Class 9 Mathematics Formulas
1. Number Systems
Irrational Numbers
- sqrt(2) is irrational: Proof by contradiction. Assume sqrt(2) = p/q in lowest terms, then p^2 = 2q^2, leading to contradiction.
- Rationalizing factor: For sqrt(2), multiply numerator and denominator by sqrt(2) to get a rational denominator.
Real Numbers
- Decimal expansion: Terminating decimals are rational, non-terminating repeating decimals are rational, and non-terminating non-repeating decimals are irrational.
2. Polynomials
- Degree of polynomial: Highest power of the variable.
- Value of polynomial: p(x) = a0 + a1x + a2x^2 + ... + anx^n
- Zero of polynomial: Value of x where p(x) = 0
- Remainder Theorem: If p(x) is divided by (x - a), remainder = p(a)
- Factor Theorem: (x - a) is a factor of p(x) if p(a) = 0
Example: For p(x) = x^3 - 6x^2 + 11x - 6, p(1) = 0, so (x - 1) is a factor.
3. Coordinate Geometry
- Distance formula: Distance between (x1, y1) and (x2, y2) = sqrt[(x2 - x1)^2 + (y2 - y1)^2]
- Section formula: Point dividing a line in ratio m:n = ((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n))
- Midpoint formula: ((x1 + x2)/2, (y1 + y2)/2)
4. Linear Equations in Two Variables
- General form: ax + by + c = 0
- Solution: Pair of values (x, y) satisfying the equation
5. Introduction to Euclid's Geometry
Basic postulates and axioms of Euclidean geometry.
6. Lines and Angles
- Complementary angles: Sum = 90 degrees
- Supplementary angles: Sum = 180 degrees
- Vertically opposite angles: Equal
- Linear pair: Adjacent angles forming a straight line, sum = 180 degrees
7. Triangles
- Pythagoras Theorem: In a right triangle, hypotenuse^2 = sum of squares of the other two sides
- Area of triangle: (1/2) x base x height
- Triangle inequality: Sum of any two sides is greater than the third side
8. Quadrilaterals
- Parallelogram properties: Opposite sides equal, opposite angles equal, diagonals bisect each other
- Rhombus: All sides equal, diagonals are perpendicular and bisect angles
- Square: All sides equal, all angles 90 degrees, diagonals equal and bisect each other
- Rectangle: Opposite sides equal, all angles 90 degrees, diagonals equal
9. Areas of Parallelograms and Triangles
- Area of parallelogram: Base x height
- Area of triangle: (1/2) x base x height
- Area of trapezium: (1/2) x (sum of parallel sides) x height
10. Circles
- Circumference: 2pi r
- Area: pi r^2
- Arc length: (theta/360) x 2pi r
- Area of sector: (theta/360) x pi r^2
11. Constructions
Construction of bisectors, perpendiculars, and other figures using a compass and ruler.
12. Heron's Formula
- Area of triangle: sqrt[s(s-a)(s-b)(s-c)] where s = (a + b + c)/2
13. Surface Areas and Volumes
- Cube: Volume = a^3, Surface area = 6a^2
- Cuboid: Volume = l x b x h, Surface area = 2(lb + bh + hl)
- Cylinder: Volume = pi r^2 h, Curved surface area = 2pi rh, Total surface area = 2pi r(r + h)
- Cone: Volume = (1/3)pi r^2 h, Curved surface area = pi rl, Total surface area = pi r(l + r)
- Sphere: Volume = (4/3)pi r^3, Surface area = 4pi r^2
14. Statistics
- Mean: Sum of observations / number of observations
- Median: Middle value when arranged in order
- Mode: Most frequent value
15. Probability
- Probability of event: Number of favorable outcomes / total possible outcomes