IIT JEE Mathematics for Class XI and XII
Subject Mathematics Medium ENGLISH
Faculty Renu Mam Status AVAILABLE
Category COMPLETE COURSE Lecture 453
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Sets and Relation

Lecture# Description Duration
01 Definition of set, Methods to represent sets :
(1) Roster form or tabular method
(2) Set builder (Property method), Inter-conversion of Roster form into set builder form or vice-versa;
Types of sets:
(1) Null Set (2) Singleton set
(3) Finite set & Cardinal number of set (4) Equivalent sets. (5) Equal sets
34 Minutes
02 Subsets, Proper subset, Total number of subsets, Idea of intervals:
(1) Close interval
(2) Open-interval
(3) Discrete interval or curly bracket,
Operation on sets (By venn-diagram)
(1) Union of 2 sets
(2) Intersection of 2 sets
(3) Set A and its complement
43 Minutes
03 (4) Set A but not B
(5) Set B but not A
(6) Neither A nor B
#Demorgan’s Law
(7) Atleast one set out of three sets A, B, C
(8) Atleast 2 sets out of 3 sets
(9) Exact 2 sets out of 3 sets
(10) Exact 1 set out of 3 sets
(11) Neither A, B nor C.
Laws of Algebra of sets
44 Minutes
04 Cartesian Product ordered pair, ordered triplets, Cartesian Product of 2 sets or 3 sets,
Introduction of Relations
52 Minutes
05 Relations, Total number of relations, types of relations:
(1) Void relation (2) Universal Relation
(3) Identity Relation (4) Reflexive Relation
(5) Symmetric Relation (6) Transitive Relation
(7) Equivalence Relation
1 Hrs 02 Minutes
06 Definition of function, Its domain and co-domain and range. 43 Minutes


Function and Inverse Trigonometric functions

Lecture# Description Duration
01 Definition of Function, Domain, Co-domain, Range, Mapping diagram, Graphical definition of function,
Rational (or Polynomial) Functions, Basic concepts, Rational inequalities, Steps to solve Rational-Inequalities.
 1 Hrs 14 Minutes
02 Solving Rational-inequalities (Non-repeated and repeated linear factors), How to take square and reciprocal
in case of inequalities.
 1 Hrs 04 Minutes
03 Modulus or Absolute value functions, Formulae of modulus-functions, Removal of Modulus-Functions, Graphs
of Modulus-Function, Modulus - Inequalities.
 1 Hrs 05 Minutes
04 Modulus-Equations and Inequalities.  55 Minutes
05 Irrational-functions, their domain and Range, Irrational Equations and inequalities, Determining domain of
irrational functions.
 1 hrs 03 Minutes
06 Irrational-Inequalities, Exponential & Logarithmic functions, their basic graphs, formulae.  1 hrs 05 Minutes
07 Formulae of Log functions, Log and exponential equations.  50 Minutes
08 Exponential and Log-inequalities when base is positive fractional or greater than one. 41 Minutes
09 (a) Log-inequalities when base is variable
(b) Log-inequalities when base is variable. Determining domain of Log-functions.

(a) 33 Minutes

(b) 48 Minutes

10 Greatest integer function (GIF), Basic graph, Formulae, Fractional Part function (FPF), Basic Graph, Formulae,
Signum-function, Basic graph. Questions.
 1 Hrs
11 (a,b) Questions on GIF, FPF and Signum functions.

(a) 39 Minutes

(b) 32 Minutes

12 (a) Trigonometric equations, General Solutions, Fundamental and General period of Basic T-Ratios,
(b) Questions the determining General and Particular solutions of T-Equations.

 (a) 1 Hr. 04 Minutes

(b) 32 Minutes.

13 (a) Questions, T-inequalities
(b) T-inequalities, Domain of T-Functions.

(a) 42 Minutes

(b) 35 Minutes

14 Inverse -trigonometric functions, condition for defining inverse of a function, classification of functions.
One-One (Injective) or many one functions, onto (Surjective) or into functions, bijective functions, Basic
Graphs of 6 inverse trigonometric - functions. Properties of ITF, Defining T (T–1(x)) or T–1 (T(x))
 1 Hrs 15 Minutes
15 Finding basic values of ITF, Domain of all types of functions.  1 hrs 06 Minutes
16 Domain of functions, Range of Functions
Method of determining Range of functions
M-1 Represent x or function of x in terms of y
M-2 Range by Using Monotonocity
 1 hrs 12 Minutes
17 M-3 Range of L / L, Q / L, L / Q,  Q / Q
M-4 Range of composite functions
 1Hrs 15 Minutes
18 Domain and Range of composite functions by defining them in one-interval or in different-different intervals.
(Using graphical method)
 1 Hrs 10 Minutes
19 Composite functions in different intervals.
Types of functions: (1) one-one (injective function)
Condition of injectivity by differentiation
(2) Onto (surjective) functions.
(3) Bijective functions. Inverse of a function
1 Hrs 17  Minutes
20 Number of 1-1 mappings, number of surjective (onto) mapping, questions on classification of functions.  1 hrs 04 Minutes
21 Questions on classification of functions and determining inverse of a function.  58 Minutes
22 Inequalities of Inverse trigonometric functions, graphs of y = T (T–1 (x)) = x (Non-Periodic Functions)
Graphs of y = T–1 (T(x)) (Periodic Functions)
 1 Hrs
23 Graphs of y = T–1 (T(x)), Questions,
Inter-conversion between various ITF’s.
 1 hrs 06 Minutes
24 Equal or Identical functions; Simplification of Miscellaneous ITF’s, Graphs.  1 hrs 11 Minutes
25 (a) Simplification of Miscellaneous ITF’s, Inverse-trigonometric functions of tan–1x ± tan–1y,
sin–1x ± sin–1y or cos–1x ± cos–1y, Questions
(b) Solving Inverse trigonometric equations.

(a) 51 Minutes

(b) 40 Minutes

26 Summation series of inverse-trigonometric functions, even or odd functions. 1 hrs 01  Minutes
27 Even or odd functions, periodic functions, fundamental or general periods of basic functions, properties
related to periodicity of functions.
1 Hrs 05 Minutes
28 Determining the fundamental period of functions, Range by period of function, functional equations to
determining period.
1 hrs 02  Minutes

(a) Functional-Equations.
(b) Questions on functional equations,

Symmetry of graphs.
Transformation of curves
(G1) Graph of y = f(x) + a
(G2) Graph of y = a f (x)
(G3) Graph of y = f (x + a)
(G4) Graph of y = f (ax)
(G5) Graph of y = –f(x)
(G6) Graph of y = f (–x)
(G7) Graph of y = | f(x)|
(G8) Graph of y = f(|x|)
(G9) Graph of y = f (–|x|)
(G10) Graph of |y| = f (x)

(a) 47 Minutes

(b) 54 Minutes 

30 Curve tracing using differential calculus.
Graph of maximum/minimum of functions between two or more than 2 functions.
1 Hrs 12 Minutes
31 Maximum-Minimum of a Curve, Miscellaneous graphs 54 Minutes

Limit, Continuity and Differentiability

Lecture# Description Duration
01 (a) Concept of Limit, Left Hand Side Limit (LHL) and Right Hand Side Limit (RHL) , Algebra on limits
(b) 7 Indeterminant forms, Steps to determining limit of a function when x→a, where to evaluate LHL & RHL separately (Doubtful points)

