$Calculate$ $I=\int_0^{\frac{1}{a}}x^p(1-ax)^qdx$

 $Q$ $$I=\int_0^{\frac{1}{a}}x^p(1-ax)^qdx$$ 

$..............(i)$                            

$solution$-

 $let$      $ 1-ax=t$

$differentiating$ $both$ $sides$

$-adx=dt$

$dx=-\frac{dt}{a}$

$Also$  $t\to 1$ $when$ $x\to 0$ & $t\to 0$ $when$ $x\to \frac{1}{a}$

$Now$ $putting$ $these$ $values$ $into$ $integral$ $(i) $

$so$ $we$ $have$

$$I=\int_1^0(\frac{1-t}{a})^p(t)^q(-\frac{dt}{a})$$ 

$$I=-\int_1^0\frac{(1-t)^p}{a^p}t^q\frac{dt}{a}$$ 

$$I=-\frac{1}{a^{p+1}}\int_1^0(1-t)^pt^qdt$$ 

$\{\int_a^bfdx=-\int_b^afdx\}$

$$I=\frac{1}{a^{p+1}}\int_0^1(1-t)^pt^qdt$$      

$$I=\frac{1}{a^{p+1}}\int_0^1(1-t)^{(p+1)-1}t^{(q+1)-1}dt$$ 

$\{\beta(m,n)=\beta(n,m)\}$    

$$I=\frac{1}{a^{p+1}}\int_0^1t^{(q+1)-1}(1-t)^{(p+1)-1}dt$$ 

$$I=\frac{1}{a^{p+1}}\beta(q+1,p+1)$$

$$I=\frac{1}{a^{p+1}}\beta(p+1,q+1)$$

$Answer. $

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Linear ordinary differential equations of first order

 

Differential equations are equation which are made of  derivative of dependent variable, say $\frac{dy}{dx}$, independent variable, say $\mathcal x$ & dependent variable, say $\mathcal y$.

Mathematically they are written as  $f(x,y,\frac{dy}{dx})=0$

As you know there are many problems in real world which are solved using differential equations.  

Linear equations

$\frac{dy}{dx}+Py=Q$

$I.F.=e^\int Pdx$ &

 $y.e^\int Pdx=\int Qe^\int Pdx +c$

$y.IF=\int Q.IF +c$

 or

 $\frac{dx}{dy}+Px=Q$

$I.F.=e^\int Pdy$ & 

$y.e^\int Pdy=\int Qe^\int Pdy +c$

$x.IF=\int Q.IF +c$

Also know Trajectories

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Trajectories

 Trajectories

 

If any curve cuts, every member curve of any family of curves, according to a rule then that curve is called trajectory of family of curves.

If curve cuts all remaining members of its own family at right angel then it is called self-orthogonal.

Finding orthogonal trajectory-

 Let equation family of curves $f(x,y,\frac{dy}{dx})=0$

·        Differentiate this equation with respect to $\mathcal x$

·        Put $-\frac{dx}{dy}$ instead of $\frac{dy}{dx}$

 

Polar form – $f(r,\theta,\frac{dr}{d\theta})=0$

•          Differentiate with respect to $\theta$
•         Put $-\frac{1}{r}\frac{dr}{d\theta}$  instead $r\frac{d\theta}{dr}$ of & $-r^2\frac{d\theta}{dr}$  instead of  $\frac{dr}{d\theta}$

Also read Linear ordinary differential equations

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Why a Mathematician can’t travel 1 mile even in infinite time?

Can you believe a mathematician can’t complete 1 mile distance even after taking infinite time. It’s very interesting to know, after all why does this happen?

When you think practically, it seems like humor. As you all know mathematics is very interesting subject. That all, which is practically possible in this world, is also possible in mathematics and which is  impossible practically, is possible in mathematics.

To understand paradox mentioned above, lets consider an example.

