Topics
Commercial Mathematics
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
Goods and Services Tax (G.S.T.)
- Tax and Its Computation
- Mathematical Study of GST (Goods and Services Tax)
- Types of GST in Mathematics
- Computational Mechanisms under GST
- GST in Trading Chain
- E-way Bill in GST Sums
- Examples based on GST Calculations
Algebra
Banking
Geometry
Shares and Dividends
Linear Inequations
Mensuration
Quadratic Equations
Trigonometry
Ratio and Proportion
Statistics
Probability
Factorisation of Polynomials
- Function and Polynomial
- Division Algorithm for Polynomials
- Remainder Theorem
- Factor Theorem
- Applications of Factor Theorem
Matrices
Arithmetic and Geometric Progression
Co-ordinate Geometry
- Foundation of Coordinate Geometry
- Advanced Concept of Reflection in Mathematics
- Invariant Points
- Combination of Reflections
- Using Graph Paper for Reflection
Similarity
Symmetry
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
Circles
Tangent and Secant Properties
Constructions
Area and Volume 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
Statistics : basic concepts, Mean, Median, Mode. Histograms and Ogive
Probability
Introduction
Every quadratic equation has two roots, but these roots may not always be of the same type. Sometimes both roots are real and different, sometimes they are real and equal, and sometimes they are not real at all. The discriminant helps us identify the nature of these roots without actually solving the equation.
The discriminant is given by:
\[ D = b^{2} - 4ac \]
By examining whether \[ D \] is positive, zero, or negative, we can quickly decide whether a quadratic equation has:
- Two distinct real roots
- Two equal real roots
- No real roots
Nature of Roots
Example
\[ \begin{array}{r l} \text{Question} & x^{2} + 2x - 9 = 0 \\ \text{Solution :} & \text{Compare } x^{2} + 2x - 9 = 0 \text{ with} \\ & ax^{2} + bx + c = 0. \\& a = 1,\ \ b = 2,\ \ c = -9. \\& \begin{aligned} \therefore\ b^{2} - 4ac &= 2^{2} - 4 \times 1 \times (-9) \\ D &= 4 - (-36) \\ D &= 40 \end{aligned} \\ & \therefore\ b^{2} - 4ac > 0 \\ & \therefore\ \text{the roots of the equation are real and unequal.} \end{array} \]
Maharashtra State Board: Class 10
CISCE: Class 10
Key Points: Nature of Roots
D = b2 – 4ac
| Condition on D | Nature of Roots |
|---|---|
| (D > 0) | Roots are real and unequal |
| (D = 0) | Roots are real and equal |
| (D < 0) | No real roots |
