Topics
Compound Interest
- Compound Interest as a Repeated Simple Interest Computation with a Growing Principal
- Use of Compound Interest in Computing Amount Over a Period of 2 Or 3-years
- Use of Formula
- Finding CI from the Relation CI = A – P
Commercial Mathematics
Goods and Services Tax (G.S.T.)
Banking
Algebra
Geometry
Shares and Dividends
Symmetry
Mensuration
Linear Inequations
Quadratic Equations
- Quadratic Equations
- Method of Solving a Quadratic Equation
- Factorisation Method
- Quadratic Formula (Shreedharacharya's Rule)
- Nature of Roots of a Quadratic Equation
- Equations Reducible to Quadratic Equations
Trigonometry
Statistics
Problems on Quadratic Equations
- Method for Solving a Quadratic Word Problem
- Problems Based on Numbers
- Problems on Ages
- Problems Based on Time and Work
- Problems Based on Distance, Speed and Time
- Problems Based on Geometrical Figures
- Problems on Mensuration
- Problems on C.P. and S.P.
- Miscellaneous Problems
Ratio and Proportion
Probability
Remainder Theorem and Factor Theorem
- Function and Polynomial
- Division Algorithm for Polynomials
- Remainder Theorem
- Factor Theorem
- Applications of Factor Theorem
Matrices
Arithmetic Progression
Geometric Progression
Reflection
- Co-ordinate Geometry
- Advanced Concept of Reflection in Mathematics
- Invariant Points
- Combination of Reflections
- Using Graph Paper for Reflection
Section and Mid-Point Formulae
Equation of a Line
Similarity
Loci
- Locus
- Points Equidistant from Two Given Points
- Points Equidistant from Two Intersecting Lines
- Summary of Important Results on Locus
- Important Points on Concurrency in a Triangle
Angle and Cyclic Properties of a Circle
Tangent Properties of Circles
Constructions
Volume and Surface Area of Solids (Cylinder, Cone and Sphere)
- Mensuration of Cylinder
- Hollow Cylinder
- Mensuration of Cones
- Mensuration of a Sphere
- Hemisphere
- Conversion of Solids
- Solid Figures
- Problems on Mensuration
Trigonometrical Identities
Heights and Distances
- Angles of Elevation and Depression
- Problems based on Elevation and Depression
Graphical Representation of Statistical Data
Measures of Central Tendency (Mean, Median, Quartiles and Mode)
Probability
Theorem: Angle at the Centre
Statement:
The angle subtended by an arc at the centre of a circle is double the angle subtended by it at any point on the remaining part of the circle.
Result:
∠AOB = 2∠ACB
Short Proof (Idea):
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Join the centre to the points on the circle.
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Radii form isosceles triangles.
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Angle at centre = sum of angles at the circle.
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Hence, the angle at the centre is double the angle at the circle.
Theorem: Angles in the Same Segment
Statement:
Angles in the same segment of a circle are equal.
Result:
∠ACB = ∠ADB
Short Proof (Idea):
-
Both angles stand on the same arc.
-
The angle at the centre is double each of them.
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Hence, both angles are equal.
Theorem: Angle in a Semi-circle
Statement:
The angle in a semicircle is a right angle.
Result:
∠ACB = 90∘
Short Proof (Idea):
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Diameter subtends an angle of 180° at the centre.
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The angle at the circle is half of it.
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Therefore, angle = 90°.
Theorem: Converse: Angle in a Circle
Statement:
If an arc of a circle subtends a right angle at any point on the remaining part of the circle, then the arc is a semicircle.
Result:
Arc AB is a semicircle
(or AB is a diameter)
Short Proof (Idea):
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Given angle at the circle ∠ACB = 90°.
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The angle at the centre is double the angle at the circle.
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Therefore, ∠AOB = 2 × 90° = 180°.
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Hence, A, O, and B lie on a straight line, so AB is a diameter.
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Therefore, arc AB is a semicircle.
