Page 1
Mathematics/Applied Mathematics (319)
2
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
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 question will be
attempted.
Page 2
Mathematics/Applied Mathematics (319)
2
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
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 question will be
attempted.
Mathematics/Applied Mathematics (319)
3
Section B1: Mathematics
UNIT I: 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 of a function. Binary operations.
2. Inverse Trigonometric Functions
Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions.
Elementary properties of inverse trigonometric functions.
UNIT II: ALGEBRA
1. Matrices
Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric
and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple
properties of addition, multiplication and scalar multiplication. Non-commutativity of multiplication
of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square
matrices of order 2). Concept of elementary row and column operations. Invertible matrices and
proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).
2. Determinants
Determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, cofactors
and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square
matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples,
solving system of linear equations in two or three variables (having unique solution) using inverse of
a matrix.
UNIT III: CALCULUS
1. Continuity and Differentiability
Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse
trigonometric functions, derivative of implicit function. Concepts of exponential, logarithmic functions.
Derivatives of log x
and e
x
. Logarithmic differentiation. Derivative of functions expressed in parametric
forms. Second-order derivatives. R o l l e ’ s and L a g r a n g e ’ s Mean Value Theorems (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 (first derivative test motivated geometrically and second derivative
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.
Page 3
Mathematics/Applied Mathematics (319)
2
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
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 question will be
attempted.
Mathematics/Applied Mathematics (319)
3
Section B1: Mathematics
UNIT I: 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 of a function. Binary operations.
2. Inverse Trigonometric Functions
Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions.
Elementary properties of inverse trigonometric functions.
UNIT II: ALGEBRA
1. Matrices
Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric
and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple
properties of addition, multiplication and scalar multiplication. Non-commutativity of multiplication
of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square
matrices of order 2). Concept of elementary row and column operations. Invertible matrices and
proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).
2. Determinants
Determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, cofactors
and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square
matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples,
solving system of linear equations in two or three variables (having unique solution) using inverse of
a matrix.
UNIT III: CALCULUS
1. Continuity and Differentiability
Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse
trigonometric functions, derivative of implicit function. Concepts of exponential, logarithmic functions.
Derivatives of log x
and e
x
. Logarithmic differentiation. Derivative of functions expressed in parametric
forms. Second-order derivatives. R o l l e ’ s and L a g r a n g e ’ s Mean Value Theorems (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 (first derivative test motivated geometrically and second derivative
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.
Mathematics/Applied Mathematics (319)
4
3. Integrals
Integration as inverse process of differentiation. Integration of a variety of functions by substitution,
by partial fractions and by parts, only simple integrals of the 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 in finding the area under simple curves, especially lines, arcs of circles/parabolas/el-
lipses (in standard form only), area between the two above said curves (the region should be cleraly
identifiable).
5. Differential Equations
Definition, order and degree, general and particular solutions of a 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 of the type –
dy
? Py ? Q ,
where P and Q are functions of x or constant
dx
dx
? Px
dy
? Q ,
where P and Q are functions of y or constant
Page 4
Mathematics/Applied Mathematics (319)
2
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
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 question will be
attempted.
Mathematics/Applied Mathematics (319)
3
Section B1: Mathematics
UNIT I: 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 of a function. Binary operations.
2. Inverse Trigonometric Functions
Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions.
Elementary properties of inverse trigonometric functions.
UNIT II: ALGEBRA
1. Matrices
Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric
and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple
properties of addition, multiplication and scalar multiplication. Non-commutativity of multiplication
of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square
matrices of order 2). Concept of elementary row and column operations. Invertible matrices and
proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).
2. Determinants
Determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, cofactors
and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square
matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples,
solving system of linear equations in two or three variables (having unique solution) using inverse of
a matrix.
UNIT III: CALCULUS
1. Continuity and Differentiability
Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse
trigonometric functions, derivative of implicit function. Concepts of exponential, logarithmic functions.
