Class 10 Mathematics- Formula

1. Real Numbers

• Euclid’s Division Lemma: For two integers aa and bb, there exist whole numbers qq and rr such that:a=bq+r,0≤r<ba=bq+r,0≤r<b

2. Polynomials

• Sum and Product of Roots for a quadratic equation ax2+bx+c=0ax2+bx+c=0:
  • Sum of roots (α+β)=−b/a(α+β)=−b/a
  • Product of roots (αβ)=c/a(αβ)=c/a

3. Pair of Linear Equations in Two Variables

• General form of a pair of linear equations:a1x+b1y+c1=0anda2x+b2y+c2=0a1​x+b1​y+c1​=0anda2​x+b2​y+c2​=0
• Condition for Consistency:
  • Unique solution: a1a2≠b1b2a2​a1​​=b2​b1​​
  • Infinite solutions: a1a2=b1b2=c1c2a2​a1​​=b2​b1​​=c2​c1​​
  • No solution: a1a2=b1b2≠c1c2a2​a1​​=b2​b1​​=c2​c1​​

4. Quadratic Equations

• Quadratic formula for the equation ax2+bx+c=0ax2+bx+c=0:x=−b±b2−4ac2ax=2a−b±b2−4ac​​

5. Arithmetic Progression

• nth term of an AP:an=a+(n−1)×dan​=a+(n−1)×d
• Sum of first nn terms:Sn=n2[2a+(n−1)×d]Sn​=2n​[2a+(n−1)×d]

6. Triangles

• Pythagoras Theorem: In a right-angled triangle:Hypotenuse2=Base2+Height2Hypotenuse2=Base2+Height2
• Area of similar triangles:Area of triangle 1Area of triangle 2=(Side of triangle 1Side of triangle 2)2Area of triangle 2Area of triangle 1​=(Side of triangle 2Side of triangle 1​)2

7. Coordinate Geometry

• Distance formula:d=(x2−x1)2+(y2−y1)2d=(x2​−x1​)2+(y2​−y1​)2​
• Section formula:(mx2+nx1m+n,my2+ny1m+n)(m+nmx2​+nx1​​,m+nmy2​+ny1​​)

8. Trigonometry

• Trigonometric ratios:sin⁡θ=Opposite sideHypotenuse,cos⁡θ=Adjacent sideHypotenuse,tan⁡θ=Opposite sideAdjacent sidesinθ=HypotenuseOpposite side​,cosθ=HypotenuseAdjacent side​,tanθ=Adjacent sideOpposite side​
• Pythagorean identity:sin⁡2θ+cos⁡2θ=1sin2θ+cos2θ=1

9. Circles

• The tangent to a circle is perpendicular to the radius at the point of contact.
• Lengths of tangents from an external point to a circle are equal.

10. Mensuration (Surface Areas and Volumes)

• Surface area of a sphere:4πr24πr2
• Volume of a sphere:43πr334​πr3
• Surface area of a cylinder:2πrh+2πr22πrh+2πr2
• Volume of a cylinder:πr2hπr2h

11. Statistics

• Mean (for grouped data):xˉ=∑fixi∑fixˉ=∑fi​∑fi​xi​​
• Median (for grouped data):Median=l+(N2−Ff)×hMedian=l+(f2N​−F​)×h
• Mode (for grouped data):Mode=l+(f1−f02f1−f0−f2)×hMode=l+(2f1​−f0​−f2​f1​−f0​​)×h

12. Probability

• Probability of an event:P(E)=Number of favorable outcomesTotal number of outcomesP(E)=Total number of outcomesNumber of favorable outcomes​

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