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Crash Course for UPSC MAINS 2021: REVISON MADE SIMPLE

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1)This course is useful to quickly revise all topics in detail for Upsc Mains exam 2) We provide day wise Brief Study material, Short Videos, and Important questions to Practice. 3) We provide a) DAY WISE Study Material for Quick Revision

b) Important questions and Answers for practice c) Short Videos on Important Concept d) Previous Year Solutions e) 30 Full Length Tests f) Daily Review g) Personal Mentorship

4) Why you need this course ? You have very less time to UPSC Mains Exam. You must practice all Important questions from various sources, solve all PYQs and take MOCK TESTS. It takes lot of time to choose different books to revise and practice. We made it SIMPLE. We provide Quality Study material, Question Bank from different books, Previous Year Solutions, and Test Series at one place. We provide mentorship for doubt clarification during this course. 5) This course will SAVE your Time. 6) The Crash Course Starts from Oct 18-2021 7) For ANY Query: WhatsApp 9346856874

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Day Topic TICK Oct 18 Day 1

Mechanics Solving Problems on Lagrange Eqns

Day 1 Numerical Analysis

Obtain (a)Quadrature Formula,

(b)Trapezoid Rule, (c)Simpson 1/3,

(d) Simpson3/8. Rule and also (e)derive their Error Formula for ALL RULES.

Derive Newton Raphson Method,

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

Oct 19 Day 2

Mechanics Solving Problems on Hamilton Eqns

Day 2 Numerical Analysis

Derive Newton Gregory Forward interpolation formula, and its Error.

Derive Newton Gregory’s Backward Interpolation formula

and its Error Derive Lagrange interpolation formula and derive its Error

formula

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

𝑃𝑛(𝑥) = ∑∅(𝑥)𝑓(𝑥𝑟)

(𝑥 − 𝑥𝑟)∅′(𝑥𝑟)

𝑛

𝑟=1

𝑤ℎ𝑒𝑟𝑒 ∅(𝑥) = ∏ (𝑥 − 𝑥𝑟)𝑛𝑟=0 .

Show that the sum of Lagrangian coefficient is unity.

Use Lagrange’s interpolation formula to express the

function 𝑥2 + 𝑥 − 3

𝑥3 − 2𝑥2 − 𝑥 + 2

𝑥2 + 6𝑥 + 1

(𝑥 − 1)(𝑥 + 1)(𝑥 − 4)(𝑥 − 6)

as sums of partial fractions.

Find the parabola passing through points (0,1) (1,3) and

(3,55) using Lagrange’s formula.

Oct 20 3

Mechanics D Alembert Principle Problems Moment of Inertia Fixed Axis Motions

3 Numerical Analysis

Derive Gauss Quadrature Formula.

Evaluate the integral 𝐼 = ∫𝑑𝑥

𝑥.

12

5 using Gauss Quadrature

n=5.

Find the value of the integral ∫ 𝑥 𝑑𝑥1

0. using Gauss

Quadrature n=4.

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

Oct 21

4 Mechanics Motion in 2Dimension

( Important questions Only)


Conversion Decimal to Octogonal and Vice versa

Decimal to Hexadeceimal and vice versa Solve Boolean Expression

Algorithms and Flow Chart

Oct 22

5 Mechanics Previous Year Paper 1992-2020


Previous Year Paper 1992-2020

Oct 23 6

Complex Analysis

Analytic Function: 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

Vector

Analysis Directional Derivative : Max/Min ,Anglemade,

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

Invariance under Transformation

Calculus and Real Analysis

Improper Integrals Limit Test

Cauchy Test Absolute Convergence and conditional convergence

Abel Test Dirichlet Test

Mean Value theorem

Rolle Theorem Lagrange Theorem ( Also proof)

