Mathematics Class 12 ISC (Science) CISCE Topics and Syllabus

• Introduction of Matrices 
• Matrices 
  • Matrices Notation 

    Matrices Notation

  • Proof of the Uniqueness of Inverse 
• Order of a Matrix 
• Equality of Matrices 
  • Determine equality of two matrices
• Types of Matrices 
  • Types of Matrices:

  1. Column matrix
  2. Row matrix
  3. Square matrix
  4. Diagonal matrix
  5. Scalar matrix
  6. Identity matrix
  7. Zero matrix
• Symmetric and Skew Symmetric Matrices 
  • Define symmetric and skew symmetric matrix
• Transpose of a Matrix 
  • Write transpose of given matrix
• Operations on Matrices 
  • Addition of Matrices 
  • Multiplication of Matrices 
    • Non-commutativity of multiplication of matrices
    • Zero matrix as the product of two non zero matrices
• Multiplication of Two Matrices 
• Elementary Transformations 
  • 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)
• Invertible Matrices 
• Introduction of Determinant 
• Determinants of Matrix of Order One and Two 
• Determinant of a Square Matrix 

  up to 3 x 3 matrices

• Determinant of a Matrix of Order 3 × 3 
  • 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)
• Properties of Determinants 
  • 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.
• Minors and Co-factors 
• Area of a Triangle 
• Inverse of Matrix 
  • Inverse of a Square Matrix by the Adjoint Method 
• Applications of Determinants and Matrices 
  • Consistent System
  • Inconsistent System
  • Solution of a system of linear equations using the inverse of a matrix
• Martin’S Rule 

  Martin’s Rule (i.e. using matrices)

(i) Matrices

Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operation on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order upto 3). 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).

(ii) Determinants

Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, co-factors 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.

Types of matrices (m × 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).

E.g.`[(1,1),(0,2),(1,1)][(1,2),(2,2)]` =AB(say) but BA is not possible.

singular and non-singular matrices.

Existence of two non-zero matrices whose product is a zero matrix.

Inverse (2x2, 3x3) `A^(-1)=(AdjA)/|A|`

Martin’s Rule (i.e. using matrices):-

a₁x + b₁y + c₁z = d₁

a₂x + b₂y + c₂z = d₂

a₃x + b₃y + c₃z = d₃

`A=[(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)]B=[(d_1),(d_2),(d_3)]X=[(x),(y),(z)]`

`AX=BrArrX=A^(-1)B`

Problems based on above.

NOTE 1:- The conditions for consistency of equations in two and three variables, using
matrices, are to be covered.

NOTE 2:- Inverse of a matrix by elementary operations to be covered.

Determinants:-

• Order.
• Minors.
• Cofactors.
• Expansion.
• Applications of determinants in finding the area of triangle and collinearity.
• Properties of determinants. Problems based on properties of determinants.

• Concept of Continuity 
• Continuous Function of Point 

  Continuous left hand limit

  Continuous right hand limit
  

• Algebra of Continuous Functions 
• Exponential and Logarithmic Functions 
• Derivatives of Composite Functions - Chain Rule 
• Derivatives of Inverse Trigonometric Functions 
• Derivatives of Implicit Functions 
• Logarithmic Differentiation 
• Derivatives of Functions in Parametric Forms 
• Mean Value Theorem 
  • Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretations
• Second Order Derivative 
• L' Hospital'S Theorem 

Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions.

Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives. Rolle's and Lagrange's Mean Value Theorems (without proof) and their geometric interpretation.

Continuity

Continuity of a function at a point x = a.

Continuity of a function in an interval.

Algebra of continues function.

Removable discontinuity.

Differentiation

Concept of continuity and differentiability of |x| , [x], 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.

Derivatives of Parametric 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^(x^x)`

Successive differentiation up to 2^nd order.

NOTE 1:- Derivatives of composite functions using chain rule.

NOTE 2:- Derivatives of determinants to be covered.

L' Hospital's theorem.

`0/0`

Rolle's Mean Value Theorem - its geometrical interpretation.

Lagrange's Mean Value Theorem - its geometrical interpretation

• Introduction to Applications of Derivatives 
• Rate of Change of Bodies or Quantities 
• Increasing and Decreasing Functions 
• Tangents and Normals 
• Approximations 
• Maxima and Minima 
  • 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
• Maximum and Minimum Values of a Function in a Closed Interval 
• Simple Problems on Applications of Derivatives 

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

Applications of derivatives:- rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in 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-lifesituations).

Equation of Tangent and Normal

Approximation.

Rate measure.

Increasing and decreasing functions.

Maxima and minima.

• Stationary/turning points.
• Absolute maxima/minima
• local maxima/minima
• First derivatives test and second derivatives test
• Point of inflexion.
• Application problems based on maxima and minima.

• Introduction of Integrals 
• Indefinite Integral 
  • 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))`
• Integrals of the Type 

