Topics
Number Systems
Real Numbers
Algebra
Polynomials
Pair of Linear Equations in Two Variables
- Introduction to linear equations in two variables
- Graphical Method
- Substitution Method
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- Cross - Multiplication Method
- Equations Reducible to a Pair of Linear Equations in Two Variables
- Consistency of Pair of Linear Equations
- Inconsistency of Pair of Linear Equations
- Algebraic Conditions for Number of Solutions
- Simple Situational Problems
- Pair of Linear Equations in Two Variables
- Relation Between Co-efficient
Quadratic Equations
- Quadratic Equations
- Solutions of Quadratic Equations by Factorization
- Solutions of Quadratic Equations by Completing the Square
- Nature of Roots of a Quadratic Equation
- Relationship Between Discriminant and Nature of Roots
- Situational Problems Based on Quadratic Equations Related to Day to Day Activities to Be Incorporated
- Application of Quadratic Equation
Arithmetic Progressions
Coordinate Geometry
Lines (In Two-dimensions)
Constructions
- Division of a Line Segment
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- Constructions Examples and Solutions
Geometry
Triangles
- Similar Figures
- Similarity of Triangles
- Basic Proportionality Theorem (Thales Theorem)
- Criteria for Similarity of Triangles
- Areas of Similar Triangles
- Right-angled Triangles and Pythagoras Property
- Similarity of Triangles
- Application of Pythagoras Theorem in Acute Angle and Obtuse Angle
- Triangles Examples and Solutions
- Concept of Angle Bisector
- Similarity of Triangles
- Ratio of Sides of Triangle
Circles
Trigonometry
Introduction to Trigonometry
- Trigonometry
- Trigonometry
- Trigonometric Ratios
- Trigonometric Ratios and Its Reciprocal
- Trigonometric Ratios of Some Special Angles
- Trigonometric Ratios of Complementary Angles
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- Proof of Existence
- Relationships Between the Ratios
Trigonometric Identities
Some Applications of Trigonometry
Mensuration
Areas Related to Circles
- Perimeter and Area of a Circle - A Review
- Areas of Sector and Segment of a Circle
- Areas of Combinations of Plane Figures
- Circumference of a Circle
- Area of Circle
Surface Areas and Volumes
- Surface Area of a Combination of Solids
- Volume of a Combination of Solids
- Conversion of Solid from One Shape to Another
- Frustum of a Cone
- Concept of Surface Area, Volume, and Capacity
- Surface Area and Volume of Different Combination of Solid Figures
- Surface Area and Volume of Three Dimensional Figures
Statistics and Probability
Statistics
Probability
Internal Assessment
Formula
Sum of first n terms of an `"AP": "S" =(n/2)[2a + (n- 1)d]` The sum of n terms is also equal to the formula where l is the last term.
Notes
Arithmetic Progression a, a + d, a + 2d, a + 3d, . . . . . . . . . . . . a +(n - 1)d
In this progression a is the first term and d is the common difference. Let’s write the sum of first n terms as Sn.
Sn = [a] + [a + d] + . . . + [a+(n-2)d] + [a+(n-1)d] ..........(eq1)
Reversing the terms and rewritting the expression again,
Sn = [a+(n-1)d] + [a+(n-2)d] + . . . + [a + d ] + [a] ........(eq2)
On adding eq1 and eq2
2Sn = [a+a+(n-1)d] + [a + d+a+(n-2)d]+ . . . + [a+(n-2)d+ a + d]+ [a+(n-1)d+a]
2Sn = [2a+(n-1)d] + [2a+(n-1)d] + . . . + [2a+(n-1)d] . . . n times.
∴2Sn = n [2a+(n-1)d]
∴Sn = n/2 [2a+(n-1)d]
Ex. Let’s find the sum of first 100 terms of A.P. 14, 16, 18, . . . .
Here a= 14, d = 2, n = 100
`Sn = n/2 [2a+(n-1)d]`
`S100= 100/2 [2× 14+ (100-1)2]`
`S100= 50 [28 + (99)2]`
`S100= 50 [28 + 198]`
`S100= 50 [226]`
`S100= 11300`
∴ Sum of first 100 terms of given A.P. is 11,300
