Class 10th Maths – Important Notes (MP Board)

 

Class 10th Maths – Important Notes (MP Board)

Chapter 1: Real Numbers

• Every composite number can be expressed as the product of primes → Fundamental Theorem of Arithmetic

• Euclid’s Division Lemma:
  For any two positive integers a and b, ∃ unique integers q, r such that
  a = bq + r, 0 ≤ r < b

• HCF × LCM = Product of Numbers (only for 2 numbers)

• Decimal expansion of rational numbers:

  • Terminates when denominator (prime factorized) is of form 2ᵐ × 5ⁿ

  • Non-terminating repeating otherwise

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Chapter 2: Polynomials

• Polynomial in x: p(x) = a₀ + a₁x + …

• Degree: Highest power of x

• Zeros of a polynomial → p(α) = 0

• Quadratic polynomial: ax² + bx + c

  • Sum of zeros = –b/a

  • Product of zeros = c/a

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Chapter 3: Pair of Linear Equations in Two Variables

Forms: a₁x + b₁y + c₁ = 0
Solutions by:

• Graphical method

• Substitution method

• Elimination method

• Cross-multiplication method

Condition of solutions:

Condition	a₁/a₂	b₁/b₂	c₁/c₂	Result
Unique solution	≠	≠	—	Intersecting lines
No solution	=	=	≠	Parallel lines
Infinite solutions	=	=	=	Coincident lines

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Chapter 4: Quadratic Equations

Standard form: ax² + bx + c = 0
Solutions:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Discriminant (D) = b² – 4ac

D	Nature of Roots
D > 0	Real & distinct
D = 0	Real & equal
D < 0	Not real

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Chapter 5: Arithmetic Progression (AP)

• General form: a, a+d, a+2d, …

• nth term: aₙ = a + (n-1)d

• Sum of n terms:

  $$Sₙ = \frac{n}{2}[2a + (n-1)d]$$

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Chapter 6: Triangles

• Basic Proportionality Theorem (BPT):
  If a line parallel to one side of a triangle intersects other two sides → divides them in same ratio

• Similarity → AAA, SSS, SAS

• Pythagoras Theorem:
  In right triangle, (Hypotenuse)² = (Base)² + (Height)²

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Chapter 7: Coordinate Geometry

Distance formula:

$$d = \sqrt{(x₂ - x₁)^2 + (y₂ - y₁)^2}$$

Mid-point:

$$\left(\frac{x₁ + x₂}{2}, \frac{y₁ + y₂}{2}\right)$$

Section formula:

$$\left(\frac{mx₂ + nx₁}{m+n}, \frac{my₂ + ny₁}{m+n}\right)$$

Area of triangle:

$$\frac{1}{2}|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|$$

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Chapter 8: Introduction to Trigonometry

$$\sin \theta = \frac{P}{H},\ \cos \theta = \frac{B}{H},\ \tan \theta = \frac{P}{B}$$

Important identities:

1. $$$$

\sin^2 \theta + \cos^2 \theta = 1
]
2. $\tan \theta = \frac{\sin \theta}{\cos \theta}$

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Chapter 9: Applications of Trigonometry

Used in finding:

• Height of object

• Distance between objects
  (using angles of elevation and depression)

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Chapter 10: Circles

• Tangent → touches circle at one point

• Radius ⟂ Tangent at point of contact

• If two tangents from same point:

  • Lengths equal

  • They subtend equal angles

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Chapter 11: Constructions

• Similar triangles

• Division of line in given ratio

• Tangents to a circle from a point outside

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Chapter 12: Areas Related to Circles

• Circumference = 2πr

• Area = πr²

• Area of sector:

  $$\frac{\theta}{360^\circ} \times \pi r^2$$

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Chapter 13: Surface Areas & Volumes

Important formulas:

• Cuboid:

  • Volume = l × b × h

• Cylinder:

  • Curved SA = 2πrh

  • Volume = πr²h

• Sphere:

  • CSA = 4πr²

  • Volume = 4/3 πr³

• Cone:

  • CSA = πrl

  • Volume = 1/3 πr²h

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Chapter 14: Statistics

• Mean = ∑fᵢxᵢ / ∑fᵢ

• Mode = value with highest frequency

• Median = middle value when data arranged

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Chapter 15: Probability

$$\text{Probability} = \frac{\text{Favourable outcomes}}{\text{Total outcomes}}$$

Values always between 0 and 1