Topics
Angle and Its Measurement
- Directed Angle
- Angles of Different Measurements
- Angles in Standard Position
- Measures of Angles with Various Systems
- Area of a Sector
- Length of an Arc
Trigonometry - 1
- Trigonometric Ratios
- Trigonometric Functions with the Help of a Circle
- Signs of Trigonometric Functions in Different Quadrants
- Range of Cosθ and Sinθ
- Trigonometric Functions of Specific Angles
- Trigonometric Functions of Negative Angles
- Important Identities and Standard Results
- Periodicity of Trigonometric Functions
- Domain and Range of Trigonometric Functions
- Graphs of Trigonometric Functions
- Polar Co-ordinate System
Trigonometry - 2
Determinants and Matrices
- Definition and Expansion of Determinants
- Minors and Cofactors of Elements of Determinants
- Properties of Determinants
- Application of Determinants
- Determinant Method (Cramer’s Rule)
- Consistency of Three Equations in Two Variables
- Area of Triangle and Collinearity of Three Points
- Concept of Matrices
- Types of Matrices
- Operation on Matrices
- Properties of Matrix Multiplication
- Transpose of a Matrix
Straight Line
- Locus of a Points in a Co-ordinate Plane
- Equations of Line in Different Forms
- Family & Concurrent Lines
Circle
Conic Sections
Measures of Dispersion
- Meaning and Definition of Dispersion
- Measures of Dispersion
- Quartiles and Range in Statistics
- Variance
- Standard Deviation
- Change of Origin and Scale of Variance and Standard Deviation
- Standard Deviation for Combined Data
- Coefficient of Variation
Probability
Complex Numbers
Sequences and Series
- Sequence, Series, and Progression
- Arithmetic Progression (A.P.)
- Geometric Progression (G. P.)
- Harmonic Progression (H. P.)
- Arithmetico Geometric Series
- Power Series
Permutations and Combination
Methods of Induction and Binomial Theorem
- Principle of Mathematical Induction
- Binomial Theorem for Positive Integral Index
- General Term in Expansion of (a + b)n
- Middle term(s) in the expansion of (a + b)n
- Binomial Theorem for Negative Index Or Fraction
- Binomial Coefficients
Sets and Relations
- Sets and Their Representations
- Classification of Sets
- Fundamental Concepts of Ordered Pairs and Relations
- Intervals
Functions
Limits
Continuity
Differentiation
- Definition of Derivative and Differentiability
- Rules of Differentiation (Without Proof)
- Derivative of Algebraic Functions
- Derivatives of Inverse Trigonometric Functions
- Derivative of Logarithmic Functions
- Derivatives of Exponential Functions
- L' Hospital'S Theorem
Estimated time: 4 minutes
- 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`.
Maharashtra State Board: Class 12
Definition: Limit
If f(x) approaches a real number l, when x approaches a, then l is called the limit of f(x).
Symbolically, \[\lim_{x\to a}f\left(x\right)=l\]
Maharashtra State Board: Class 12
Definition: Left Hand Limit
When we have the values of f near x to the left of a, i.e.
\[\lim_{x\to a^{-}}f\left(x\right)\] is the expected value of f at x = a.
Maharashtra State Board: Class 12
Definition: Right Hand Limit
When we have the values of f near x to the right of a i.e.
\[\lim_{x\to a^{+}}f\left(x\right)\] is the expected value of f at x = a.
Maharashtra State Board: Class 12
Theorem: Sandwich theorem (Squeeze theorem)
If f(x) ≤ g(x) ≤ h(x) and \[\lim_{x\to a}\mathrm{f}(x)=l=\lim_{x\to a}\mathrm{h}(x)\]
\[\therefore\lim_{x\to a}g(x)=l\]
Related QuestionsVIEW ALL [125]
In problems 1 – 6, using the table estimate the value of the limit.
`lim_(x -> 2) (x - 2)/(x^2 - x - 2)`
| x | 1.9 | 1.99 | 1.999 | 2.001 | 2.01 | 2.1 |
| f(x) | 0.344820 | 0.33444 | 0.33344 | 0.333222 | 0.33222 | 0.332258 |