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

• Integration as an Inverse Process of Differentiation 

  Derivatives	Integrals (Anti derivatives)
  `d/(dx) (x^(n+1)/(n+1)) = x^n`	`int x^n dx = x^(n+1)/(n+1) + "C"`, n ≠ –1
  `d/(dx)`(x) = 1	`int dx` = x + C
  `d/(dx)`(sin x) = cos x	`int` cos x dx = sin x +C
  `d/(dx)` (-cos x) = sin x	`int`sin x dx = -cos x +C
  `d/(dx)` (tan x) = sec2x	`int sec^2 x` dx = tanx + C
  `d/(dx)`(-cot x) = `cosec^2x `	`int cosec^2x` dx = -cot x +C
  `d/(dx)` (sec x) = sec x tan x	`int` sec x tan x dx = sec x +C
  `d/(dx)` (-cosecx) = cosec x cot x	`int` cosec x cot x dx = -cosec x +C
  `d/(dx) (sin^-1) = 1/(sqrt(1-x^2))`	`int (dx)/(sqrt(1-x^2))= sin^(-1) x +C `
  `d/(dx) (-cos^(-1)) = 1/(sqrt (1-x^2))`	`int (dx)/(sqrt (1-x^2))= -cos^(-1) x + C `
  `d/(dx) (tan^(-1) x) = 1/(1+x^2)`	`int (dx)/(1+x^2)= tan^(-1) x + C `
  `d/(dx) (-cot^(-1) x) = 1/(1+x^2)`	`int (dx)/(1+x^2)= -cot^(-1) x + C `
  `d/(dx) (sec^(-1) x) = 1/(x sqrt (x^2 - 1))`	`int (dx)/(x sqrt (x^2 - 1))`= `sec^(-1)` x + C
  `d/(dx) (-cosec^(-1) x) = 1/(x sqrt (x^2 - 1))`	`int (dx)/(x sqrt (x^2 - 1))=-cosec^(-1) x + C `
  `d/(dx)(e^x) = e^x`	`int e^x dx = e^x + C`
  `d/(dx) (log|x|) = 1/x`	`int 1/x dx = log|x| +C`
  `d/(dx) ((a^x)/(log a)) = a^x`	`int a^x dx = a^x/log a` +C

• Methods of Integration: Integration by Substitution 
  • ∫ 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
• Methods of Integration: Integration Using Partial Fractions 

  No	From of the rational function	Form of the partial fraction
  1	`(px + q )/((x-a)(x-b))`a ≠ b	`A/(x-a) + B/(x-b)`
  2	`(px+q)/(x-a)^2`	`A/(x-a) + B/(x-a)^2`
  3	`((px)^2 + qx +r)/((x-a)(x-b)(x-c))`	`A/(x-a)+B/(x-b) + C /(x-c)`
  4	`((px)^2 + qx + r)/((x-a)^2 (x-b))`	` A/(x-a) + B/(x-a)^2 +C/(x-b)`
  5	`((px)^2 + qx +r)/((x-a)(x^2 + bx +c))`	`A/(x-a) + (Bx + C)/ (x^2 + bx +c)`,

• Methods of Integration: Integration by Parts 
  • `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`
• Some Properties of Indefinite Integral 
• Anti-derivatives of Polynomials and Functions 

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

• Evaluation of Simple Integrals of the Following Types and Problems 
• Definite Integral as the Limit of a Sum 
• Fundamental Theorem of Calculus 

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

• Definite Integrals 
• Properties of Definite Integrals 
  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
• Evaluation of Definite Integrals by Substitution 

Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.

Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

Indefinite integral:-

Integration as the inverse of differentiation.

Anti-derivatives of polynomials and functions (ax +b)ⁿ , sinx, cosx, 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 tanx, cotx, secx, cosecx.

Integration by parts.

Integration using partial fractions.

Expression of the form`f(x)/g(x)` when degree of f(x) < degree of g(x)

`(x+2)/((x-2)(x-1)^2)=A/(x-1)+B/(x-1)^2 +C/(x-2)`

`(x+1)/((x^2+3)(x-1))=(Ax+B)/(x^2+3)+c/(x-1)`

When degree of `f (x) >=` degree of g(x),

Integrals of the type:-

`int(dx)/(x^2+-a^2)`

and

`intsqrt(a^2+-x^2)dx`

`int(dx)/(acosx+bsinx),`

`int(dx)/(a+bcosx),int(dx)/(a+bsinx)int(dx)/(acosx+bsinx+c),`

`int((acosx+bsinx)dx)/(`

`int(dx)/(acos^2x+bsin^2x+c)`

`int(1+-x^2)/(1+x^4)dx,`

`int(dx)/(1+x^4),intsqrt(tanxdx),intsqrt(cotxdx)`

Definite Integral:-

Definite integral as a limit of the sum.

Fundamental theorem of calculus (without proof)

Properties of definite integrals.

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

`int_a^bf(x)dx=int_a^bf(t)dt`

`int_a^bf(x)dx=-int_b^af(x)dx`

`int_a^bf(x)dx=int_a^cf(x)dx+int_c^bf(x)dx`

where a < c < b

`int_a^bf(x)dx=int_a^bf(a+b-x)dx`

`int_0^af(x)dx=int_0^af(a-x)dx`

`int_0^(2a)f(x)dx={(2int_0^af(x),`

`int_(-a)^af(x)dx={(2int_0^af(x)dx,`

• Basic Concepts of Differential Equation 
• Order and Degree of a Differential Equation 
• General and Particular Solutions of a Differential Equation 
• Formation of a Differential Equation Whose General Solution is Given 
• Formation of Differential Equation by Eliminating Arbitary Constant 
• Methods of Solving First Order, First Degree Differential Equations 
  • Differential Equations with Variables Separable Method 
  • Homogeneous Differential Equations 
• Solutions of Linear Differential Equation 
  • 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`.
• Application on Growth and Decay 
• Solve Problems on Velocity, Acceleration, Distance and Time 
• Solve Population Based Problems on Application of Differential Equations 
• Application on Coordinate Geometry 

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 solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:

`(dy)/(dx)+py=q,`

where p and q are functions of x or constants.

`(dx)/(dy)+px=q,`

where p and q are functions of y or constants.

• Differential equations, order and degree.
• Formation of differential equation by eliminating arbitrary constant(s).
• 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.
• Solve problems of application on growth and decay.
• Solve problems on velocity, acceleration, distance and time.
• Solve population based problems on application of differential equations.
• Solve problems of application on coordinate geometry.

NOTE 1:- Equations reducible to variable separable type are included.

NOTE 2:- The second order differential equations are excluded.