Suppose there is a boy named John who wants to travel 1 mile. Now you would be thinking is there any rocket science in travelling 1 mile distance? You are right, of course there is no rocket science.

Then why, John can’t travel 1 mile even in infinite time?

Before travelling, there is a condition which must have to be fulfilled by John.

Condition is as follows

Suppose John is at point A, which is 0 and he wants to go at point B, which is 1. First time john must have to travel half of the total distance. Besides this, he must have to always travel half of the remaining distance.

Mathematically you can understand this as:

John has been given distance of 1 mile. According to condition, John must have to travel half of the distance.

So, distance travelled by john in first time would be

 

               Total distance / 2 = 1 / 2 = 0.5 mile

Now consider second condition: John must have to always travel half of the remaining distance.  

So,

 Remaining distance = total distance - travelled distance

                                      =   1 – 0.5

                                      =    0.5 mile

Now distance travelled by John in second time would be

                         Remaining distance/ 2 = 0.5 / 2 = 0.25 mile

So,

 Remaining distance = total distance - total travelled distance

                                  = 1 – ( 0.5 + 0.25)

                                  = 1- ( 0.75)

                                  =  0.25 mile

 Now John would travel in third time

                       Remaining distance / 2 =  0.25 / 2 = 0.125 mile

So,

Remaining distance = total distance - total travelled distance

                                    = 1 – ( 0.5 + 0.25 + 0.125 )

                                   =  1 – ( 0.875)

                                   =  0.125 mile

Now John would travel in fourth time

             Remaining distance / 2 = 0.125 / 2 = 0.0625 mile

So,

 Remaining distance = total distance – total travelled distance

                                     = 1 – ( 0.5 + 0.25 + 0.125 + 0.0625 )

                                     = 1 – ( 0.9375 )

                                     =  0.0625 mile 

Now John would travel in fifth time

                 Remaining distance / 2 = 0.0625 / 2 = 0.03125 mile

So,

Remaining distant = total distance - total travelled distance

                             = 1 – ( 0.5 + 0.25 + 0.125 + 0.0625 + 0.03125 )

                             = 1 – ( 0.96875 )

                             = 0.03125 mile

.

.

.

.

.

.

.

.

.

.

.

.

Continue in this way, on and on.

 

Now, lets see how much distance John has travelled so far  

 

John travelled total distance in 1^st time = 0.5 mile

John travelled total distance in 2^nd time = 0.75 mile

John travelled total distance in 3^rd time = 0.875 mile

John travelled total distance in 4^th time = 0.93755 mile

John travelled total distance in 5^th time = 0.96875 mile

John travelled total distance in 6^th time = 0.984375 mile

John travelled total distance in 7^th time = 0.9921875 mile

John travelled total distance in 8^th time = 0.99609375 mile

.

.

.

.

.

After continuing in this way you can see John is not reaching exactly at B, although he is reaching approximately at B.

 

 

 

 

 

 

 

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Csir net mathematics 2020 latest syllabus pdf download

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CSIR-UGC National Eligibility Test 

(NET) for Junior Research 

Fellowship and Lecturer-ship 

SYLLABUS FOR

MATHEMATICAL SCIENCES

PAPER I AND PAPER II

             UNIT – 1 

Analysis: 

Elementary set theory, finite, countable and uncountable sets, Real number 

system as a complete ordered field, Archimedean property, supremum, infimum. 

Sequences and series, convergence, limsup, liminf. 

 Bolzano Weierstrass theorem, Heine Borel theorem. 

Continuity, uniform continuity, differentiability, mean value theorem. 

Sequences and series of functions, uniform convergence. 

 Riemann sums and Riemann integral, Improper Integrals. 

Monotonic functions, types of discontinuity, functions of bounded variation, Lebesgue 

measure, Lebesgue integral. 

Functions of several variables, directional derivative, partial derivative, derivative as a 

linear transformation. 

Metric spaces, compactness, connectedness. Normed Linear Spaces. Spaces of 

Continuous functions as examples. 