(a) 52 Minutes

(b) 36 Minutes

02 (a) Identify type of indeterminant forms, Method of solving Limits
(i) Factorisation (ii) Rationalization
(b) Questions on factorisation and Rationalisation
 (a-50 Min., b-25 Min.)
03 (a) M-3- Evaluate of limit when x →∞ or x→ –∞
(b) Questions based on method no.3
 (a-34 Min., b-33 Min.)
04 (a) M-4- Series expansion by Maclaurin’s Series, Series Expansion of Basic functions,
(b) Determining unknown parameters by series expansion.
M-5- Standard - Limits
(a-37 Min., b-27 Min.)
05 (a) Formulae of standard-limits, Questions based on standard limits.
(b) Standard limits using substitution method.
M-6- Limit in form of 1
 (a-47 Min., b-28 Min.)
06 (a) Questions on 1 form. L’Hospital’s rule (LH-Rule).
(b) Questions based on LH-Rule
 (a-36 Min., b-22 Min.)
07 (a) 0° or ∞° forms.
(b) Miscellaneous questions of limit
(a-41 Min., b-36 Min.)
08 Sandwitch Theorem ( or Squeeze - Play Theorem)
Continuity of a function y = f(x) at point x = a
Types of discontinuity:
(1) First kind of discontinuity (removable discontinuity) (In this case limit exist)
(A) Missing point discontinuity.
(B) Isolated point discontinuity.
(2) Non-Removable Discontinuity (Limit does not exist)
(A) Finite Non-removable discontinuity, Jump of discontinuity = | RHL – LHL |
(B) Infinite Non-removable discontinuity.
(C) Oscillating discontinuity.
Jump of discontinuity = | RHL – LHL |
 55 Minutes
09 (a, b) Continuity at a point,
Continuity in an interval, determining unknown parameters using concept of continuity at a point.
 (a-32 Min., b-18 Min.)
10 (a, b) Differentiability of a function at a point, Equation of tangent at a point,
Questions to check continuity and differentiability at a point
 (a-45 Min., b-20 Min.)
11 (a) Determining unknown parameters using concepts of continuity and differentiability at a point.
Continuity and differentiability of higher order derivatives.
(b) Questions based on LH rule and differentiation.
 (a-38 Min., b-30 Min.)
12 (a, b) Differentiability in an interval, questions based to check continuity and differentiability in an interval.  (a-29 Min., b-27 Min.)
13 (a) Graphical method to check differentiability,
Differentiability of maximum-minimum of two or more than 2 functions.
(b) Graphical method to check differentiability
 (a-32 Min., b-30 Min.)
14 (a) Determination of a function using differentiation
(b) Miscellaneous questions based on LCD.
(a-25 Min., b-24 Min.)
15 (a, b) Miscellaneous questions based on LCD.  (a-33 Min., b-34 Min.)



Lecture# Description Duration
01 (a) Some basic differentiation by using first principle (AB-Initio method), Rules of differentiation
(b) Formulae of differentiation, Properties of differentiation , Differentiation of Product of two functions,
Chain Rule, Differentiation of
u/v, Differentiation of composite functions,
Differentiation of Parametric functions, Differentiation of one function w.r.t. other functions.
 (a-30 Min., b-41.22 Min.)
02 Questions of Differentiation of functions.  55 Minutes
03 (a, b) Differentiation of Log-functions.  (a-29 Min., b-23 Min.)
04 (a) Derivative of inverse - functions.
(b) Derivative of inverse - functions by substitution method.
(a-16 Min., b-38 Min.)
05 (a) Derivative of Inverse - Functions by substitution method
(b) Derivative of Inverse - Functions and derivative of higher order Inverse functions.
(c) Questions based on differentiation of ITFs, Parametric differentiation
(a-25 Min., b-33 Min., c-25 Min.)
06 (a,b) Parametric Differentiation, Differentiation of Implicit functions.  (a-37 Min., b-21 Min.)
07 (a) Derivative of functions represented by infinite series, Differentiation of determinants.
(b) Higher order derivatives.
 (a-28 Min., b-25 Min.)
08 (a,b) Higher order derivatives.  (a-24 Min., b-25 Min.)


Application of Derivatives

Lecture# Description Duration
01 (a) Brief Revision of Straight Line and Tangent-Normal:
Equation of tangent and Normal to the curve y = f (x) at a point, Length of tangent,
Length of subtangent, Length of normal, Length of subnormal, Tangent to the curve at (0, 0)
(b) Questions based on concept of tangent and normal when point lies on the curve.
(a-27 Min., b-42 Min.)
02 (a) Questions based on tangent and normal when curve given in parametric form.
(b) Tangent and normal from an external point.
(a-26 Min., b-34 Min.)
03 (a) Questions based on tangents and normals from an external point.
(b) Tangent on the curve - intersecting the curve again.
(a-35 Min., b-23 Min.)
04 (a) Common-tangents.
(b) Angle of intersection of two curves; shortest -distance between 2 non-intersecting curves.
(a-36 Min., b-39 Min.)
05 (a) Rate of change
(b) Approximate value of a number, Monotonocity of a function, strictly increasing (SI),
Strictly decreasing (SD), Monotonically increasing (MI), Monotonically decreasing (MD) functions,
Monotonocity at a point and in an interval, Condition for monotonocity for differentiable functions,
Monotonocity of discontinuous functions.
(a-26 Min., b-46 Min.)
06 (a, b) Questions on monotonicity of function at a point or in an interval. (a-35 Min., b-39 Min.)
07 (a) Questions of Monotonocity.
(b) Proving inequalities by using monotonocity.
(a-35 Min., b-32 Min.)
08 (a) Concavity, Convexity and point of inflexion (POI) of curve.
(b) Curve tracing by using concept of differential calculus.
(a-30 Min., b-29 Min.)
09 (a, b) Rolle’s theorem, Langrange’s Mean Value theorem (LMVT) (a-30 Min., b-35 Min.)
10 (a, b, c) Maxima and minima at a point, local maxima and local minima and absolute maxima and absolute
minima. Range of a function in an interval. Using concept of maxima and minima.
(a-28 Min., b-20 Min., c-29 Min.)
11 (a, b) Questions. (a-28 Min., b-28 Min.)
12 (a) Questions of Maxima and Minima based on location of roots.
Theory of equations using maxima and minima.
(b) Questions.
(c) Optimization of Geometrical problems by maxima and minima.
(a-33 Min., b-40 Min., c-55 Min.)
13 (a, b) Geometry Problems. (a-43 Min., b-41 Min.)
14 Geometry Problems.  33 Minutes

Indefinite Integration

Lecture# Description Duration
01 (a) Concept of integration, Standard formulae
(b) Defining all standard formulae.
(a-34 Min., b-23 Min.)
02 (a, b) Basic integration directly formulae based. (a-39 Min., b-39 Min.)
03 (a) Substitution method; Formulae of some standard substitution.
(b) Questions based on substitution method.
(a-27 Min., b-33 Min.)
04 (a) Integral in the form of : ∫sinm x cosn x dx ; ∫ tanm x secn x dx
(b) Integral in the form of : ∫ xm(a + bxn )dx , Questions on substitution method.
(a-40 Min., b-31 Min.)
05 (a) Questions on substitution method in irrational functions.
(b) Questions on substitution method.
(a-34 Min., b-38 Min.)
06 (a) Integration by parts.
(b) Integration by parts, Using
(A) ∫ex (f(x) + f '(x))dx = f(x)ex + C   OR   (B) ∫(f(x) + xf '(x))dx = xf(x) + C
(a-35 Min., b-36 Min.)
07 (a) Questions based on integration by parts.
(b) Questions based on integration by parts, Integration of Rational function - by partial fraction method-
(i) When non-repeated linear factors in denominator
(ii) Repeated linear factors in denominator
(iii) Quadratic factors in denominator (D<0)
(a-29 Min., b-38 Min.)