Derivatives of log x
and e
x
. Logarithmic differentiation. Derivative of functions expressed in parametric
forms. Second-order derivatives. R o l l e ’ s and L a g r a n g e ’ s Mean Value Theorems (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 (first derivative test motivated geometrically and second derivative
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.
Mathematics/Applied Mathematics (319)
4
3. Integrals
Integration as inverse process of differentiation. Integration of a variety of functions by substitution,
by partial fractions and by parts, only simple integrals of the 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 in finding the area under simple curves, especially lines, arcs of circles/parabolas/el-
lipses (in standard form only), area between the two above said curves (the region should be cleraly
identifiable).
5. Differential Equations
Definition, order and degree, general and particular solutions of a 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 of the type –
dy
? Py ? Q ,
where P and Q are functions of x or constant
dx
dx
? Px
dy
? Q ,
where P and Q are functions of y or constant
Mathematics/Applied Mathematics (319)
4
UNIT IV: VECTORS AND THREE-DIMENSIONAL GEOMETRY
1. Vectors
Vectors and scalars, magnitude and direction of a vector. Direction cosines/ratios of vectors. Types
of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a
vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position
vector of a point dividing a line segment in a given ratio. Scalar (dot) product of vectors, projection
of a vector on a line. Vector (cross) product of vectors, scalar triple product.
2. Three-dimensional Geometry
Direction cosines/ratios of a line joining two points. Cartesian and vector equation of a line, coplanar
and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane. Angle
between (i) two lines, (ii) two planes, (iii) a line and a plane. Distance of a point from a plane.
Unit V: Linear Programming
Introduction, related terminology such as constraints, objective function, optimization, different types
of linear programming (L.P.) problems, mathematical formulation of L.P. problems, graphical method
of solution for problems in two variables, feasible and infeasible regions, feasible and infeasible
solutions, optimal feasible solutions (up to three non-trivial constrains).
Unit VI: Probability
Multiplications theorem on probability. Conditional probability, independent events, total probability,
B a y e ’ s theorem. Random variable and its probability distribution, mean and variance of haphazard
variable. Repeated independent (Bernoulli) trials and Binomial distribution.
Section B2: Applied Mathematics
Unit I: Numbers, Quantification and Numerical Applications
A. Modulo Arithmetic
? Define modulus of an integer
? Apply arithmetic operations using modular arithmetic rules
B. Congruence Modulo
? Define congruence modulo
? Apply the definition in various problems
C. Allegation and Mixture
? Understand the rule of allegation to produce a mixture at a given price
? Determine the mean price of a mixture
? Apply rule of allegation
D. Numerical Problems
? Solve real life problems mathematically
Page 5
Mathematics/Applied Mathematics (319)
2
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
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 question will be
attempted.
Mathematics/Applied Mathematics (319)
3
Section B1: Mathematics
UNIT I: 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 of a function. Binary operations.
2. Inverse Trigonometric Functions
Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions.
Elementary properties of inverse trigonometric functions.
UNIT II: ALGEBRA
1. Matrices
Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric
and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple
properties of addition, multiplication and scalar multiplication. Non-commutativity of multiplication
of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square
matrices of order 2). Concept of elementary row and column operations. Invertible matrices and
proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).
2. Determinants
Determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, cofactors
and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square
matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples,
solving system of linear equations in two or three variables (having unique solution) using inverse of
a matrix.
UNIT III: CALCULUS
1. Continuity and Differentiability
Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse
trigonometric functions, derivative of implicit function. Concepts of exponential, logarithmic functions.
Derivatives of log x
and e
x
. Logarithmic differentiation. Derivative of functions expressed in parametric
forms. Second-order derivatives. R o l l e ’ s and L a g r a n g e ’ s Mean Value Theorems (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 (first derivative test motivated geometrically and second derivative
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.
