Subject Mathematics Medium ENGLISH
Faculty Renu Mam Status AVAILABLE
Category TOPIC BASED COURSE Lecture 51
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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

Questions based on types of matrices and Algebra of Matrices.
Questions based on Matrix - multiplication, transpose of matrix, properties of transpose.

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


Questions (1), (2) and (3)
Solutions of questions No. (1), (2) and (3)
Question based on square matrices.

Introduction of determinants,
Expansion of 2x2 and 3x3 order determinants,
Properties of determinants.

(a) Questions on determinants
(b) Questions on determinants, product of 2 determinants, questions based on product of determinants.

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.

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,

(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) 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)


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

(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,

Cross product (Vector - product) of two vectors,
Geometrical - interpretation, properties of cross-product,

Direction cosines (DC’s) and direction -Ratios (DR’s) of a line segment, questions.

Vector equation of a line (parametric & non parametric form), Symmetrical form of a line (3-D Form)
Point of intersection of 2 lines,

Questions based on line.

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.

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.

Questions based on line & plane.

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

Questions, skew-lines, shortest distance (SD) between 2 skew-lines, condition for lines to be intersecting,
distance between two parallel lines.

Angle bisectors of two lines, Acute or obtuse angle bisectors. Questions

Scalar triple product (STP) of 3 vectors. Geometrical interpretation. Volume of parallelopiped. Properties of
STP. Vector-triple product of three vectors (VTP). Geometrical - Interpretation.

Questions on STP and VTP, Tetrahedron, its centroid, volume of tetrahedron, angle between any 2 faces of
regular tetrahedron.

(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.


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.

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.

Questions based on Conditional probability,
Questions based on dependent or independent events,
Law’s of total probability.

Baye’s theorem (Reverse theorem).

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.

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