Mathematics and Statistics 11th HSC Commerce Maharashtra State Board Topics and Syllabus

• Creating a Set
• Creating Set using List or Tuple
• Set Operations
• Programs using Sets

1. Roster method
2. Set-Builder method
3. Venn Diagram

1. Open Interval
2. Closed Interval
3. Semi-closed Interval
4. Semi-open Interval

• Types of Sets:

1.  Empty Set
2. Singleton set 
3. Finite set
4. Infinite set 
5. Subset 
6. Superset 
7. Proper Subset 
8. Power Set
9. Equal sets 
10. Equivalent sets 
11. Universal set

1. Complement of a set
2. Union of Sets
3. Intersection of sets
4. Distributive Property

1. Ordered Pair
2. Cartesian Product of two sets
3. Number of elements in the Cartesian product of two finite sets
4. Relation (Definition)

1. One-One Relation(Injective)
2. Many-one relation
3. Into relation
4. Onto relation (Surjective)

Reflexive, symmetric, transitive, not reflexive, not symmetric and not transitive.

• Function, Domain, Co-domain, Range
• Types of function
  1. One-one or One to one or Injective function
  2. Onto or Surjective function
• Representation of Function
• Graph of a function
• Value of funcation
• Some Basic Functions - Constant Function, Identity function, Power Functions, Polynomial Function, Radical Function, Rational Function, Exponential Function, Logarithmic Function, Trigonometric function

• one-one (or injective)
• many-one
• onto (or surjective)
• one-one and onto (or bijective)

• Arrow form/Venn Diagram Form
• Ordered Pair(x, y)
• Rule / Formula
• Tabular Form 
• Graphical form

Evaluation of function

Constant Function
- Identity function
- Linear Function
- Quadratic Function
- Function of the form - Square Function, Cube Function
- Polynomial Function
- Rational Function
- Exponential Function 
- Logarithmic Function

• Composition of Functions
• Inverse functions
• Piecewise Defined Functions
  1) Signum function
  2) Absolute value function (Modulus function)
  3) Greatest Integer Function (Step Function)
  4) Fractional part function

• Signum function
• Absolute value function (Modulus function)
• Greatest Integer Function (Step Function)

• Imaginary number
• Complex Number

• Equality of two Complex Numbers 
• Conjugate of a Complex Number 
• Properties of `barz`
• Addition of complex numbers - Properties of addition, Scalar Multiplication
• Subtraction of complex numbers - Properties of Subtraction
• Multiplication of complex numbers - Properties of Multiplication
• Powers of i in the complex number
• Division of complex number - Properties of Division

• Properties of 1, w, w²

• Finite sequence 
• Infinite sequence
• Progression

Properties of Geometric Progression.

• Expressing recurring decimals as rational numbers

• Arithmetic mean (A. M.)
• Geometric mean (G. M.) 
• Harmonic mean (H. M.)

Properties of Sigma Notation

• Equation of a locus

• Inclination of a line
• Slope of a line
• Perpendicular Lines
• Angle between intersecting lines

• Slope-Point Form
• Slope-Intercept  form
• Two-points  Form
• Double-Intercept  form
• Normal  Form 
• General  form 

• Point  of  intersection  of  lines  
• The  distance  of  the  Origin  from  a  Line
• Distance  of  a  point  from  a  line
• Distance  between  parallel  lines

1.1.1 Recall

• Matrix 
• Order of a matrix
• General form of a Matrix
• Types of matrices

1. Row matrix
2. Column matrix
3. Zero matrix (or) Null matrix 
4. Square matrix
5. Triangular matrix
6. Diagonal matrix
7. Scalar matrix
8. Unit matrix (or) Identity matrix
9. Multiplication of a matrix by a scalar
10. Negative of a matrix
11. Equality of matrices
12. Addition and subtraction of matrices
13. Multiplication of matrices
14. Transpose of a matrix

1.1.2 Minors 

1.1.3 Cofactors

1.1.4 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.