Linear Algebra: 

Vector spaces, subspaces, linear dependence, basis, dimension, algebra 

of linear transformations. 

Algebra of matrices, rank and determinant of matrices, linear equations. 

Eigenvalues and eigenvectors, Cayley-Hamilton theorem. 

Matrix representation of linear transformations. Change of basis, canonical forms, 

diagonal forms, triangular forms, Jordan forms. 

Inner product spaces, orthonormal basis. 

Quadratic forms, reduction and classification of quadratic forms.

UNIT – 2 

Complex Analysis: 

Algebra of complex numbers, the complex plane, polynomials, 

Power series, transcendental functions such as exponential, trigonometric and hyperbolic 

functions. 

Analytic functions, Cauchy-Riemann equations. 

 Contour integral, Cauchy’s theorem, Cauchy’s integral formula, Liouville’s theorem, 

Maximum modulus principle, Schwarz lemma, Open mapping theorem. 

Taylor series, Laurent series, calculus of residues. 

 Conformal mappings, Mobius transformations. 

Algebra: 

Permutations, combinations, pigeon-hole principle, inclusion-exclusion 

principle, derangements. 

Fundamental theorem of arithmetic, divisibility in Z, congruences, Chinese Remainder 

Theorem, Euler’s Ø- function, primitive roots. 

Groups, subgroups, normal subgroups, quotient groups, homomorphisms, 

cyclic groups, permutation groups, Cayley’s theorem, class equations, Sylow theorems. 

Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, 

principal ideal domain, Euclidean domain. 

Polynomial rings and irreducibility criteria. 

Fields, finite fields, field extensions.

UNIT – 3 

Ordinary Differential Equations (ODEs): 

Existence and Uniqueness of solutions of initial value problems for first order ordinary 

differential equations, singular solutions of first order ODEs, system of first order ODEs. 

General theory of homogenous and non-homogeneous linear ODEs, variation of 

parameters, Sturm-Liouville boundary value problem, Green’s function. 

Partial Differential Equations (PDEs): 

Lagrange and Charpit methods for solving first order PDEs, Cauchy problem for first 

order PDEs. 

Classification of second order PDEs, General solution of higher order PDEs with 

constant coefficients, Method of separation of variables for Laplace, Heat and Wave 

equations. 

Numerical Analysis :

Numerical solutions of algebraic equations, Method of iteration and Newton-Raphson 

method, Rate of convergence, Solution of systems of linear algebraic equations using 

Gauss elimination and Gauss-Seidel methods, Finite differences, Lagrange, Hermite and 

spline interpolation, Numerical differentiation and integration, Numerical solutions of 

ODEs using Picard, Euler, modified Euler and Runge-Kutta methods. 

Calculus of Variations: 

Variation of a functional, Euler-Lagrange equation, Necessary and sufficient conditions 

for extrema. Variational methods for boundary value problems in ordinary and partial 

differential equations. 

Linear Integral Equations: 

Linear integral equation of the first and second kind of Fredholm and Volterra type, 

Solutions with separable kernels. Characteristic numbers and eigenfunctions, resolvent 

kernel. 

Classical Mechanics: 

Generalized coordinates, Lagrange’s equations, Hamilton’s canonical equations, 

Hamilton’s principle and principle of least action, Two-dimensional motion of rigid 

bodies, Euler’s dynamical equations for the motion of a rigid body about an axis, theory 

of small oscillations.

UNIT – 4 

Descriptive statistics, exploratory data analysis

Sample space, discrete probability, independent events, Bayes theorem. Random 

variables and distribution functions (univariate and multivariate); expectation and 

moments. Independent random variables, marginal and conditional distributions. 

Characteristic functions. Probability inequalities (Tchebyshef, Markov, Jensen). Modes 

of convergence, weak and strong laws of large numbers, Central Limit theorems (i.i.d. 

case). 

Markov chains with finite and countable state space, classification of states, limiting 

behaviour of n-step transition probabilities, stationary distribution. 