(a) Questions on partial fraction method
Integration in the form of : ∫ dx ÷ ax2 + bx + c

Integration in the form of : ∫ (px+q)dx ÷ ax2+bx+c

(b) Integration in the form of : ∫ (x2 ± a2)dx ÷ x4+kx2+a4 or ∫ dx ÷ x4+kx2+a4

Integration in the form of : (a) ∫ dx ÷ x(xn + 1) (b) ∫ dx ÷ xn (1+xn)1/n (c) ∫ dx ÷ x2(xn+1)n-1/n

(a-44 Min., b-32 Min.)

(a) Integration of Irrational Functions
Integration in the form of : ∫ dx ÷ √ax2+bx+c OR ∫ √ax2+bx+c dx

Integration in the form of : ∫ (px+q)dx ÷ √ax2+bx+c OR ∫(px+q) √ax2+bx+c dx

(b) Integration in the form of :

(A) ∫ dx ÷ (px+q)√ax+b       (B)  ∫ dx ÷ (px2+qx+r)√ax+b

(C) ∫ dx ÷ (px+q)√ax2+bx+c (D)  ∫ dx ÷ (px2+qx+r)√ax2+bx+c

(c) Questions based on Integration of Irrational functions.
Integration in the form of : ∫ dx ÷ a + b sin2 x OR ∫ dx ÷ a + b cos2 x OR ∫ dx ÷ a cos2 x + b sin2 x OR ∫ dx ÷ a + b cos2 x + c sin2 x OR ∫ dx ÷ (a sin x + b cos x)2 OR ∫ f(tan x)dx ÷ a sin x + b sin x cos x + c cos2 x

(a-35 Min., b-25 Min.)

(a) Integration in the form of : ∫ dx ÷ a + bsin x OR ∫ dx ÷ a + bcos x

∫ dx ÷ asinx ± bcos x OR ∫ dx ÷ a sinx ± b cos x + c OR ∫ (p sin x + qcos x + r) ÷ (a cos x + b sin x + c) * dx

Integration in the form of :

∫ (a sin x + b) dx ÷ (a+b sin x)2 OR ∫ (a cos x+b) dx ÷ (a+b cos x)2

Integration in the form of ∫(sinx + cos x)f(sin2x)dx

(b) Integration in the form of :

∫ f(eax )dx OR ∫ (aex + be-x ) ÷ (pex + qe-x )*dx , Reduction Formulae.

(a-42 Min., b-38 Min.)
11 (a, b) Miscellaneous Questions (a-25 Min., b-38 Min.)
12 (a, b) Miscellaneous Questions (a-33 Min., b-29 Min.)


Definite Integration

Lecture# Description Duration

(a, b) Introduction of definite integral (DI), Geometrical interpretation of definite integral,
                         b              a
Property No. 1:  ∫ f(x)dx =- ∫ f(x)dx
                         a              b


                         b             b
Property No. 2:  ∫ f(x)dx = ∫ f(t)dt , Questions.
                         a             a

(a-49 Min., b-35 Min.)

(a, b) Questions based on P1, P2 and Concepts of indefinite integration.

(a-38 Min., b-33 Min.)

                                                    b             c          b
(a, b) Questions, property no. 3:  ∫ f(x)dx =  ∫ f(x)dx+∫ f(x)dx where a < c < b
                                                    a             b          c

(a-33 Min., b-38 Min.)

                                                                                           b             b
  Questions based on P-3, Property no. 4(King-Property): ∫ f(x)dx =  ∫ f(a+b-x)dx,
                                                                                           a             a

                                       a             a
Modified property no. 4 : ∫ f(x)dx =  ∫ f(a-x)dx
                                       0             0

Questions based on P4.

(a-44 Min., b-40 Min.)

(a, b) Questions based on P4,

Questions based on P5, P6.

(a-41 Min., b-33 Min.)

(a, b) Property No. 7 (Based on periodicity of function) :


 nT            T
 ∫ f(x)dx = n ∫ f(x)dx (where T = Period of function y = f(x))
 0              0

Walle’s formulae, Leibnitz theorem, Modified Leibnitz theorem.

(a-37 Min., b-52 Min.)
07 (a) Questions based on Leibnitz theorem.
(b) Definite Integrals as the limit of a sum (AB-initio method).
(a-27 Min., b-47 Min.)
08 Questions based on integral as Limit of a sum. (a-35 Min.)

Area Under the Curve

Lecture# Description Duration
01 (a,b) Quadrature, How to evaluate area under the curve with x-axis or with y-axis, area bounded by the
two intersecting curves, area bounded by the curves in different-2 conditions.
(a-37 Min., b-17 Min.)
02 (a, b, c) Questions based on area under the curves. (a-28 Min., b-24 Min., c-29 Min.)
03 (a, b) Questions, Questions based on determining parameters. (a-36 Min., b-29 Min.)
04 (a, b) Questions based on determining the parameters, area under the curves using inequalities. (a-36 Min., b-39 Min.)
05 (a, b) Area under the curves using functional inequalities, area bounded with f(x) and its inverse f–1 (x).
Miscellaneous Questions.
(a-30 Min., b-30 Min.)


Differential equation

Lecture# Description Duration
01 (a, b, c) Introduction of DE, Ordinary Differential Equation (ODE) and Partial Differential Equations (PDE),
Order and degree of DE, about constants, arbitrary constants and essential arbitrary constants,
Formation of differential equations, Methods of solving differential equations.
General solutions and particular solutions of differential equations.
Method no.1 : Variable separable form, in the form of dy÷dx= f(x).g(y).
(a-47 Min., b-18 Min., c-22 Min.)
02 (a, b) Method no. 2: (a) Reduces to variable separable form, i.e. in the form of dy÷dx = f(ax+by+c).
(b) Substitution method: in x2 + y2 = r2 , put x = r cos θ, y = r sin θ,
and in x2 – y2 = r2 , put x = r sec θ, y = r tan θ,
Method no. 3: Solution of Homogeneous differential equations, in the form of dy÷dx = f(y÷x) or dx÷dy=f(x÷y), Questions
(a-27 Min., b-34 Min.)
03 (a, b, c) Questions on method no. 3,
Method No. 4 :
Reduces to Homogeneous Differential equation, i.e. in the form of dy÷dx=ax+by+c÷Ax+By+k , Questions
Method no. 5 : Exact (direct) differential equations. Questions based on method no. 5.
(a-25 Min., b-34 Min., c-23 Min.)
04 (a, b) Method no. 6 : Linear differential equation, i.e. in the form of dy÷dx+Py=Q OR dx÷dy+Px=Q Method No.7 : Reduces to linear differential equations (Bernoulli’s equations) (a-40 Min., b-33 Min.)
05 (a, b, c) Geometrical applications of differential equations,
Tangent and normal to the curve y = f(x) at point (x, y), length of tangent,
Length of subtangent, Length of Normal, Length of subnormal, Radius-vector,
Higher Degree & order of differential equations, orthogonal trajectory (OT) of curves,
Clairaut’s differential equations.
(a-29 Min., b-35 Min., c-32 Min.)