• One-Sided Limit
• Right-hand Limit
• Left-hand Limit
• Existence of a limit of a function at a point x = a

1. `lim_(x → 0) ((e^x - 1)/x) = log e = 1`

2. `lim_(x → 0) ((a^x - 1)/x) = log a (a > 0, a ≠ 0)`

3. `lim_(x → 0) [ 1 + x]^(1/x) = e`

4. `lim_(x → 0) (log(1 + x)/x) = 1`

5. `lim_(x → 0) ((e^(px) - 1)/(px)) = 1`, (p constant)

6. `lim_(x → 0) ((a^(px) - 1)/(px)) = log a`, (p constant)

7. `lim_(x → 0) (log(1 + px)/(px)) = 1`, (p constant)

8. `lim_(x → 0) [ 1 + px]^(1/(px)) = e`, (p constant)

• Continuity of a function at a point
• Definition of Continuity
• Continuity from the right and from the left
• Examples of Continuous Functions
• Properties of continuous functions
• Types of Discontinuities
• Jump Discontinuity
• Removable Discontinuity
• Infinite Discontinuity
• Continuity over an interval
• The intermediate value theorem for continuous functions

Discontinuous Function

• Derivative by the method of the first Principle
• Derivatives of some standard functions
• Relationship between differentiability and continuity

• Theorem 1. Derivative of Sum of functions
• Theorem 2. Derivative of Difference of functions.
• Theorem 3. Derivative of Product of functions.
• Theorem 4. Derivative of Quotient of functions.

• Demand function
• Marginal Demand (MD)
• Supply function(S)
• Total cost function (C)
• Marginal Cost (MC)
• Revenue and Profit Functions
• Total Revenue (R)

• Computing Median for Ungrouped Data
• Computing Median for Grouped Data

• Meaning of Correlation
• Types of correlation

1. Positive Correlation
2. Negative Correlation
3. Simple correlation

• Scatter Diagram
• Karl Pearson’s Correlation Coefficient

Properties of correlation coefficient r(x, y)

• Tree Diagram 
• Addition Principle 
• Multiplication principle

Properties of the factorial function

• ⁿC_r , ⁿC_n =1, ⁿC₀ = 1, ⁿC_r = ⁿC_n–r, ⁿC_x = ⁿC_y, then x + y = n or x = y, ⁿ⁺¹C_r = ⁿC_r-1 + ⁿC_r
• When all things are different
• When all things are not different.
• Mixed problems on permutation and combinations.

• Properties of Combinations:
  1. Consider ⁿC_{n - r} = ⁿC_r for 0 ≤ r ≤ n.
  2. ⁿC₀ = `(n!)/(0!(n - 0)!) = (n!)/(n!) = 1, because 0! = 1` as has been stated earlier.
  3. If ⁿC_r = ⁿC_s, then either s = r or s = n - r.
  4. `"" ^nC_r = (""^nP_r)/(r!)`
  5. ⁿC_r + ⁿC_{r - 1} = ^{n + 1}C_r
  6. ⁿC₀ + ⁿC₁ + ......... ⁿC_n = 2ⁿ 
  7. ⁿC₀ + ⁿC₂ + ⁿC₄ + ...... = ⁿC₁ + ⁿC₃ + ⁿC₅ + ....... = 2^{(n - 1)}
  8. ⁿC_r = `"" (n/r) ^(n - 1)C_(r- 1) = (n/r)((n - 1)/(r - 1)) ^(n - 2)C_(r - 2) = ....`
  9. ⁿC_r has maximum value if (a) r = `n/2 "when n is even (b)" r = (n - 1)/2 or (n + 1)/2` when n is odd.

• Random experiment
• Outcome
• Equally likely outcomes
• Sample space
• Event

1. Union of two events
2. Exhaustive Events
3. Intersection of two events
4. Mutually Exclusive Events

Representation of solution of linear inequality in one variable on the number line

Consider linear inequalities in two variables x and y

• Meaning of Depreciation
• Features of Depreciation