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

## Syllabus

1.1 Angle and Its Measurement
1.2 Trigonometry - 1
1.3 Trigonometry - 2
1.4 Determinants and Matrices
• Definition and Expansion of Determinants
• Value of a Determinant
• Determinant of order 3
• Expansion of Determinant
• Minors and Cofactors of Elements of Determinants
• 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.
• Application of Determinants
• Introduction to Matrices
• Types of 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
• Algebra of Matrices
• Transpose of a Matrix
• Symmetric Matrix
• Skew-Symmetric Matrix
• Equality of Two matrices
• Scalar Multiplication of a Matrix
• Multiplication of Two Matrices
• Matrices
1.5 Straight Line
1.6 Circle
1.7 Conic Sections
• Double Cone
• Conic Sections
• Parabola
• Standard equation of the parabola
• Tracing of the parabola y2 = 4 ax (a>0)
• Parametric expressions of standard parabola y2 = 4ax
• General forms of the equation of a parabola
• Tangent
• Condition of tangency
• Tangents from a point to a parabola
• Ellipse
• Standard equation of the ellipse
• Special cases of an ellipse
• Tangent to an ellipse
• Equation of tangent to the ellipse
• Condition for tangency
• Tangents from a point to the ellipse
• Locus of point of intersection of perpendicular tangents
• Auxilary circle and director circle of the ellipse
• Hyperbola
• Standard equation of the hyperbola
• Tangent to a hyperbola
• Tangent at a point on a hyperbola
• Equation of tangent to the hyperbola
• Condition for tangency
• Tangents from a point to the hyperbola
• Locus of point of intersection of perpendicular tangents
• Auxiliary Circle, Director Circle
• Asymptote
1.8 Measures of Dispersion
1.9 Probability
2.1 Complex Numbers
2.2 Sequences and Series
2.3 Permutations and Combination

2.5 Sets and Relations
• Sets and Their Representations

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

• Types of Sets
• 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
• Operations on Sets
1. Complement of a set
2. Union of Sets
3. Intersection of sets
4. Distributive Property
• Intervals
1. Open Interval
2. Closed Interval
3. Semi-closed Interval
4. Semi-open Interval
• Concept of Relations
• 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
2.6 Functions
• Concept of Functions
• 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
• Algebra of Functions
• 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
2.7 Limits
• Concept of Limits
• 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.
• Factorization Method
• Rationalization Method
• Limits of Trigonometric Functions
• Substitution Method
• Limits of Exponential and Logarithmic Functions

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
• Limit at infinity
• Infinite Limits
2.8 Continuity
• Continuous and Discontinuous Functions
• 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
2.9 Differentiation