### MATHEMATICS/ APPLIED MATHEMATICS (319) Syllabus

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Note: There will be one Question Paper which will contain Two Sections i.e. Section A and Section B [B1 and B2]. Section A will have 15 questions covering both i.e. Mathematics/Applied Mathematics which will be compulsory for all candidates Section B1 will have 35 questions from Mathematics out of which 25 questions need to be attempted. Section B2 will have 35 questions purely from Applied Mathematics out of which 25 questions will be attempted. |

### SECTION A

1. Algebra |

(i) Matrices and types of Matrices |

(ii) Equality of Matrices, transpose of a Matrix, Symmetric and Skew Symmetric Matrix |

(iii) Algebra of Matrices |

(iv) Determinants |

(v) Inverse of a Matrix |

(vi) Solving of simultaneous equations using Matrix Method |

2. Calculus |

(i) Higher order derivatives |

(ii) Tangents and Normals |

(iii) Increasing and Decreasing Functions |

(iv). Maxima and Minima |

3. Integration and its Applications |

(i) Indefinite integrals of simple functions |

(ii) Evaluation of indefinite integrals |

(iii) Definite Integrals |

(iv). Application of Integration as area under the curve |

4. Differential Equations |

(i) Order and degree of differential equations |

(ii) Formulating and solving of differential equations with variable separable |

5. Probability Distributions |

(i) Random variables and its probability distribution |

(ii) Expected value of a random variable |

(iii) Variance and Standard Deviation of a random variable |

(iv). Binomial Distribution |

6. Linear Programming |

(i) Mathematical formulation of Linear Programming Problem |

(ii) Graphical method of solution for problems in two variables |

(iii) Feasible and infeasible regions |

(iv). Optimal feasible solution |

### Section B1: Mathematics

**UNITI: RELATIONS AND FUNCTIONS**

1. Relations and Functions

Types Of Relations:Reflexive,symmetric,transitive and equivalence relations.One to one and onto functions,composite functions,inverse function.Binary Operations.

2. InverseTrigonometricFunctions

Definition,range, domain, principal value branches. Graphs of inverse trigonometric functions. Elementary Properties Of Inverse trigonometric functions.

**UNITII: ALGEBRA **

1. Matrices

Concept,notation,order, equality,types of matrices, zero matrix,transpose of matrix,symmetric and skew symmetric matrices.Addition,multiplication and scalar multiplication of matrices,simple properties of addition,multiplication scalar multiplication.Non-commutativity of multiplication of matrices and existence of non-zero matrices whose whose products the zero matrix (restrictto square matrices of order 2). Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness inverse,if it exists;(Here all matrices will have real entries).

2. Determinants

Determinant Of a square matrix (upto3×3matrices),properties of determinants,minors, cofactors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix.Consistency,inconsistencyandnumberofsolutionsofsystemoflinearequationsbyexamples, solving system of linear equations in two or three variables(having unique solution)using inverse of matrix.

**UNIT III: CALCULUS**

1. Continuity and Differentiability

Continuity And differentiability,derivative of composite functions, chain rule, derivatives of inverse trigonometric functions,derivative implicit function.Concepts Exponential,logarithmic functions. Derivatives Of Log x ande x .Logarithmic Differentiation.Derivative Functions expressed parametric forms. Second-order derivatives.Rolle’s and Lagrange’s Mean ValueTheorems(without proof) and their geometric interpretations.

2. Applications of Derivatives

Applications Of Derivatives:Rate Of change,increasing/decreasing functions,tangents and normals, approximation,maxima and minima(firstderivativetestmotivatedgeometricallyandsecondderivative test given as a provable tool).Simple problems(that illustrate basic principles and understanding of the subject as well as real-life situations). Tangent and Normal.

3. Integrals

Integration As Inverse Process Of Differentiation.Integration of variety of a function by substitution, by partial fractions and by parts, only simple integrals ofthe type –

to be evaluated. Definite integrals as a limit of a sum. Fundamental Theorem of Calculus(without proof). Basic properties of definite integrals and evaluation of definite integrals.

4. Applications of the Integrals

Applications Of finding the area under simple curves, especially lines, arcs of circles/parabolas/ellipses(in standard form only), area between the two above said curves(the region should be clearly identifiable).

5. DifferentialEquations

Definition,order and degree, general and particular solutions of differential equation.Formation Of differential equation whose general solution is given.Solution of differential equations by method of separation of variables, homogeneous differential equations of first order and first degree. Solutions of linear differential equation ofthe type –

### Tips For MATHEMATICS:-

## Practice as much as you can

It doesn't matter your skill or present thesis. Politics as a whole involves many processes and relationships that no one person can fully comprehend. But don't give up! As a developing political scientist, you should read obscure economic books and political theorists. It pays to question these writings to gain information. Not even a private blog or notebook will do. Making time to broaden your horizons will aid you in your main concentration. Others' theories can often impact those who come after them. The same is true for political thought. It won't be wasted.

## Start by solving examples

Don't start with complex issues. Doing challenging math problems after reading the chapter will just discourage you. It may even make you despise Math. Start simply. Solve the textbook examples. Don't check up the answer beforehand. Compare your solution to the textbook solution or the reference book. Examine all of your steps, not just the major answer. After you master the steps, you can go on to the easier difficulties. Then you can go on to the harder ones.

## Clear all your doubts

It's easy to become trapped in Math. Don't allow doubts fester, have them cleared up right away. The faster you clear your doubts, the faster you improve. Ask your teacher, friends, or an app.

## Note down all formulae

You remember things even if they are unconscious when you see them enough. That's why some people choose to pin schematics or formulas to their walls. Make flash cards of all the equations in your textbook and hang them up in your room until the exam!

## Understand the derivation

You could believe the derivation isn't important for the exam, but it is. You need to grasp the rationale behind a formula to memorise it. Instead of memorising the formula, you need to understand why speed is a function of distance and time. Routine learning may cause exam forgetfulness, but clear concepts are usually easy to recall.

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