Class 10 Mathematics Formulas
1. Real Numbers
- Euclid's Division Algorithm: For any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 <= r < b
- Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of primes, and this factorization is unique.
2. Polynomials
- Quadratic polynomial: ax^2 + bx + c
- Sum of zeros: -b/a
- Product of zeros: c/a
- Quadratic formula: x = [-b +/- sqrt(b^2 - 4ac)] / (2a)
- Discriminant: D = b^2 - 4ac
3. Pair of Linear Equations in Two Variables
- Substitution method: Solve one equation for one variable, then substitute into the other.
- Elimination method: Eliminate one variable by adding or subtracting equations.
- Cross-multiplication method: x/(b1c2-b2c1) = y/(c1a2-c2a1) = 1/(a1b2-a2b1)
4. Quadratic Equations
- Standard form: ax^2 + bx + c = 0
- Solutions: [-b +/- sqrt(D)] / (2a) where D = b^2 - 4ac
- Nature of roots: If D > 0, roots are real and distinct; if D = 0, roots are real and equal; if D < 0, roots are complex.
5. Arithmetic Progressions
- General term: a_n = a + (n - 1)d
- Sum of first n terms: S_n = n/2 [2a + (n - 1)d] = n/2 (a + l)
6. Triangles
- Similarity criteria: AAA, AA, SSS, SAS
- Pythagoras theorem: a^2 + b^2 = c^2 for a right triangle
- Area ratio: (AB^2/AC^2) = (AD/DB) for similar triangles
7. Coordinate Geometry
- Distance formula: sqrt[(x2 - x1)^2 + (y2 - y1)^2]
- Section formula: ((m x2 + n x1)/(m + n), (m y2 + n y1)/(m + n))
- Area of triangle: (1/2)|x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|
- Condition for collinearity: Area = 0
8. Introduction to Trigonometry
- sin theta: opposite / hypotenuse
- cos theta: adjacent / hypotenuse
- tan theta: opposite / adjacent
- cosec theta: 1 / sin theta
- sec theta: 1 / cos theta
- cot theta: 1 / tan theta
Identities: sin^2 theta + cos^2 theta = 1, 1 + tan^2 theta = sec^2 theta, 1 + cot^2 theta = cosec^2 theta
9. Some Applications of Trigonometry
- Height and distance problems: Use trigonometric ratios to find unknown heights and distances.
10. Circles
- Tangent: A line touching a circle at one point
- Tangent length: sqrt(external secant x whole secant)
- Angle in semicircle: 90 degrees
- Angle between tangent and chord: Equal to the angle in the alternate segment
11. Constructions
Construction of tangents, division of line segments, and related geometric figures.
12. Areas Related to Circles
- Area of sector: (theta/360) x pi r^2
- Area of segment: Area of sector - area of triangle
- Length of arc: (theta/360) x 2pi r
13. Surface Areas and Volumes
- Frustum of cone: Volume = (1/3)pi h(r1^2 + r2^2 + r1r2), Lateral surface area = pi(r1 + r2)l
- Spherical shell: Volume = (4/3)pi(R^3 - r^3)
14. Statistics
- Mean of grouped data: sum(fi xi) / sum(fi)
- Median of grouped data: l + [(n/2 - cf)/f] x h
- Mode of grouped data: l + [(f1 - f0)/(2f1 - f0 - f2)] x h
15. Probability
- Probability: P(E) = n(E)/n(S)
- Complementary events: P(E') = 1 - P(E)