Matrices and Determinants

Lecture# Description Duration

Definition of Matrix A = [ai j ]m x n
Its order, basic questions of formation of a matrix and based on its order.
Types of Matrices:
1. Row Matrix
2. Column Matrix
3. Null Matrix
4. Square Matrix : (a) Diagonal elements (b) Trace of square matrix and its properties
5. Diagonal Matrix: (a) Scalar Matrix (b) Identity or unit matrix and its properties.
6. Upper triangular matrix
7. Lower triangular matrix

# Algebra of matrices
(1) Comparable matrices
(2) Equal matrices
(3) Multiplication of scalar to a Matrix
(4) Addition and subtraction of matrices
(5) Multiplication of 2 matrices and properties of matrix multiplication

1:19 Hrs.
02 Questions based on types of matrices and Algebra of Matrices.
Questions based on Matrix - multiplication, transpose of matrix, properties of transpose.
(a-32 Min., b-42 Min.)

Questions based on Transpose and multiplication, some special types of square matrices :
(1) Symmetric matrix
(2) Skew - symmetric matrix
Properties of symmetric and skew symmetric matrices.
(3) Orthogonal matrix
(4) Nilpotent matrix
(5) Idempotent matrix
(6) Involutary matrix


1 Hr. 15 Min.
04 Questions (1), (2) and (3)
Solutions of questions No. (1), (2) and (3)
Question based on square matrices.
 54 Min.
05 Introduction of determinants,
Expansion of 2x2 and 3x3 order determinants,
Properties of determinants.
1 Hr. 35 Min.
06 (a) Questions on determinants
(b) Questions on determinants, product of 2 determinants, questions based on product of determinants.
(a-58 Min., b-45 Min.)
07 Questions on product of 2 determinants,
Differentiation and integration of determinants,
Summation of determinants,
System of Non-Homogenous Linear equations in 3 variables,
Cramer’s rule.
1 Hr. 2 Min.
08 System of linear equations in 2-variables,
Consistency and Inconsistency of linear equations,
Homogenous system of linear equations,
Trivial and Non-trivial solutions of Homogenous linear equations,
1 Hr. 1 Min.
09 (a) Adjoint of square matrix, inverse of a square matrix,
Properties of adjoint and Inverse of matrix,
Cancellation Law.
System of Linear equations by matrix method, questions.
(b) Questions, Elementary transformations along row (column),
Introduction of Rank of a matrix.
(c) Determination of Rank of a matrix.
(a-55 Min., b-39 Min., c-20 Min.)
10 (a) Consistency and Non-consistency of system of Linear equations by Rank method,
Solution of 3 equations in two variables.
(b) Matrices polynomial, characteristic matrix,
Caley-Hamilton theorem.
Inverse of a non-singular matrix by elementary transformation (along Row / Column) (Board Topic)
(a-52 Min., b-37 Min.)

Vectors - 3D

Lecture# Description Duration
01 Introduction of vector, types of vectors:
(1) Null vectors
(2) Unit Vector
Law’s of addition/subtraction in a parallelogram.
(3) Position vector (PV)
(4) Equal vectors
(5) Parallel or collinear vectors
1 Hr. 13 Min.
02 (a) (6) Coplanar vectors
(7) Reciprocal vectors
Geometry on vectors
(1) Distance formula
(2) Section formula (Internal section division and External section Division)
(3) Centroid
(4) Incentre.
Dot product (scalar-product) of two vectors.
Geometrical interpretation, projection of vector.
Component of vector.
(b) Projection and component of vector along and perpendicular to other vector,
Properties of dot product,
(a-55 Min., b-39 Min.)
03 Cross product (Vector - product) of two vectors,
Geometrical - interpretation, properties of cross-product,
 (1 Hr. 2 Min.)
04 Direction cosines (DC’s) and direction -Ratios (DR’s) of a line segment, questions.  (1 Hr. 20 Min.)
05 Vector equation of a line (parametric & non parametric form), Symmetrical form of a line (3-D Form)
Point of intersection of 2 lines,
50 Minutes
06 Questions based on line.  38 Minutes
07 Questions, Plane, Vector equation of a plane passing through a point and whose direction alongn n ,
General equation of plane, equation of a plane passing through 3 points,
Intercept form of plane, Condition of coplanarity of 4 points, angle between 2 planes,
Equation of plane parallel to given plane, Distance between two parallel planes, Perpendicular distance, Foot
of perpendicular, Image of a point w.r.t. plane. Angle bisectors of two planes.
 57 Minutes
08 Condition of acute or obtuse angle bisectors, position of points w.r.t. plane or angle bisector containing a
points; Angle between two planes, condition of line perpendicular to plane and condition of a line parallel to
Questions based on line and plane.
(1 Hr. 3 Min.)
09 Questions based on line & plane.  57 Minutes
10 Family of planes passing through line of intersection of 2 planes, symmetrical form of line, unsymmetrical
form of line, reduction of unsymmetrical form of line into symmetrical form.
Questions, Condition of co-planarity of two lines.
Equation of plane containing 2 lines. Questions
 56 Minutes
11 Questions, skew-lines, shortest distance (SD) between 2 skew-lines, condition for lines to be intersecting,
distance between two parallel lines.
 49 Minutes
12 Angle bisectors of two lines, Acute or obtuse angle bisectors. Questions  46 Minutes
13 Scalar triple product (STP) of 3 vectors. Geometrical interpretation. Volume of parallelopiped. Properties of
STP. Vector-triple product of three vectors (VTP). Geometrical - Interpretation.
(1 Hr. 11 Min.)
14 Questions on STP and VTP, Tetrahedron, its centroid, volume of tetrahedron, angle between any 2 faces of
regular tetrahedron.
(1 Hr. 5 Min.)
15 (a,b) Circum-radius and inradius of regular tetrahedron. Questions, Reciprocal-system of vectors,
Linearly Independent and Linearly dependent vectors (LILD), Sphere, Types of sphere,
Section of Sphere intersected by a plane, Questions of sphere.
(a-47 Min., b-60 Min.)



