r .

Show that the sum of Lagrangian coefficient is unity.

Use Lagrange’s interpolation formula to express the function  x 2 + x - 3 x 3 - 2 x 2 - x +2

as sums of partial fractions.

Find the parabola passing through points (0,1) (1,3) and (3,55) using Lagrange’s formula

Gauss Quadrature Formula.

Derive formula for n=3,4,5

Solve ODE Problems

a) Euler

b) Euler Modified

c)Runge Kutta Order 1,  4

Solve Linear Eqns

a) Gauss Elimination

b) Gauss Jordan

c) Gauss Seidel

d) Gauss Jacobi

Use Gauss Jordan to Find Inverse

Conversion

Decimal to Octogonal and Vice versa

Decimal to Hexadeceimal and vice versa

Solve Boolean Expression

Algorithms and Flow Chart

Krishna Series Numerical Analysis

Numerical Analysis by Iyenger (Few Questions)

Numerical Analysis by Sastry

Flow Charts: Free Study Material by SuccessClap

Mechanics

24

Dec 10 2023

Solving Problems on Lagrange Eqns

Solving Problems on Hamilton Eqns

Classical Mechanics by JC Upadhyay,

Krishna: Rigid Dynamics Vol1

25

Jan 21 2024

D Alembert Principle Problems

Moment of Inertia

Fixed Axis Motions( Important questions Only)

Motion in 2Dimension ( Important questions Only)

Krishna : Rigid Dynamics Vol 1 and Vol 2

Fluid Dynamics

26

Jan 28 2024

Learn Basics: Del,div,curl in catesian, spherical, cylindrigal, general coordinate system

Kinematics

Continuity

Equation of motion

Boundary Problems

Path Line

Vector Potential

Vortex Line, vector

Euler Eqns

Energy Conservation

Bernoulli Eqns and Application

Stream Fns

Source Sink problem

Fluid Dynamics: Raisinghania

Chapter 1,2,3,4,5

27

Feb 4 2024

Irrotational Motion

Circular Cylinder Motion

Irrotational 3D

Vortex

Navier Stoke

Laminar Flow

Fluid Dynamics: Raisinghania

Chapter 6, 7,10,11,14,16

Dynamics

28

Feb 11 2024

Rectilinear Motion

SHM

Projectile Motion

Constrained Motion

Motion in a Plane, Circle

Work, Energy, Impulse

Krishna Series Dynamics

29

Feb 18 2024

Kepler Laws,

Problems on Central Forces

Planetary Motion

Motion in resisting medium

Krishna Series Dynamics

STATICS

30

Feb 25

2024

Force Eqns, Moments

Equilibrium of Rigid Body

Frictions

Catenary

Krishna Series Statics

31

Mar 3 2024

Virtual Work

Stable and Unstable Equilibrium

Krishna Series Statics

Abstract Algebra

32

Mar 10 2024

Group Theory

Groups

SubGroups

Cosets, Lagrange Theorem

Cyclic Group

Normal Subgroups

Quotient Groups

Homomorphism

Fundamental Theorem of Homomorphism 1,2,3

Permutation Groups

Cayley’s theorem

Any State B.Sc Text Book

Group Theory by R Kumar

SuccessClap Free Study material on Algebra

Don’t waste time on Galian, Khanna Bhambri or any book

33

Mar 17 2024

Ring Theory

Rings

SubRings

Ring Homomorphism

Embedness

Max Ideal

Prime Ideal

PID

Divisibility in Rings, prime, Irreducible element

Euclidean Domain, UFD,

Einstein criteria

Any State B.Sc Text Book

Ring Theory by R Kumar

SuccessClap Free Study material on Algebra

PRELIMS BREAK

34

June 9, 2024

Full Length Test Paper 1

FLT1

35

June 23 2024

Full Length Test Paper 2

FLT2

36

July 7, 2024

Full Length Test Paper 1

FLT3

37

July 21, 2024

Full Length Test Paper 2

FLT4

38

Aug 4, 2024

Full Length Test Paper 1

FLT5

39

Aug 18, 2024

Full Length Test Paper 2

FLT6

40

41

Aug 25, 2024

Full Length Test Paper 1

Full Length Test Paper 2

FLT7

FLT8

42

43

Sep 1, 2024

Full Length Test Paper 1

Full Length Test Paper 2

FLT9

FLT10

44

45

Sep 8, 2024

Full Length Test Paper 1

Full Length Test Paper 2

FLT11

FLT12