Cauchy MVT Function Increasing/Decreasing

Generalized MVT

Oct 25

7 Complex Analysis

Complex Integration: Cauchy Integrals

Zeroes, Singularity,Poles, Rouche

Vector Analysis

Green Theorem Stoke Theorem


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

Oct 26

8 Complex Analysis

Series Expansion Expand in Taylor Series

Expand in Laurent Series

Vector Analysis

Divergence Theorem Work done


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

Multiple Integrals

Change of Order of Integration

Evaluation of Integrals

Oct 27

9 Complex Analysis

Use Cauchy Residue Theorem, to Evaluate the Integral Contour Integrations

Vector Analysis

Derive Serret Frennet formula Show curve lie in Plane Find Curvature Vector

Problems on finding Radius of Curvature, Torsion

Calculus

and Real Analysis

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

Oct 28

10 Complex Analysis

Previous Year Paper 1992-2020

Vector Analysis

Previous Year paper 1992-2020

Oct 29 11

Linear Algebra

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

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

LPP Formulation of LPP Graphical Method of Solution

Simplex Method Big M Method


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

Oct 30 12

Linear Algebra

Theorems and Problems from Krishna Series Eigen Values, Vectors

Orthogonality Similarity

Diagonalizable Quadratic Form

LPP Big M Method Construct Dual and Solve


Length of Arc Areas

Cartesian Polar

Loop/Asymptotes Volumes Cartesion

Polar Spherical System

Surfaces

Nov 1

13 Linear

Algebra Vector Spaces, Basis, Dimensions, Nullity

LPP Transportation Problem


Uniform Convergence Cauchy Principle

Mn Test Weirstrass Test

Abel Test Dirichlet Test Properties on

a)Sum b) Differntiability

c) Integrability

Functions of Several Variables Limit

Continuity Differentiability

Nov 2 14

Linear Algebra

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

LPP Assignment Problem


Jacobians Polar Coordinates

Root equation problems

MVT Taylor Maclurin Expansion Partial Differentiation

PD Eulers Total Differentiation

Asymptote General Method

Inspection method Intersection of Curve and Asymptote

Nov 3 15

Linear Algebra

Previous Year Problems 1992-2020

LPP Assignment Problem

Nov 4 16

PDE Formation of PDE of a given function Lagrange Eqns : Various Methods

Surface Orthogonal to a given Surface

ODE Formation of DE Eqn of 1st Order, 1st Degree

Linear Differential Eqns Eqn reducible to Linear form

Bernauli Eqn


PYQ Calculus & Real Analysis

Nov 5

17 PDE Charpit method

Clairaut equations Jacobi method

ODE 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 Tarajectory/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 Caurait form

Singular Soln Dynamics Constrained Motion

Motion in a Plane Work, Energy,Impulse

Nov 6 18

PDE Homogeneous Linear PDE with Constant Coefficient Linear PDE of Second Order with Constant Coefficient

ODE Linear Differential Eqns with Consatnt Coefficients Method of variation of parameter

Analytic Geometry

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

Nov 8 19

PDE Cauchy Method of Characteristic Strip ( Total 10 Qns ) Canonical Form

ODE Linear Eqn of Second Order Reduction with One known integral

Reduction to Normal Changing independent variable

Analytic Geometry

Planes Eqn of plan through 3 points

Angle between planes Variable Plane Problems

Bisecting Planes Combined Eqn of Planes

Projection of planes 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

Nov 9 20

PDE 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 Insullated

ODE Laplace , Inverse Laplace

Nov 10 21

PDE Laplace Eqns a)Three sides Temerature is 0, Other side f(x)

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

ODE Application to initial value problems for 2nd order linear equations with constant coefficients

Nov 11 22

PDE Previous Year Paper 1992-2020

ODE Previous year Solution 1992-2020

Nov 12 23

Analytic Geometry

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

Dynamics Rectilinear Motion SHM

Projectile Motion

Nov 13 24

Analytic Geometry

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 Conicoids Prove 6 normals to ellipsoid

Find cubic curve through feet of normal Diametral Plane

Problems on Semi conjugate diameter properties

Dynamics Problems on Central Forces

Nov 12 23

Statics Common Catenary

Analytic Geometry

Paraboloid Generating Lines

Nov 13 24

Fluid Dynamics

Kinematics Continuity

Boundary Problems

Statics Frictions

Modern Algebra

Groups SubGroups

Cosets, Lagrange Theorem

Nov 15 25

Fluid Dyanamics

Path Line Vector Potential

Euler Eqns

Statics Virtual Work

Nov 16 26

Modern Algebra

Cyclic Group Normal Subgroups

Homomorphism

Statics Stable and Unstable Equilibrium

Fluid Dynamic

Energy Conservation Bernoulli Eqns

Nov 18 27

Fluid Dynamics

Stream Fns Source Sink problem

Modern Algebra

FundamentalTheorem of Homomorphism Rings

SubRings

Nov 19 28

Modern Algebra

Ring Homomorphism Embedness Max Ideal

Fluid Dynamic

Irrotational Motion

Nov 20 29

Fluid Dynamic

Navier Stoke Laminar Flow

Modern Algebra

Prime Ideal PID

Divisibility in Rings, prime, Irreducible element

Nov 22 30

Modern Algebra

Euclidean Domain, UFD, Einstein criteria

REVISION and Topic wise Solving Very Important Questions Only

Nov 23

Linear Algebra Calculus

Nov 24

Analytic Geometry ODE

Nov 25 Vector Analysis Real Analysis

Nov 26 Complex Analysis PDE

Nov 27 Mechanics Numerical Analysis

Nov 29 Fluid Dynamics Statics and Dynamics

Nov 30 Modern Algebra LPP

Dec 1 Linear Algebra Calculus

Dec 2 Analytic Geometry ODE

Dec 3 Vector Analysis Real Analysis

Dec 4 Complex Analysis PDE

Dec 6 Mechanics Numerical Analysis

Dec 7 Fluid Dynamics Statics and Dynamics

Dec 8 Modern Algebra LPP


Dec10 Analytic Geometry ODE



Dec 14 Mechanics Numerical Analysis

Dec 15 Fluid Dynamics Statics and Dynamics

Dec 16 Modern Algebra LPP


Mechanics Numerical Analysis


Analytic Geometry ODE


Fluid Dynamics

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