 IIT JEE Mathematics for Class XII
Subject Medium Mathematics Renu Mam AVAILABLE COMPLETE COURSE 232 XI XII XIII QUESTION BANK ATTACHED You May Pay in Installments through Credit Card
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Syllabus

### Mathematics Topic List

#### 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,
Questions
(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
29

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

Graphs:
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.)

#### Differentiation

 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.) 08 (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.) 09 (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.) 10 (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 01 (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.) 02 (a, b) Questions based on P1, P2 and Concepts of indefinite integration. (a-38 Min., b-33 Min.) 03 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.) 04 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.) 05 (a, b) Questions based on P4, Questions based on P5, P6. (a-41 Min., b-33 Min.) 06 (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 01 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.) 03 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 #Submatrix 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, Questions. 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. #Questions 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, Questions. (a-55 Min., b-39 Min.) 03 Cross product (Vector - product) of two vectors, Geometrical - interpretation, properties of cross-product, Questions. (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, Questions. 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 plane. 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.)

#### Probability

 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.)