# ICSE Board Math Syllabus for Class 12

## ICSE Board Syllabus for Class 12 Math

COURSE STRUCTURE

MATHEMATIC

CLASS 12

SECTION A

Determinants and Matrices

(i) Determinants

• Order.

• Minors.

• Cofactors.

• Expansion.

• Properties of determinants.

• Product of determinants (without proof).

• Simple problems using properties of

determinants e.g. evaluate b

• Cramer's Rule

. Solving simultaneous equations in

2 or 3 variables,

. Consistency, inconsistency.

. Dependent or independent.

NOTE: the consistency condition for three

equations in two variables is required to be

covered.

(ii) Matrices

Types of matrices (m x n; m, n ≤ 3),

order; Identity matrix, Diagonal matrix.

• Symmetric, Skew symmetric.

• Operation – addition, subtraction,

multiplication of a matrix with scalar, multiplication

of two matrices

(the compatibility).

• Singular and non-singular matrices.

•Existence of two non-zero matrices

whose product is a zero matrix.

• Martin’s Rule (i.e. using matrices)

- a 1x + b 1y + c 1z = d 1.AdjA/A

a 2x + b 2y + c 2z = d 2 .

a 3x + b 3y + c 3z = d 3 .

AX = B ⇒ X = A −1 B

NOTE: The conditions for consistency of

equations in two and three variables, using

matrices, are to be covered

2 Boolean Algebra

Boolean algebra as an algebraic structure,

principle of duality, Boolean function.

Switching circuits, application of Boolean

algebra to switching circuits.

3 Conics

• As a section of a cone.

• Definition of Foci, Directrix, Latus Rectum.

. Simple problems based on above.

• PS = ePL where P is a point on the conics, S

is the focus, PL is the perpendicular distance

of the point from the directrix.

(i) Parabola

•e =1, y2 = 4ax, x2 = 4ay, y2 = -4ax,

x2 = -4ay, (y -β)2 = 4a (x - α),

(x - α)2 = 4a (y - β).

•Rough sketch of the above.

•The latus rectum; quadrants they lie

in; coordinates of focus and vertex;

and equations of directrix and the

axis.

•Finding equation of Parabola when

Foci and directrix are given.

•Simple and direct questions based on

the above.

(ii) Ellipse

•Cases when a > b and a < b.

•Rough sketch of the above.

•Major axis, minor axis; latus rectum;

coordinates of vertices, focus and

centre; and equations of directrices

and the axes.

•Finding equation of ellipse when

focus and directrix are given.

•Simple and direct questions based on

the above.

•Focal property i.e. SP + SP′ = 2a.

(iii) Hyperbola

x2 y 2 − = 1 , e > 1, b 2 = a 2 (e 2 − 1) a 2 b2

• Cases when coefficient y2 is negative

and coefficient of x2 is negative.

• Rough sketch of the above.

• Focal property i.e. SP - S’P = 2a.

• Transverse and Conjugate axes; Latus

rectum; coordinates of vertices, foci

and centre; and equations of the

directrices and the axes.

x2 y2+ 2 = 1 , e <1, b2 = a 2 (1 − e2 )2a b

• General second degree equation

ax 2 + 2hxy + by 2 + 2 gx + 2 fy + c = 0

represents a parabola if h2 = ab,

ellipse if h2 < ab, and hyperbola if h2 >ab.

Condition that y = mx + c is a tangent

to the conics.

. Inverse Trigonometric Function

• Principal values.

• sin-1x, cos-1x, tan-1x etc. and their graphs.

sin-1x = cos −1 1 − x 2 = tan −1

. Addition formulae.

(cos x ± cos y = cos ( xy msin-1x ± sin-1 y = sin -1 x 1 − y2 ± y 1 − x2-1

similarly tan-1x ± tan-1 y = tan-1

. Similarly, establish

formulae for 2sin-1x, 2cos-1x, 2tan-1x,

3tan-1x etc. using the above formula.

• Application of these formulae.

5 Calculus

(i) Differential Calculus

•Revision of topics done in Class XI -

mainly the differentiation of product of

two functions, quotient rule, etc.

• Derivatives of trigonometric functions.

• Derivatives of exponential functions.

• Derivatives of logarithmic functions.

• Derivatives of inverse trigonometric

functions - differentiation by means of

substitution.

• Derivatives of implicit functions and

chain rule for composite functions.

• Differentiation of a function with

respect to another function e.g.

differentiation of sinx3 with respect to x3.