Mathematics/Applied Mathematics (319)
4
3. Integrals
Integration as inverse process of differentiation. Integration of a variety of functions by substitution,
by partial fractions and by parts, only simple integrals of the 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 in finding the area under simple curves, especially lines, arcs of circles/parabolas/el-
lipses (in standard form only), area between the two above said curves (the region should be cleraly
identifiable).
5. Differential Equations
Definition, order and degree, general and particular solutions of a 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 of the type –
dy
? Py ? Q ,
where P and Q are functions of x or constant
dx
dx
? Px
dy
? Q ,
where P and Q are functions of y or constant
Mathematics/Applied Mathematics (319)
4
UNIT IV: VECTORS AND THREE-DIMENSIONAL GEOMETRY
1. Vectors
Vectors and scalars, magnitude and direction of a vector. Direction cosines/ratios of vectors. Types
of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a
vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position
vector of a point dividing a line segment in a given ratio. Scalar (dot) product of vectors, projection
of a vector on a line. Vector (cross) product of vectors, scalar triple product.
2. Three-dimensional Geometry
Direction cosines/ratios of a line joining two points. Cartesian and vector equation of a line, coplanar
and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane. Angle
between (i) two lines, (ii) two planes, (iii) a line and a plane. Distance of a point from a plane.
Unit V: Linear Programming
Introduction, related terminology such as constraints, objective function, optimization, different types
of linear programming (L.P.) problems, mathematical formulation of L.P. problems, graphical method
of solution for problems in two variables, feasible and infeasible regions, feasible and infeasible
solutions, optimal feasible solutions (up to three non-trivial constrains).
Unit VI: Probability
Multiplications theorem on probability. Conditional probability, independent events, total probability,
B a y e ’ s theorem. Random variable and its probability distribution, mean and variance of haphazard
variable. Repeated independent (Bernoulli) trials and Binomial distribution.
Section B2: Applied Mathematics
Unit I: Numbers, Quantification and Numerical Applications
A. Modulo Arithmetic
? Define modulus of an integer
? Apply arithmetic operations using modular arithmetic rules
B. Congruence Modulo
? Define congruence modulo
? Apply the definition in various problems
C. Allegation and Mixture
? Understand the rule of allegation to produce a mixture at a given price
? Determine the mean price of a mixture
? Apply rule of allegation
D. Numerical Problems
? Solve real life problems mathematically
Mathematics/Applied Mathematics (319)
4
E. Boats and Streams
? Distinguish between upstream and downstream
? Express the problem in the form of an equation
F. Pipes and Cisterns
? Determine the time taken by two or more pipes to fill or
G. Races and Games
? Compare the performance of two players w.r.t. time,
? distance taken/ distance covered/ Work done from the given data
H. Partnership
? Differentiate between active partner and sleeping partner
? Determine the gain or loss to be divided among the partners in the ratio of their investment
with due
? consideration of the time volume/ surface area for solid formed using two or more shapes
I. Numerical Inequalities
? Describe the basic concepts of numerical inequalities
? Understand and write numerical inequalities
UNIT II: ALGEBRA
A. Matrices and types of matrices
? Define matrix
? Identify different kinds of matrices
B. Equality of matrices, Transpose of a matrix, Symmetric and Skew symmetric matrix
? Determine equality of two matrices
? Write transpose of given matrix
? Define symmetric and skew symmetric matrix
UNIT III: CALCULUS
A. Higher Order Derivatives
? Determine second and higher order derivatives
? Understand differentiation of parametric functions and implicit functions Identify dependent
and independent variables
B. Marginal Cost and Marginal Revenue using derivatives
? Define marginal cost and marginal revenue
? Find marginal cost and marginal revenue
C. Maxima and Minima
? Determine critical points of the function
? Find the point(s) of local maxima and local minima and corresponding local maximum and
local minimum values
? Find the absolute maximum and absolute minimum value of a function
UNIT IV: PROBABILITY DISTRIBUTIONS
A. Probability Distribution
? Understand the concept of Random Variables and its Probability Distributions
? Find probability distribution of discrete random variable
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