Lecture# Description Duration
01 Some definitions : (1) Experiment (2) Sample - space (3) Event (E)
Types of Events:
(a) Happening or occurance of an event
(b) Compliment (Non-occurance) of event,
Definition of Probability : p(A) =
Favourable elements of event A / Total elements
(c) Simple events
(d) Compound or mixed events
(e) Exclusive: Events
(f) Exhaustive events
(g) Equally likely events
(h) Independent events or dependent events
Questions based on permutation and combination.
(a- 47 Min., b-28 Min., c-26 Min., d-41 Min.)
02 Algebra of events:
(1) Event A
(2) Complement of event A
(3) Events A & B both
(4) Atleast event A or B
(5) Event A but not event B
(6) Event B but not event A
(7) Exactly one event out of 2 events
(8) None of events A or B
(9) Event A or B but not both
(10) Atleast one of the events A, B, C
(11) Exactly one event out of 3 events
(12) Exactly 2 events out of 3 events
(13) None of events out of 3 events.
(14) Occurance of events A & B but not C.
Questions based on Algebra of events,
Conditional probability, Multiplication theorems for dependent or Independent events, Complement Law,
Questions on Conditional Probability.
(a-34 Min., b-35 Min., c-25 Min., d-24 Min.)
03 Questions based on Conditional probability,
Questions based on dependent or independent events,
Law’s of total probability.
(a-26 Min., b-29 Min., c-31 Min., d-39 Min.)
04 Baye’s theorem (Reverse theorem). (a-27 Min., b-40 Min., c-24 Min., d-4 Min.)
05 Discrete - Random variable,
Probability - Distribution, Mean & Variance of discrete - random variable X,
Variance, Standard derivation,
#Binomial - Distribution, Mean and Variance of Binomial Distribution,
Questions based on them.
(a-35 Min., b- 32 Min., c-26 Min.)


Set Relation

Lecture# Description Duration
1 Definition of set, Methods to represent sets :
(1) Roster form or tabular method
(2) Set builder (Property method), Inter-conversion of Roster form into set builder form or vice-versa;
Types of sets:
(1) Null Set (2) Singleton set
(3) Finite set & Cardinal number of set (4) Equivalent sets. (5) Equal sets
 34 Minutes
02 Subsets, Proper subset, Total number of subsets, Idea of intervals:
(1) Close interval
(2) Open-interval
(3) Discrete interval or curly bracket,
Operation on sets (By venn-diagram)
(1) Union of 2 sets
(2) Intersection of 2 sets
(3) Set A and its complement
 43 Minutes
03 (4) Set A but not B
(5) Set B but not A
(6) Neither A nor B
#Demorgan’s Law
(7) Atleast one set out of three sets A, B, C
(8) Atleast 2 sets out of 3 sets
(9) Exact 2 sets out of 3 sets
(10) Exact 1 set out of 3 sets
(11) Neither A, B nor C.
Laws of Algebra of sets
 44 Minutes
04 Cartesian Product ordered pair, ordered triplets, Cartesian Product of 2 sets or 3 sets,
Introduction of Relations
 52 Minutes
05 Relations, Total number of relations, types of relations:
(1) Void relation (2) Universal Relation
(3) Identity Relation (4) Reflexive Relation
(5) Symmetric Relation (6) Transitive Relation
(7) Equivalence Relation
 1 Hrs 02 Minutes
06 Definition of function, Its domain and co-domain and range.  43 Minutes

Fundamentals of Mathematics

Lecture# Description Duration
01 Number systems:
(1) Natural numbers (2) Whole numbers (W) (3) Integers (I or Z)
(4) Prime Numbers (5) Composite numbers
(6) Co-prime numbers (Relatively prime numbers)
(7) Twin-prime numbers
(8) Rational numbers : (a) Terminating rational numbers (b) Repeating rational numbers
(9) Irrational numbers (Q’ or Qc)
(10) Real numbers (R)
(11) Complex numbers (C or Z)
Algebra of complex numbers, converting into a + ib (i = √-1) form, square root of a complex number.
 1 Hrs 24 Minutes
02 Basics of Mathematics - About the concept helpful to solve inequalities, Domain of a function,
About the functions - (1) Rational functions (2) Irrational functions (3) Polynomial functions
 58 Minutes
03 Some important identities, Factor or remainder theorems, Cramer’s method to solve linear equations in two
variables, Ratios and Proportion, Squaring in case of inequalities.
 53 Minutes
04 When we cross multiply the denominator incase of inequalities?
Rational (Polynomial) Inequalities - Steps to solving inequalities
(For Non-repeated and repeated linear factors), Questions
 54 Minutes
05 Questions of Rational inequalities containing repeated linear factors, Modulus functions (Absolute - Value
functions) , Domain, Range and Graphs of basic modulus functions, Removal of modulus functions, Properties
of Modulus functions, Equations based on |x| = a (a≥0)
 55 Minutes
06 Defining modulus functions, Removal of modulus, Basic Graphs and Graphs of combination of Modulus
functions, Modulus equations.
 a-14 Min., b-51 Min.
07 Modulus inequalities.  1 Hrs
08 Questions of Modulus - inequalities.  56 Minutes
09 (a) Irrational function - domain, Range and Graph of y = √x , Irrational equations.
(b) Irrational Inequalities.
 1 Hrs 02 Minutes
10 Exponential and Logarithmic functions, domain-range and graph of basic exponential & log functions,
Properties formulae, Simplification of log functions.
 53 Minutes
11 Basic questions to simplify the Log functions, Log-equations.  a-35 Min., b-19 Min.
12 Logarithmic and Exponential equations.  46 Minutes
13 Exponential and Log inequalities.  41 Minutes
14 Log-inequalities when base is variable, Domain of functions including irrational or log functions.  a-33 Min., b-48 Min.
15 Greatest integer function (GIF), Domain-Range and basic graph of GIF, Properties, Fractional-part function
(FPF), Domain-Range and Basic Graph, Properties, Signum function, Domain-Range and Graph.
 1 Hrs 01 Minutes
16 Questions based on GIF, FPF and Signum function.  a-39 Min., b-32 Min.


Quadratic Equation

Lecture# Description Duration
01 Polynomial functions, Roots (zeros), Solutions, Equation vs Identity, Questions,
Methods of finding roots (i) Factorisation
 1 hrs 08 Minutes
02 Methods of finding roots- (ii) Transformation method. (iii) Dharacharya Method (Perfect square),
 1 hrs 07 Minutes
03 Questions based on finding roots.  1 Hrs 02 Minutes

Nature of roots : in ax2 + bx + c = 0 (a≠0)
(1) When a, b, c, ∈ R
(2) When a, b, c, ∈ Q
(3) When a = 1, b, c, ∈ I and D is Perfect square of integer
(4) when a, b, c ∉ R
(5) when D1 + D2 ≥ 0 (in a1x2 + b1x+ c1 = 0 and a2x2 + b2x+ c2 = 0 where

D1 = b12 –4a1c1 and D2 = b2 –4a2c2)
(6) Intermediate Mean Value Theorem (IMVT)
Questions based on nature of roots.