. Logarithmic Differentiation - Finding

dy/dx when y = x

•Successive 2nd order.

• L'Hospital's theorem.

Rolle's Mean Value Theorem - its

geometrical interpretation.

•Lagrange's Mean Value Theorem - its

geometrical interpretation.

• Maxima and minima.

(ii) Integral Calculus

• Revision of formulae from Class XI.

• Integration of 1/x, ex, tanx, cotx, secx,

cosecx.

• Integration by parts.

• Integration by means of substitution.

• Properties of definite integrals.

Problems based on the following

properties of definite integrals are to be

covered.

• Application of definite integrals - area

bounded by curves, lines and coordinate

axes is required to be covered.

6 Correlation and Regression

• Definition and meaning of correlation and

regression coefficient.

• Coefficient of Correlation by Karl Pearson.

• Rank correlation by Spearman’s (Correction

included).

• Lines of regression of x on y and y on x.

NOTE: Scatter diagrams and the following

topics on regression are required.

i) The method of least squares.

ii) Lines of best fit.

iii) Regression coefficient of x on y and y on x.

iv) b xy × b yx = r 2 , 0 ≤ b xy × b yx ≤ 1

v) Identification of regression equations

7. Probability

• Random experiments and their outcomes.

• Events: sure events, impossible events,

mutually

exclusive events, independent

events and dependent events.

• Definition of probability of an event.

• Laws of probability: addition and

multiplication laws, conditional probability

(excluding Baye’s theorem).

8 Complex umbers

• Argument and conjugate of complex numbers.

• Sum, difference, product and quotient of two

complex numbers additive and multiplicative

• Simple locus question on complex number;

proving and using -

• Triangle inequality.

• Square root of a complex number.

• Demoivre’s applications.

• Cube roots of unity: 1, ω , ω 2 ; application

problems.

9 Differential Equations

• Differential equations, order and degree.

• Solution of differential equations.

• Variable separable.

• Homogeneous equations and equations

reducible to homogeneous form.

• Linear form functions of x only. Similarly for dx/dy.

NOTE: Equations reducible to variable

separable type are included. The second order

differential equations are excluded.

SECTIO B

10. Vectors

• Scalar (dot) product of vectors.

• Cross product - its properties - area of a

triangle, collinear vectors.

• Scalar triple product - volume of a

parallelopiped, co-planarity.

Proof of Formulae (Using Vectors)

• Sine rule.

• Cosine rule

• Projection formula

• Area of a ∆ = ½absinC

NOTE: Simple geometric applications of the

above are required to be covered.

11. Co-ordinate geometry in 3-Dimensions

(i) Lines

NOTE: Symmetric and non-symmetric forms of

lines are required to be covered.

(ii) Planes

• Cartesian and vector equation of a

plane.

• Direction ratios of the normal to the

plane.

• One point form.

• Normal form.

• Intercept form.

• Distance of a point from a plane.

• Angle between two planes, a line and a

plane.

• Equation of a plane through the

intersection of two planes i.e. -

P1 + kP2 = 0.

Simple questions based on the above.

12 Probability

Baye’s theorem; theoretical probability

distribution, probability distribution function;

binomial distribution – its mean and variance.

12. Cartesian and vector equations of a line

through one and two points.

Coplanar and skew lines.

Conditions for intersection of two lines.

Shortest distance between two lines.

NOTE: Theoretical probability distribution is to

be limited to binomial distribution only.

SECTION C

13 Discount

True discount; banker's discount; discounted

value; present value; cash discount, bill of

exchange.

NOTE: Banker’s gain is required to be covered.

14. Annuities

Meaning, formulae for present value and

amount; deferred annuity, applied problems on

loans, sinking funds, scholarships.

NOTE: Annuity due is required to be covered.

15. Linear Programming

Introduction, definition of related terminology

such as constraints, objective function,

optimization, isoprofit, isocost lines; advantages

of linear programming; limitations of linear

programming; application areas of linear

programming; 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, optimum feasible solution.

16. Application of derivatives in Commerce and

Economics in the following

Cost function, average cost, marginal cost,

revenue function and break even point.

17. Index numbers and moving averages

• Price index or price relative.

• Simple aggregate method.

• Weighted aggregate method.

• Simple average of price relatives.

• Weighted average of price relatives

(cost of living index, consumer price index).

NOTE: Under moving averages the following

are required to be covered:

• Meaning and purpose of the moving averages.

• Calculation of moving averages with the

given periodicity and plotting them on a

graph.

