Mathematics ISC (Commerce) Class 12 CISCE Syllabus 2025-26

• Matrices
• Determinants
• Cramer’s Rule
• Application in Economics

• Determine equality of two matrices

• Row Matrix
• Column Matrix
• Zero or Null matrix
• Square Matrix
• Diagonal Matrix
• Scalar Matrix
• Unit or Identity Matrix
• Upper Triangular Matrix
• Lower Triangular Matrix
• Triangular Matrix
• Symmetric Matrix
• Skew-Symmetric Matrix
• Determinant of a Matrix
• Singular Matrix
• Transpose of a Matrix

• Define symmetric and skew symmetric matrix

• Write transpose of given matrix

• Interchange of any two rows or any two columns
• Multiplication of the elements of any row or column by a non-zero scalar
• Adding the scalar multiples of all the elements of any row (column) to corresponding elements of any other row (column)

up to 3 x 3 matrices

• 1st, 2nd and 3rd Row
• 1st, 2nd and 3rd Columns
• Expansion along the first Row (R₁)
• Expansion along the second row (R₂)
• Expansion along the first Column (C₁)

• Property 1 - The value of the determinant remains unchanged if its rows are turned into columns and columns are turned into rows.
• Property 2 -  If any two rows  (or columns)  of a determinant are interchanged then the value of the determinant changes only in sign.
• Property 3 - If any two rows ( or columns) of a  determinant are identical then the value of the determinant is zero.
• Property  4  -  If each element of a row (or column)  of a determinant is multiplied by a  constant k then the value of the new determinant is k times the value of the original determinant.
• Property  5  -  If each element of a row (or column) is expressed as the sum of two numbers then the determinant can be expressed as the sum of two determinants
• Property  6  -  If a constant multiple of all elements of any row  (or column)  is added to the corresponding elements of any other row  (or column  )  then the value of the new determinant so obtained is the same as that of the original determinant. 
• Property 7 -  (Triangle property) - If all the elements of a  determinant above or below the diagonal are zero then the value of the determinant is equal to the product of its diagonal elements.

• Consistent System
• Inconsistent System
• Solution of a system of linear equations using the inverse of a matrix

Martin’s Rule (i.e. using matrices)

Continuous left hand limit

Continuous right hand limit

• Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretations

• First and Second Derivative test
• 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

Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations)

• Integration as the inverse of differentiation
• Anti-derivatives of polynomials and functions (ax + b)ⁿ, sin x, cos x, sec²x, cosec²x etc.
• Integrals of the type sin²x, sin³x, sin⁴x, cos²x, cos³x, cos⁴x.
• Integration of 1/x, e^x
• Integration by substitution.
• Integrals of the type f'(x) [f (x)]ⁿ, `(f'(x))/f(x)`.
• Integration of tan x, cot x, sec x, cosec x.
• Integration by parts.
• Integration using partial fractions
  Expressions of the form `(f(x))/(g(x))` when degree of f(x) < degree of g(x)
  E.g. `(x + 2)/((x - 3)(x + 1)) = A/(x - 3 ) + B/(x + 1)`
          `(x + 2)/((x - 2)(x - 1)^2) = A/(x - 1) + B/(x - 1)^2 + C/(x - 2)` 
  When degree of f (x) ≥ degree of g(x),
  e.g.
  `(x^2 + 1)/(x^2 + 3x + 2) = 1 - ((3x + 1)/(x^2 + 3x + 2))`

`int dx/(x^2 +- a^2), int (px + q)/(ax^2 + bx + c) dx`

• ∫ tan x dx = log | sec x |  + C
• ∫ cot x dx = log | sin x | + C
• ∫ sec x dx = log | sec x + tan x | + C
• ∫ cosec x dx = log | cosec x – cot x | + C

• `int(u.v) dx = u intv dx - int((du)/(dx)).(intvdx) dx`
• Integral of the type ∫ e^x [ f(x) + f'(x)] dx = e^xf(x) + C
• Integrals of some more types

1. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
2. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
3. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`

Anti-derivatives of polynomials and functions (ax +b)ⁿ , sinx, cosx, sec²x, cosec²x etc .

Area function, First fundamental theorem of integral calculus and Second fundamental theorem of integral calculus

1. `int_a^a f(x) dx = 0`
2. `int_a^b f(x) dx = - int_b^a f(x) dx`
3. `int_a^b f(x) dx = int_a^b f(t) dt`
4. `int_a^b f(x) dx = int_a^c f(x) dx + int_c^b f(x) dx`
   where a < c < b, i.e., c ∈ [a, b]
5. `int_a^b f(x) dx = int_a^b f(a + b - x) dx`
6. `int_0^a f(x) dx = int_0^a f(a - x) dx`
7. `int_0^(2a) f(x) dx = int_0^a f(x) dx + int_0^a f(2a - x) dx`
8. `int_(-a)^a f(x) dx = 2. int_0^a f(x) dx`, if f(x) even function
   = 0,  if f(x) is odd function

• Solutions of linear differential equation of the type:

1. `dy/dx` + py = q, where p and q are functions of x or constants.
2. `dx/dy` + px = q, where p and q are functions of y or constants.

• Differential equations, order and degree.
• Solution of differential equations.
• Variable separable.
• Homogeneous equations
• Linear form `dy/dx` + Py = Q where P and Q are functions of x only. Similarly, for `dx/dy`.