 1 Hrs 03 Minutes
05 Plotting of quadratic expression (Graph) when a > 0 or a < 0
in y = ax2 + bx + c (a≠0), Range of y = ax2 + bx + c when x ∈ R
Sign of a, b, c, D, Range in an interval x ∈[x1, x2],
 1 Hrs 11 Minutes

Sign of quadratic expression, Range of
y =L/Q , y

Q/ Q

 1 hrs 10 Minutes
07 Range by substitution, condition of common roots-
(1) when 1 root common (2) when both the roots are common
Location of roots-
(1) When both the roots are greater than k (k∈R)
(2) When both the roots are less than k
(3) When 1 root < k and other root > k
(4) When both the roots lies in interval (k1, k2)
(5) When only 1 root lies in (k1, k2)
 1 Hrs 27 Minutes
08 Questions based on location of Roots,
Pseudo-Quadratic equation, Questions based on it.
 1 Hrs  26 Minutes


Sequence and Series

Lecture# Description Duration
01 Arithmetic progression (AP), Standard terms, General term or last term (tn or 𝓁) of AP, Condition for 3 terms
in AP, Arithmetic mean (AM) of n numbers, Middle terms, Sum of First n terms (sn) of AP, Properties of AP,
n Arithmetic means inserted between 2 numbers, sum of n Arithmetic mean inserted between two numbers,
Properties of AP.
 45 Minutes
02 Questions based on Arithmetic progression and their properties.  1 Hrs 07 Minutes
03 Summation series based on AP, Geometric progression (GP), Standard terms, General term of GP, Sum of
first n terms of GP, Sum of ∞ terms of GP, supposition of terms in GP, n Geometric means between 2 positive
numbers, Properties of GP.
 1 Hrs 09 Minutes
04 Questions based on GP and their properties.  41 Minutes
05 Summation Series based on G.P., Harmonic Progression (HP), General term of Harmonic Progression,
Harmonic Means of n numbers, Questions based on Harmonic Progression.
 59 Minutes
06 Relation between AM, GM, HM, Solving inequalities based on AM ≥ GM ≥ HM.
Arithmetic Geometric Progression (AGP), General Term, Sum of first n terms of AGP,
Sum of ∞ numbers of terms in AGP, Summation series of AGP.
 a- 43 Min., b-42 Min.

(a) Summation of series based on product of terms in GP but with non-AP; Summation of series, i.e.

             n                                                                             n      n      n        n 
     Sn = tr,  (Vn - Vn-1 )method, Evaluating the value of  1,  ∑r,  r2,  ∑r3,
             r=1                                                                         r=1    r=1   r=1    r=1

(b) Method of differences
(1) First difference in AP. (2) Second difference in AP
(3) First difference in GP. (4) Second difference in GP;
Questions Based on method of differences.

 a-37 Min., b-35 Min.
08 Miscellaneous Series  1 Hrs
09 Miscellaneous Series  34 Minutes



Lecture# Description Duration
01 Basic Trigonometric Ratios (T-Ratios), and Identities, Questions based on Basic Trigonometry identities,
elimination of angle θ.
 57 Minutes
02 Trigonometry Ratios of allied angles, General solutions on coordinate axes, values of Trigonometry Ratios in
[0, 90°],Questions based on allied angles, Graph of Trigonometry ratios, their domain-range and fundamental
 1 Hrs 17 Minutes
03 Compound angles of trigonometry ratios, transformation formulae, values of trigonometry ratios at 15°(π÷12),75°(5π÷12), Questions  a-35 Min., b-42 Min.

Multiple and sub-multiple angles,
Values of Trigonometry Ratios at θ = π÷8, θ = π÷24

θ = 52*10÷2, θ = 142*10÷2, value of sin 180 (180 = π÷10), cos360(360 = π÷5), Questions.

 a-53 Min., b-38 Min.
05 Questions based on multiple and sub-multiple angles.  60 Minutes
06 Questions.
Conversion of sin5A in terms of sinA and Conversion of cos5A in terms of cosA.
 a-32 Min., b-32 Min.
07 Conditional identities and Range of Trigonometric functions.  a-25 Min., b-34 Min.
08 Range by using concept of differentiation .  a-40 Min., b-19 Min.

Trigonometric series-
(1) Cosine product series,
(2) (A) Cosine summation series (B) Sine summation series
Questions, Trigonometric Equations,
General solutions on coordinate axes, General solution of sinθ = sin α, cosθ = cos α, tanθ = tan α.and

sin2 θ = sin2 α
cos2 θ = cos2 α
tan2 θ = tan2 α

 a-29 Min., b-38 Min.
10 Basic Trigonometric equations directly formula based.  a-24 Min., b-27 Min.
11 Trigonometric equations based on trigonometric identities,
Questions based on Boundary values, solving simultaneous trigonometric equations.
a-33 Min., b-25 Min.
12 Advanced Level Trigonometric equations.  a-34 Min., b-38 Min.
13 Advanced Level Trigonometric equations, Trigonometric-Inequalities. a-25 Min., b-41 Min.
14 Domain of trigonometric functions.  40 Minutes


Solutions of triangles

Lecture# Description Duration
01 About the triangle,
(1) Sine rule
(2) Area of ΔABC.
(3) Napier’s analogy (Law’s of tangent)
(4) Cosine-formula
(5) Projection formula
(6) T-Ratios of half- angles, Questions
 43 Minutes
02 Questions  a-53 Min.
03 Questions, m-n rule, circles connected to a triangle-
(1) Circumcircle
(2) Incircle
(3) Ex-circles
(4) Centroid
(5) orthocentre
(6) Circum-centre.
 a-31 Min., b-40 Min., c-34 Min.
04 (1) Length of angle Bisectors.
(2) Length of Medians.
(3) Length of altitudes,
Distances of special points from vertices (A, B, C) and sides (AB, AC, BC)
(1) Circumcentre (O), (2) Incentre (I) (3) Centroid (G) (4) Excentres (I1, I2, I3)
 49 Minutes
05 Questions a-32 Min., b-22 Min.
06 Questions, Pedal-triangle (ΔLMN), its all parameters.  a-44 Min., b-34 Min.
07 Ex-central-triangle (ΔI1 I2 I3), its all parameters,
Distance between two special points-
(1) Distance between circumcentre (o) & orthocentre (H),
(2) Distance between circumcentre (0) and Incentre (I)
(3) Distance between circumcentre and excentres (I1, I2, I3)
(4) Distance between orthocentre (H) and Incentre (I)
(5) Distance between centroid (G) and circumcentre (o)
 a-35 Min., b- Min.


Binomial theorem

Lecture# Description Duration
01 About factorial n (n!,⌊n ), Domain-Range and Properties of factorial n. About nCr, nPr, formulae based on n! ,
nCr and nPr, Binomial expansion (for n ∈ N), Pascal-Triangle, General term, mth term from ending, middle term
(for n odd, n even), Questions based on Binomial expansion and determining terms in Binomial expansion.
 a-50 Min., b-20 Min.
02 Questions based to determine middle term in Binomial expansion, Questions based to determine coefficient
of xr in Binomial expansion, Questions based to determine the term independent of x.
 a-36 Min., b-31 Min.
03 Questions based on determining coefficient in product of 2 Binomial expansions, Multinomial theorem.  a-25 Min., b-39 Min.
04 Coefficient determining by concept of permutation and combination and by using multinomial theorem; total
number of terms in multinomial expansion; Number of terms free from fractional or irrational powers in
Binomial expansion.
a-34 Min., b-28 Min.
05 Numerically-Greatest term in the expansion of (x + a)n (n ∈ N), Algebraically - Greatest and least term in the
expansion of (x + a)n (n ∈ N); Questions based on Ι + ƒ .
 a-31 Min., b-35 Min.
06 Questions based on Ι + ƒ , exponent of prime number (p) in ⌊n ; Questions based on divisibility and remainder,
Last digit by cyclicity, Last digit, Last two digits, Last 3 digits in a number.
 a-29 Min., b-43 Min.
07 Summation of series.  a-42 Min., b-33 Min.
08 Questions based on Sigma, Summation of Binomial Coefficients taken two at a time, Summation when
upper index is variable.
 a-47 Min., b-35 Min.
09 Questions based on summation of Binomial coefficients taken two at a time when upper index is variable.  32 Minutes
10 Double-Sigma, Binomial expansion for negative or fractional power, Some-important expansions,
Questions based on determining
Coefficient in negative or fractional power in Binomial expansion.
a-43 Min., b-34 Min.