• If the period is even, then the centered moving

average is to be found out and plotted.

MATHEMATIC

CLASS 12

SECTION A

Determinants and Matrices

(i) Determinants

• Order.

• Minors.

• Cofactors.

• Expansion.

• Properties of determinants.

• Product of determinants (without proof).

• Simple problems using properties of

determinants e.g. evaluate b

• Cramer's Rule

. Solving simultaneous equations in

2 or 3 variables,

. Consistency, inconsistency.

. Dependent or independent.

NOTE: the consistency condition for three

equations in two variables is required to be

covered.

(ii) Matrices

Types of matrices (m x n; m, n ≤ 3),

order; Identity matrix, Diagonal matrix.

• Symmetric, Skew symmetric.

• Operation – addition, subtraction,

multiplication of a matrix with scalar, multiplication

of two matrices

(the compatibility).

• Singular and non-singular matrices.

•Existence of two non-zero matrices

whose product is a zero matrix.

• Martin’s Rule (i.e. using matrices)

- a 1x + b 1y + c 1z = d 1.AdjA/A

a 2x + b 2y + c 2z = d 2 .

a 3x + b 3y + c 3z = d 3 .

AX = B ⇒ X = A −1 B

NOTE: The conditions for consistency of

equations in two and three variables, using

matrices, are to be covered

2 Boolean Algebra

Boolean algebra as an algebraic structure,

principle of duality, Boolean function.

Switching circuits, application of Boolean

algebra to switching circuits.

3 Conics

• As a section of a cone.

• Definition of Foci, Directrix, Latus Rectum.

. Simple problems based on above.

• PS = ePL where P is a point on the conics, S

is the focus, PL is the perpendicular distance

of the point from the directrix.

(i) Parabola

•e =1, y2 = 4ax, x2 = 4ay, y2 = -4ax,

x2 = -4ay, (y -β)2 = 4a (x - α),

(x - α)2 = 4a (y - β).

•Rough sketch of the above.

•The latus rectum; quadrants they lie

in; coordinates of focus and vertex;

and equations of directrix and the

axis.

•Finding equation of Parabola when

Foci and directrix are given.

•Simple and direct questions based on

the above.

(ii) Ellipse

•Cases when a > b and a < b.

•Rough sketch of the above.

•Major axis, minor axis; latus rectum;

coordinates of vertices, focus and

centre; and equations of directrices

and the axes.

•Finding equation of ellipse when

focus and directrix are given.

•Simple and direct questions based on

the above.

•Focal property i.e. SP + SP′ = 2a.

(iii) Hyperbola

x2 y 2 − = 1 , e > 1, b 2 = a 2 (e 2 − 1) a 2 b2

• Cases when coefficient y2 is negative

and coefficient of x2 is negative.

• Rough sketch of the above.

• Focal property i.e. SP - S’P = 2a.

• Transverse and Conjugate axes; Latus

rectum; coordinates of vertices, foci

and centre; and equations of the

directrices and the axes.

x2 y2+ 2 = 1 , e <1, b2 = a 2 (1 − e2 )2a b

• General second degree equation

ax 2 + 2hxy + by 2 + 2 gx + 2 fy + c = 0

represents a parabola if h2 = ab,

ellipse if h2 < ab, and hyperbola if h2 >ab.

Condition that y = mx + c is a tangent

to the conics.

. Inverse Trigonometric Function

• Principal values.

• sin-1x, cos-1x, tan-1x etc. and their graphs.

sin-1x = cos −1 1 − x 2 = tan −1

. Addition formulae.

(cos x ± cos y = cos ( xy msin-1x ± sin-1 y = sin -1 x 1 − y2 ± y 1 − x2-1

similarly tan-1x ± tan-1 y = tan-1

. Similarly, establish

formulae for 2sin-1x, 2cos-1x, 2tan-1x,

3tan-1x etc. using the above formula.

• Application of these formulae.

5 Calculus

(i) Differential Calculus

•Revision of topics done in Class XI -

mainly the differentiation of product of

two functions, quotient rule, etc.

• Derivatives of trigonometric functions.

• Derivatives of exponential functions.

• Derivatives of logarithmic functions.

• Derivatives of inverse trigonometric

functions - differentiation by means of

substitution.

• Derivatives of implicit functions and

chain rule for composite functions.

• Differentiation of a function with

respect to another function e.g.

differentiation of sinx3 with respect to x3.

. Logarithmic Differentiation - Finding

dy/dx when y = x

•Successive 2nd order.