Standard discrete and continuous univariate distributions. Sampling distributions. 

Standard errors and asymptotic distributions, distribution of order statistics and range. 

Methods of estimation. Properties of estimators. Confidence intervals. Tests of 

hypotheses: most powerful and uniformly most powerful tests, Likelihood ratio tests. 

Analysis of discrete data and chi-square test of goodness of fit. Large sample tests. 

Simple nonparametric tests for one and two sample problems, rank correlation and test 

for independence. Elementary Bayesian inference. 

Gauss-Markov models, estimability of parameters, Best linear unbiased estimators, tests 

for linear hypotheses and confidence intervals. Analysis of variance and covariance. 

Fixed, random and mixed effects models. Simple and multiple linear regression. 

Elementary regression diagnostics. Logistic regression. 

Multivariate normal distribution, Wishart distribution and their properties. Distribution of 

quadratic forms. Inference for parameters, partial and multiple correlation coefficients 

and related tests. Data reduction techniques: Principle component analysis, Discriminant 

analysis, Cluster analysis, Canonical correlation. 

Simple random sampling, stratified sampling and systematic sampling. Probability 

proportional to size sampling. Ratio and regression methods. 

Completely randomized, randomized blocks and Latin-square designs. Connected, 

complete and orthogonal block designs, BIBD. 2K factorial experiments: confounding 

and construction. 

Series and parallel systems, hazard function and failure rates, censoring and life 

testing. 

Linear programming problem. Simplex methods, duality. Elementary queuing and 

inventory models. Steady-state solutions of Markovian queuing models: M/M/1, M/M/1 

with limited waiting space, M/M/C, M/M/C with limited waiting space, M/G/1.

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Download upsc mathematics optional syllabus 2020 pdf

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MATHEMATICS Optional syllabus 2020

                              PAPER I

(1) Linear Algebra :

Vector spaces over R and C, linear dependence and independence, subspaces, bases, 

dimensions, Linear transformations, rank and nullity, matrix of a linear transformation.

Algebra of Matrices; Row and column reduction, Echelon form, congruence’s and similarity; 

Rank of a matrix; Inverse of a matrix; Solution of system of linear equations; Eigenvalues and 

eigenvectors, characteristic polynomial, Cayley-Hamilton theorem, Symmetric, skew-symmetric, 

Hermitian, skew-Hermitian, orthogonal and unitary matrices and their eigenvalues. 

(2) Calculus :

Real numbers, functions of a real variable, limits, continuity, differentiability, mean-value 

theorem, Taylor’s theorem with remainders, indeterminate forms, maxima and minima, 

asymptotes; Curve tracing; Functions of two or three variables; Limits, continuity, partial 

derivatives, maxima and minima, Lagrange’s method of multipliers, Jacobian.

Riemann’s definition of definite integrals; Indefinite integrals; Infinite and improper 

integral; Double and triple integrals (evaluation techniques only); Areas, surface and volumes.

(3) Analytic Geometry :

Cartesian and polar coordinates in three dimensions, second degree equations in three 

variables, reduction to Canonical forms; straight lines, shortest distance between two skew 

lines, Plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and 

their properties.

(4) Ordinary Differential Equations :

Formulation of differential equations; Equations of first order and first degree, integrating 

factor; Orthogonal trajectory; Equations of first order but not of first degree, Clairaut’s equation, 

singular solution.

Second and higher order liner equations with constant coefficients, complementary function, 

particular integral and general solution.

Section order linear equations with variable coefficients, Euler-Cauchy equation; Determination of complete solution when one solution is known using method of variation of 

parameters.

Laplace and Inverse Laplace transforms and their properties, Laplace transforms of 

elementary functions. Application to initial value problems for 2nd order linear equations with 

constant coefficients.

(5) Dynamics and Statics :

Rectilinear motion, simple harmonic motion, motion in a plane, projectiles; Constrained 

motion; Work and energy, conservation of energy; Kepler’s laws, orbits under central forces.