Straight lines

Lecture# Description Duration
01 Point, Rectangular - Cartesian Coordinate system, Parametric and Polar Coordinates of a point, distance
between 2 points, section formula, Harmonic conjugate, Questions, About the Quadrilateral, Area of triangle,
Condition of collinearity of three points, Concurrent lines, condition of concurrency of three lines.
 a-44 Min., b-38 Min.
02 Area of quadrilateral, Area of n sided polygon.  29 Minutes
03 Special points of triangle :
(1) Centroid (G) (2) Incentre (I) (3) Excentres (I1, I2, I3)
(4) Orthocentre (H) (5) Circum-centre(o)
Types of straight lines-
(1) General equation
(2) Slope - intercept form
(3) (a) Equation of a line parallel to x-axis.
(b) Equation of line perpendicular to x-axis.
(c) Equation of line coincident to x-axis.
(d) Equation of line coincident to y-axis.
(e) Equation of coordinates axes.
(4) Slope point form
(5) Two points form
(6) Determinant form
(7) Intercept form
(8) Normal or Perpendicular form.
Angle between two lines, condition of two lines to be parallel or perpendicular.
 a-43 Min., b-46 Min.
04 Condition of lines to be intersecting, Parallel, Coincident, Measurement of interior angle of Δ,
Questions based on point, special points and types of lines.
a-41 Min., b-49 Min.
05 Questions based on special points and types of lines.  a-45 Min., b-37 Min.
06 Equations of lines passing through P(x1, y1) and making an angle α with the line y = mx + C, slope of a line
equally inclined to the two given lines, Questions.
Parametric or distance form of a line.
 a-35 Min., b-49 Min.
07 Perpendicular distance, foot of perpendicular, foot of perpendicular, image of a point (x1, y1) w.r.t. line
ax + by + c = 0, distance between two parallel lines, Area of parallelogram, About the parallelogram, positions
of two points w.r.t. line/plotting of linear-inequations, condition that a point lies inside of a triangle.
 a-48 Min., b-37 Min.
08 Questions based on perpendicular distance, foot of perpendicular and image.  a-45 Min., b-34 Min.
09 Locus, Steps how to evaluate locus of a point, questions, angle bisectors, types of angles bisectors, how to
identify type of angle bisector, angle bisectors containing a point P(x1, y1).
 a-43 Min., b-51 Min.
10 Questions based on angle-bisectors, family of lines (concurrent lines), Questions based on family of lines.  a-47 Min., b-24 Min.
11 Pair of lines (combined or joint equations), Non-homogenous equation of second degree, homogeneous
equation of second degree, angle between pair of lines, separate equations from second degree, condition
that second degree non-homogenous equations represents pair of lines, point of intersection of pair of lines,
combined equations of angles bisectors of pair of lines.
 a-39 Min., b-34 Min.
12 Questions, distance between two parallel pairs of lines, Homogenisation.  a-39 Min., b-20 Min.


Lecture# Description Duration
01 Definition of Circle, Types of Circles-
(1) Centre - Radius form
(2) General equation : Equation of Circle passing through 3 non-collinear points.
 39 Minutes
02 Basic questions on circle, types of circles :
(3) Diameter form
(4) Standard equation of circle
(5) Parametric Form
(6) Point - Circle,
Intercepts formed by circle on coordinate axes, position of points w.r.t. circle, Some Important notes related
to Circle, Different-2 positions of circles, Questions.
 a- 60 Min., b- 25 Min.
03 Questions  a-40 Min., b-25 Min.
04 Position of Line w.r.t. Circle, Length of chord Intercepted by the circle on, Tangent, Types of tangent-
(1) Slope - Form,
(2) Point - Form, Normal of Circle
(3) Parametric - Form
(4) Equation of tangent to the curve at (0, 0), number of tangents to the circle,
Questions, Application of tangents -
(1) Length of tangents
(2) Power of points P(x1, y1) w.r.t. circle
(3) Area of quadrilateral PACB
(4) Angle between two tangents
(5) Chord of contact
(6) Equation of chord whose mid point is given
(7) Director circle
(8) Separate equations of tangents
(9) Combined equations or pair of tangents
(10) Equation of circle circumscribing the ΔPAB
(11) PA.PB = PC. PD = PT2
(12) OA.OB = OC.OD
(13) Area of triangle formed by pair of tangents with their chord of contact, Questions
 a-45 Min., b-45 Min., c-37 Min
05 Questions  a-38 Min., b-32 Min.
06 Questions, Position of 2 circles and their common tangents-
(1) When 2 circles are separated of each other, length of external and internal common tangent
(2) When two circles touches externally
(3) When two circles intersect at two real and distinct points, common chord of two circles, equation of
common chord and its length, maximum length of common chord, angle of intersection of 2 circles, orthogonal
circles and condition of orthogonality,
(4) When two circles touches internally
(5) When one circle lies completely inside of other, Questions.
 a-58 Min., b-38 Min.
07 Questions, Family of Circles-
(1) Equations of family of circles passing through the point of intersection of circles, s = 0 and line L = 0
(2) Equation of family of circles passes through 2 points A & B.
(3) Equation of family of circles passes through point of intersection of 2 circles.
(4) Equation of family of circles touching a curve at a point, Questions
 a-44 Min., b-30 Min.
08 Questions, Radical axis/Radical centre, Equation of circle cuts three given circles orthogonally, pole and
 a-30 Min., b-32 Min.

