ICSE Board Math Syllabus for Class 12

ICSE Board Syllabus for Class 12 Math




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

(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

•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

• 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,
• Integration by parts.
• Integration by means of substitution.

• Properties of definite integrals.

  Problems based on the following
  properties of definite integrals are to be

• 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

• 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,
 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

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.


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
• Direction ratios of the normal to the
• One point form.
• Normal form.
• Intercept form.
• Distance of a point from a plane.
• Angle between two planes, a line and a
• 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.


13 Discount
 True discount; banker's discount; discounted
 value; present value; cash discount, bill of
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

• If the period is even, then the centered moving
  average is to be found out and plotted.

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