Equilibrium of a system of particles; Work and potential energy, friction, Common catenary; 

Principle of virtual work; Stability of equilibrium, equilibrium of forces in three dimensions.

(6) Vector Analysis :

Scalar and vector fields, differentiation of vector field of a scalar variable; Gradient, 

divergence and curl in cartesian and cylindrical coordinates; Higher order derivatives; Vector 

identities and vector equation.

Application to geometry : Curves in space, curvature and torsion; Serret-Furenet's 

formulae.

Gauss and Stokes’ theorems, Green's indentities.

                              PAPER II

(1) Algebra : 

Groups, subgroups, cyclic groups, cosets, Lagrange’s Theorem, normal subgroups, quotient 

groups, homomorphism of groups, basic isomorphism theorems, permutation groups, Cayley’s 

theorem.

Rings, subrings and ideals, homomorphisms of rings; Integral domains, principal ideal 

domains, Euclidean domains and unique factorization domains; Fields, quotient fields.

(2) Real Analysis :

Real number system as an ordered field with least upper bound property; Sequences, limit of 

a sequence, Cauchy sequence, completeness of real line; Series and its convergence, absolute 

and conditional convergence of series of real and complex terms, rearrangement of series. 

Continuity and uniform continuity of functions, properties of continuous functions on compact 

sets.

Riemann integral, improper integrals; Fundamental theorems of integral calculus.

Uniform convergence, continuity, differentiability and integrability for sequences and series of 

functions; Partial derivatives of functions of several (two or three) variables, maxima and minima.

(3) Complex Analysis :

Analytic function, Cauchy-Riemann equations, Cauchy's theorem, Cauchy's integral formula, 

power series, representation of an analytic function, Taylor’s series; Singularities; Laurent’s series; 

Cauchy’s residue theorem; Contour integration.

(4) Linear Programming : 

Linear programming problems, basic solution, basic feasible solution and optimal solution; Graphical method and simplex method of solutions; Duality.

Transportation and assignment problems.

(5) Partial Differential Equations :

Family of surfaces in three dimensions and formulation of partial differential equations; 

Solution of quasilinear partial differential equations of the first order, Cauchy’s method of 

characteristics; Linear partial differential equations of the second order with constant coefficients, 

Divergence and canonical form; Equation of a vibrating string, heat equation, Laplace 

equation and their solutions.

(6) Numerical Analysis and Computer Programming :

Numerical methods: Solution of algebraic and transcendental equations of one variable by 

bisection, Regula-Falsi and Newton-Raphson methods, solution of system of linear equations by 

Gaussian Elimination and Gauss-Jorden (direct), Gauss-Seidel (iterative) methods. Newton’s 

(forward and backward) and interpolation, Lagrange’s interpolation.

Numerical integration: Trapezoidal rule, Simpson’s rule, Gaussian quadrature formula.

Numerical solution of ordinary differential equations : Eular and Runga Kutta methods.

Computer Programming : Binary system; Arithmetic and logical operations on numbers; Octal 

and Hexadecimal Systems; Conversion to and from decimal Systems; Algebra of binary numbers.

Elements of computer systems and concept of memory; Basic logic gates and truth tables, 

Boolean algebra, normal forms.

Representation of unsigned integers, signed integers and reals, double precision reals and 

long integers.

Algorithms and flow charts for solving numerical analysis problems.

(7) Mechanics and Fluid Dynamics : 

Generalised coordinates; D’Alembert’s principle and Lagrange’s equations; Hamilton 

equations; Moment of inertia; Motion of rigid bodies in two dimensions.

Equation of continuity; Euler’s equation of motion for inviscid flow; Stream-lines, path of a 

particle; Potential flow; Two-dimensional and axisymmetric motion; Sources and sinks, vortex 

motion; Navier-Stokes equation for a viscous fluid

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