Conic sections

Lecture# Description Duration
01 Introduction of Conic Section, Definition of Conic-Section, General equation of conic section, Locus of a
moving point P will be conic when focus(s) lies on directrix and does not lies on directrix,
Questions, some definitions related to conic -section
(1) Focus (2) Directrix (3) Axis (4) Vertex (5) Centre
(6) Focal- chord (7) Double- ordinate (8) Latus-Rectum (LR)
Standard parabola - Its all parameters, two questions.
a-36 Min., b-32 Min., c-25 Min.
02 Questions based on parameters of parabola, position of point w.r.t. parabola, Questions. a-25 Min., b-27 Min., c-25 Min.
03 (1) Parametric equation of a chord
(2) Length of parametric chord
(3) Focal chord
(4) Minimum length of focal chord
(5) Focal distance
(6) 𝓁 (LR) = 2 (HM of 𝓁1 & 𝓁2), where 𝓁1 = PS, 𝓁2 = QS and P & Q are 2 moving points on parabola, S = focus,
(7) (a) If focal chord of parabola makes ∠angle with its axis then 𝓁(LR) = 4a cosec2 α.
(b) Length of focal chord at a distance p from vertex is  4a3÷p2
(8) If P1Q1 and P2Q2 are two focal chords of parabola y2 = 4ax then chords P1P2 & Q1Q2 intersect on its
(9) If P1P2 and Q1Q2 are two focal chord of parabola are at right angle then area of quadrilateral P1Q1 and P2Q2
is minimum when chords are inclined at an angle π/4 to the axis and its minimum area is 32a2.
(10) The circle described on any focal chord of parabola as diameter touches its directrix.
(11) A line having slope (m) passes through focus(s) cuts the parabola at two real & distinct points
if m ∈ R-{0}, Questions
 a-27 Min., b-33 Min.
04 Questions, Position of line w.r.t. Parabola, Condition of tangency
Types of tangent - (1) Point form (2) Parametric form
Questions based on tangents.
a-31 Min., b-40 Min., c-23 Min.
05 Questions based on tangents, common tangents of two curves,
Properties of tangents : P1, P2, P3, P4
a-40 Min., b-40 Min.
06 Properties of tangents: P5, P6, P7, P8
Questions based on Properties of tangents, Normal, Types-
(1) Point form (2) Parametric form (3) Slope- form, condition of normality,
Questions based on normals, properties of normal, P1, P2, P3 (a, b, c, d), P4, P5 .
 a-32 Min., b-29 Min., c-28 Min.
07 Properties of Normal- P6 , P7 (a, b), P8, P9- Reflection property,
P10, P11 (a, b, c)
P-11- Condition of three real & distinct normal to parabola, Questions based on normal and its properties,
(1) Number of tangents to a parabola,
(2) Pair of tangents
(3) Director - Circle
(4) Chord of contact
(5) Chord whose mid point is given, Questions
 a-38 Min., b-20 Min., c-42 Min., d-34 Min.
08 Introduction of ellipse and hyperbola, standard ellipse (when a > b and a < b) and standard hyperbola and
conjugate hyperbola, its basic parameters, auxiliary - Circle/Parametric coordinates of ellipse and hyperbola,
Alternate definition of ellipse and hyperbola, Some important notes, Questions determining the basic parameters
of ellipse and hyperbola.
 a-38 Min., b-31 Min., c-30 Min., d-25 Min.
09 Basic questions on ellipse and hyperbola, Questions based on Locus,
Questions based on Parametric coordinates.
 a-36 Min., b-34 Min., c-32 Min., d-18 Min.
10 Parametric equation of chord of ellipse and hyperbola, questions on parametric chord, position of line w.r.t.
ellipse Hyperbola, Condition of tangency, types of tangent-
(1) Slope form (2) Point form (3) Parametric Form,
Properties of tangents, Questions based on tangents.
 a- 38 Min., b-47 Min.
11 Questions based on tangents and its properties, Pair of tangents, Equation of chord of contact, Equation of
chord whose mid point is given
#Director Circle, Questions, Normal of ellipse and Hyperbola, Types-
(1) Point Form (2) Parametric Form (3) Slope Form.
 a-43 Min., b-39 Min., c-14 Min.
12 Questions based on normal of ellipse Hyperbola, Reflection Property of ellipse - Hyperbola,
Asymptotes of Hyperbola and conjugate Hyperbola, Properties of Asymptotes,
 a-26 Min., b-44 Min., c-34 Min.
13 Rectangular (Equilateral) Hyperbola, Rectangular Hyperbola considered coordinate axes as its asymptotes,
its all parameters, tangents and normals, Questions.
 a-32 Min., b-31 Min.

Permutations and combinations

Lecture# Description Duration
01 Introduction of factorial n ( ⌊n or n!) , nCr, nPr, Physical interpretation of n!, nCr, nPr.  19 Minutes
02 Fundamental - Principles of counting
(i) Multiplication - Rule (ii) Addition- Rule
Basic Questions based on multiplication and addition-Rule; Sample-space.
 a-51 Min., b-49 Min.
03 Questions, Number Problems.  a-40 Min., b-35 Min.
04 Number problems based on divisible by 3, 4, 5, 25,
Theorem-1: Selection and Permutation of r things out of n.
Theorem-2 : Permutation of n things in which some things are of same kind.
 a-46 Min., b-28 Min.
05 Questions considering the word “RAKESH MODI” Questions, problem of forming words of 7 letters taking 3
vowels and 4 consonants using letters of word : “DIFFERENTIATION”.
Problem of forming the words each consisting 3 consonants and 3 vowels by using letters of words
 a-42 Min., b-25 Min., c-35 Min.
06 Rank (Position) of a word or numbers, sum of the numbers formed (No repetition or when repetition allowed),
Circular Permutation.
 a-35 Min., b-33 Min., C-35 Min.
07 Number of selection of r consecutive things out of n distinct things, Geometrical Problems Number of total
lines, number of diagonals, number of triangles
(a) One side common with given polygon
(b) Two sides common with given polygon
(c) Three sides common with given polygon.
(d) None of the side common with given polygon.
Chess board problems - Number of total rectangles, Number of total squares.
Problem based on moving from left bottom corner to the right top corner in a chess board.
 a-48 Min., b-57 Min.
08 Groupings & distribution of n differents things into groups or bundles.  a-30 Min., b-29 Min., c-27 Min.
09 Selection of none, one or more things when given things are different or identical, Total number of divisors,
Total number of proper divisors, Sum of total divisors, Number of ways in which a number (N) can be resolved
as a product of two factors which are relatively prime or co-prime.
 a-48 Min., b-45 Min., c-14 Min.
10 Multinomial theorem of permutation and combination, Beggar’s Method  a-45 Min., b-43 Min.
11 Questions based on multinomial theorem, Dearrangement of n different things.  a-31 Min., b-23 Min.
12 Miscellaneous questions  a-33 Min., b-34 Min.
13 Miscellaneous questions a-30 Min., b-17 Min.

Complex number

Lecture# Description Duration
01 Introduction of complex number, about iota (i), Algebra of complex numbers-
(1) Addition/subtraction (2) Multiplication
(3) Conjugate of a complex number (4) Division
(5) Equality of two complex numbers (6) Square root of a complex numbers,
Questions to solving complex equations.
a-43 Min., b-28 Min., c-23 Min.
02 Questions, Representation of Complex number (Geometrical interpretation of Complex number)
(1) Cartesian form
(2) Polar or parametric form
(3) Euler’s form
# Demoiver’s theorem, Questions.
a-48 Min., b-38 Min.
03 Properties of modulus/conjugate, Modulus - Inequalities (Triangular Inequalities), Properties of argument of
complex number, Interconversion of complex number (z) into Cartesian form (x, y) or vice-versa.
 a-41 Min., b-29 Min.
04 Questions based on Properties of Modulus, conjugate, argument of complex number and modulus inequalities  a-46 Min., b-47 Min.
05 Geometrical meaning of arg(z) = θ.
Solving questions graphical, cube-roots of unity, cube-roots of –1, Properties,
Questions based on cube roots of 1 and cube roots of –1.
 a-46 Min., b-39 Min., c-29 Min.
06 nth roots of unity, Properties, Questions based on nth roots of unity, rotation theorem (Geometrical interpretation
of ei θ).
Questions based on Rotation theorem.
 a-41 Min., b-21 Min., c-44 Min.
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