• L'Hospital's theorem.

Rolle's Mean Value Theorem - its

geometrical interpretation.

•Lagrange's Mean Value Theorem - its

geometrical interpretation.

• Maxima and minima.

(ii) Integral Calculus

• Revision of formulae from Class XI.

• Integration of 1/x, ex, tanx, cotx, secx,

cosecx.

• Integration by parts.

• Integration by means of substitution.

• Properties of definite integrals.

Problems based on the following

properties of definite integrals are to be

covered.

• Application of definite integrals - area

bounded by curves, lines and coordinate

axes is required to be covered.

6 Correlation and Regression

• Definition and meaning of correlation and

regression coefficient.

• Coefficient of Correlation by Karl Pearson.

• Rank correlation by Spearman’s (Correction

included).

• Lines of regression of x on y and y on x.

NOTE: Scatter diagrams and the following

topics on regression are required.

i) The method of least squares.

ii) Lines of best fit.

iii) Regression coefficient of x on y and y on x.

iv) b xy × b yx = r 2 , 0 ≤ b xy × b yx ≤ 1

v) Identification of regression equations

7. Probability

• Random experiments and their outcomes.

• Events: sure events, impossible events,

mutually

exclusive events, independent

events and dependent events.

• Definition of probability of an event.

• Laws of probability: addition and

multiplication laws, conditional probability

(excluding Baye’s theorem).

8 Complex umbers

• Argument and conjugate of complex numbers.

• Sum, difference, product and quotient of two

complex numbers additive and multiplicative

• Simple locus question on complex number;

proving and using -

• Triangle inequality.

• Square root of a complex number.

• Demoivre’s applications.

• Cube roots of unity: 1, ω , ω 2 ; application

problems.

9 Differential Equations

• Differential equations, order and degree.

• Solution of differential equations.

• Variable separable.

• Homogeneous equations and equations

reducible to homogeneous form.

• Linear form functions of x only. Similarly for dx/dy.

NOTE: Equations reducible to variable

separable type are included. The second order

differential equations are excluded.

SECTIO B

10. Vectors

• Scalar (dot) product of vectors.

• Cross product - its properties - area of a

triangle, collinear vectors.

• Scalar triple product - volume of a

parallelopiped, co-planarity.

Proof of Formulae (Using Vectors)

• Sine rule.

• Cosine rule

• Projection formula

• Area of a ∆ = ½absinC

NOTE: Simple geometric applications of the

above are required to be covered.

11. Co-ordinate geometry in 3-Dimensions

(i) Lines

NOTE: Symmetric and non-symmetric forms of

lines are required to be covered.

(ii) Planes

• Cartesian and vector equation of a

plane.

• Direction ratios of the normal to the

plane.

• One point form.

• Normal form.

• Intercept form.

• Distance of a point from a plane.

• Angle between two planes, a line and a

plane.

• Equation of a plane through the

intersection of two planes i.e. -

P1 + kP2 = 0.

Simple questions based on the above.

12 Probability

Baye’s theorem; theoretical probability

distribution, probability distribution function;

binomial distribution – its mean and variance.

12. Cartesian and vector equations of a line

through one and two points.

Coplanar and skew lines.

Conditions for intersection of two lines.

Shortest distance between two lines.

NOTE: Theoretical probability distribution is to

be limited to binomial distribution only.

SECTION C

13 Discount

True discount; banker's discount; discounted

value; present value; cash discount, bill of

exchange.

NOTE: Banker’s gain is required to be covered.

14. Annuities

Meaning, formulae for present value and

amount; deferred annuity, applied problems on

loans, sinking funds, scholarships.

NOTE: Annuity due is required to be covered.

15. Linear Programming

Introduction, definition of related terminology

such as constraints, objective function,

optimization, isoprofit, isocost lines; advantages

of linear programming; limitations of linear

programming; application areas of linear

programming; 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, optimum feasible solution.

16. Application of derivatives in Commerce and

Economics in the following

Cost function, average cost, marginal cost,

revenue function and break even point.

17. Index numbers and moving averages

• Price index or price relative.

• Simple aggregate method.

• Weighted aggregate method.

• Simple average of price relatives.

• Weighted average of price relatives

(cost of living index, consumer price index).

NOTE: Under moving averages the following

are required to be covered:

• Meaning and purpose of the moving averages.

• Calculation of moving averages with the

given periodicity and plotting them on a

graph.

• If the period is even, then the centered moving

average is to be found out and plotted.

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