Mathematics and Statistics 11th HSC Science (General) Maharashtra State Board Topics and Syllabus

• Zero angle
• One rotation angle
• Straight angle
• Right angle

• Angle in a Quadrant
• Quadrantal Angles
• Co-terminal angles

• Sexagesimal system (Degree measure)
• Circular system (Radian measure)
• Theorem:The radian so defined is independent of the radius of the circle used and πc = 1800.

• Trigonometric ratios of any angle

• Angle of measure 0°
• Angle of measure 90° or (π/2)^c
• Angle of measure 360° or (2π)^c
• Angle of measure 120° or (2π/3)^c
• Angle of measure 225° or (5π^c/4)^c
• Angle of measure –60° or - π/3

• Domain and Range of Trignometric Functions and Their Graphs

1. The graph of sine function
2. The graph of cosine function
3. The graph of tangent function

1. For any two angles A and B, cos (A -B) = cos A cos B + sin A sin B

2. For any two angles A and B, cos (A + B) = cos A cos B − sin A sin B

3. For any two angles A and B, sin (A − B) = sin A cos B − cos A sin B

4. For any two angles A and B, sin (A + B) = sin A cos B + cos A sin B

5. For any two angles A and B, tan (A + B) =` (tan A + tan B)/(1 –tan A tan B)`

6. For any two angles A and B, tan (A -B) = `(tan A -tan B)/(1 + tan A tan B)`

• Value of a Determinant
• Determinant of order 3
• Expansion of Determinant

• 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.

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

• 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

• Transpose of a Matrix
• Symmetric Matrix
• Skew-Symmetric Matrix
• Equality of Two matrices
• Addition of Two Matrices
• Scalar Multiplication of a Matrix
• Multiplication of Two Matrices

• Locus
• Equation of Locus
• Shift of Origin

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

1. The distance of the Origin from a Line
2. The distance of the point (x₁,y₁) from a line
3. The distance between two parallel lines

• Standard form
• Centre-radius form
• Diameter Form

The general equation of a circle is of the form x² + y² + 2gx + 2fy + c = 0, if g² + f² − e > 0.

• The equation of tangent to a standard circle x² + y² = r² at point P(x₁, y₁) on it.

1. Variance and Standard Deviation for raw data
2. Variance and Standard Deviation for ungrouped frequency distribution
3. Variance and Standard Deviation for grouped frequency distribution

• Random Experiment
• Outcome
• Sample space
• Favourable Outcome

• Elementary Event
• Certain Event
• Impossible Event
• Algebra of Events

• Union of Two Events
• Exhaustive Events
• Intersection of Two Events
• Mutually Exclusive Events

• Equally likely outcomes
• Probability of an Event
• Elementary Properties of Probability

1. Using the definition of probability
2. Using Venn diagram

• Partition of a sample space
• Theorem of total probability

• 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

• Solution of a Quadratic Equation in complex number system

• Modulus of z
• Argument of z
• Argument of z in different quadrants/axes - Properties of modulus of complex numbers, Properties of arguments 
• Polar & Exponential form of C.N.

• Properties of 1, w, w²

• Finite sequence 
• Infinite sequence
• Progression

• N^th Term of Geometric Progression (G.P.)
• General Term of a Geometric Progression (G.P.)
• Sum of First N Terms of a Geometric Progression (G.P.)
• Sum of infinite terms of a G.P.
• Geometric Mean (G.M.)

• n^th term of A.G.P.
• Sum of n terms of A.G.P.
• Properties of Summation

• Tree Diagram 
• Addition Principle 
• Multiplication principle

• Properties of the factorial notation:
  For any positive integers m, n.,
  1) n! = n × (n - 1)!
  2) n > 1, n! = n × (n - 1) × (n - 2)!
  3) n > 2, n! = n × (n - 1) × (n - 2) × (n - 3)!
  4) (m + n)! is always divisible by m! as well as by n!
  5) (m × n)! ≠ m! × n!
  6) (m + n)! ≠ m! + n!
  7) m > n, (m - n)! ≠ m! - n! but m! is divisible by n!
  8) (m ÷ n)! ≠ m! ÷ n

• 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.

1) Roster or Tabular method or List method
2) Set-Builder or Rule Method
3) Venn Diagram

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

• Ordered Pair
• Cartesian Product of two sets
• Cartesian product of a set with itself
• Definitions of relation, Domain, Co-domain, and Range of a Relation
• Binary relation on a set
• Identity Relation
• Types of relations 
• Equivalence relation 
• Congruence Modulo

• 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

• 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

• Definition of Limit
• One-Sided Limit
• Left-hand Limit
• Right-hand Limit
• Existence of a limit of a function at a point x = a
• Algebra of limits:
  Let f(x) and g(x) be two functions such that
  `lim_(x→a) f(x) = l and lim_(x → a) g(x) = m, then`
  1. `lim_(x → a) [f(x) ± g(x)] = lim_(x → a) f(x) ± lim_(x → a) g(x) = l ± m`
  2. `lim_(x → a) [f(x) xx g(x)] = lim_(x→ a) f(x) xx lim_(x→ a) g(x) = l xx m`
  3. `lim_(x → a) [kf(x)] = k xx lim_(x→ a) f(x) = kl, "where" ‘k’ "is a constant"`
  4. `lim_(x → a) f(x)/g(x) = (lim_(x → a) f(x))/(lim_(x → a) g(x)) = l/m "where" m≠ 0`.

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)

• Limit at infinity
• Infinite Limits

• 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

• 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.