Notes for Class 10 Mathematics 

                                                              

Real Numbers

Real numbers include rational and irrational numbers, with key concepts like Euclid’s Division Lemma, the Fundamental Theorem of Arithmetic, and decimal expansions. Euclid’s Lemma helps find the HCF of two numbers, while the Fundamental Theorem ensures unique prime factorization. Decimal expansions distinguish rational numbers (terminating or non-terminating recurring) from irrational ones. Study steps include practicing Euclid’s algorithm, proving irrationality, and solving NCERT exercises 1.1–1.4. Supplement with RD Sharma for deeper practice.

Formulas:

  • Euclid’s Division Lemma: a = bq + r, where 0 ≤ r < b
  • HCF using Euclid’s algorithm: Apply lemma repeatedly until remainder is 0
  • LCM = (a × b)/HCF(a, b)

Example:
NCERT Exercise 1.1, Q1: Find HCF of 135 and 225. Apply Euclid’s algorithm: 225 = 135 × 1 + 90, 135 = 90 × 1 + 45, 90 = 45 × 2 + 0. Thus, HCF = 45.



Polynomials

Polynomials are expressions like ax² + bx + c, with focus on finding zeros, relating zeros to coefficients, and using the division algorithm. Zeros are values where p(x) = 0, and for quadratics, the sum and product of roots relate to coefficients. The division algorithm ensures p(x) = g(x)q(x) + r(x). Study steps include factorizing polynomials, verifying zeros, and solving NCERT exercises 2.1–2.4, with RS Aggarwal for extra practice.

Formulas:

  • Sum of roots (quadratic): -b/a
  • Product of roots (quadratic): c/a
  • Division algorithm: p(x) = g(x)q(x) + r(x)

Example:
NCERT Example 2: Find zeros of p(x) = x² - 3x + 2. Factorize: (x - 1)(x - 2) = 0, so zeros are x = 1, 2. Verify: Sum = 1 + 2 = 3 = -(-3)/1, Product = 1 × 2 = 2 = 2/1.


Pair of Linear Equations in Two Variables

This topic involves solving equations like ax + by = c using substitution, elimination, or cross-multiplication, and checking consistency via ratios a₁/a₂, b₁/b₂, c₁/c₂. Graphical solutions involve plotting lines to find intersections. Study steps include mastering algebraic methods, plotting graphs, and solving word problems in NCERT exercises 3.1–3.7. Practice with Oswaal sample papers for variety.

Formulas:

  • Substitution: Solve one equation for x or y, substitute into the other
  • Elimination: Add/subtract equations to eliminate one variable
  • Consistency: a₁/a₂ ≠ b₁/b₂ (unique), a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (infinite), a₁/a₂ = b₁/b₂ = c₁/c₂ (no solution)

Example:
NCERT Example 3: Solve 2x + 3y = 11, 2x - 4y = -24. Substitution: From first, x = (11 - 3y)/2, substitute into second, solve to get y = 7, x = -5.


Quadratic Equations

Quadratic equations (ax² + bx + c = 0) are solved by factorization, completing the square, or the quadratic formula. The discriminant (D = b² - 4ac) determines root nature: real (D ≥ 0), equal (D = 0), or no real roots (D < 0). Study steps include memorizing the formula, practicing factorization, and solving word problems in NCERT exercises 4.1–4.4, with RD Sharma for advanced problems.

Formulas:

  • Quadratic formula: x = [-b ± √(b² - 4ac)]/2a
  • Discriminant: D = b² - 4ac
  • Sum of roots: -b/a, Product of roots: c/a

Example:
NCERT Example 4: Solve x² - 3x - 10 = 0. Factorize: (x - 5)(x + 2) = 0, roots x = 5, -2. Verify: D = 9 + 40 = 49, x = [3 ± √49]/2 = 5, -2.


Arithmetic Progressions

An arithmetic progression (AP) is a sequence with constant difference d. The nth term and sum of n terms are key. Study steps include identifying APs, calculating terms, and solving word problems in NCERT exercises 5.1–5.4. Practice real-life applications like savings or seating arrangements.

Formulas:

  • nth term: aₙ = a + (n-1)d
  • Sum of n terms: Sₙ = n/2 [2a + (n-1)d]

Example:
NCERT Example 5: Find 10th term of AP 5, 8, 11, …. Here, a = 5, d = 3, so a₁₀ = 5 + (10-1) × 3 = 32.



Triangles

Triangles focus on similarity (AA, SSS, SAS), Pythagoras theorem, and Thales’ theorem. Similar triangles have proportional sides and equal angles. Study steps include learning theorems, proving similarity, and solving NCERT exercises 6.1–6.6, with previous year papers for practice.

Formulas:

  • Pythagoras theorem: a² + b² = c² (for right triangles)
  • Thales’ theorem: DE/BC = AD/DB = AE/EC (if DE || BC)
  • Area ratio of similar triangles: (Side₁/Side₂)²

Example:
NCERT Exercise 6.3, Q1: In a right triangle with legs 3 cm, 4 cm, find hypotenuse. Apply Pythagoras: AC² = 3² + 4² = 25, so AC = 5 cm.


Coordinate Geometry

Coordinate geometry uses formulas to find distances, section points, and triangle areas. Study steps include memorizing formulas, solving collinearity problems, and practicing NCERT exercises 7.1–7.4, with RS Aggarwal for advanced problems.

Formulas:

  • Distance: √[(x₂ - x₁)² + (y₂ - y₁)²]
  • Section formula: [(m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂)]
  • Area of triangle: ½ |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|

Example:
NCERT Example 3: Find distance between (2, 3) and (4, 1). Apply formula: √[(4-2)² + (1-3)²] = √(4 + 4) = 2√2.


Introduction to Trigonometry

Trigonometric ratios (sin, cos, tan) and identities are key, with standard angle values. Study steps include memorizing ratios, practicing identities, and solving NCERT exercises 8.1–8.4, with RD Sharma for complex problems.

Formulas:

  • sin²θ + cos²θ = 1
  • 1 + tan²θ = sec²θ
  • 1 + cot²θ = cosec²θ

Example:
NCERT Example 5: Evaluate sin 60° cos 30° + cos 60° sin 30°. Use values: (√3/2)(√3/2) + (1/2)(1/2) = 3/4 + 1/4 = 1.


Some Applications of Trigonometry

This chapter applies trigonometry to heights and distances using angles of elevation/depression. Study steps include understanding angles, solving triangle problems, and practicing NCERT exercise 9.1, with Oswaal for applications.

Formulas:

  • tan θ = Height/Distance
  • sin θ = Height/Hypotenuse, cos θ = Distance/Hypotenuse

Example:
NCERT Example 3: Find height of a pole with angle of elevation 60°, distance 20 m. Use tan 60° = h/20, so h = 20√3 m.


Circles

Circles involve tangent properties (e.g., perpendicular to radius) and number of tangents from a point. Study steps include learning theorems, solving tangent length problems, and practicing NCERT exercises 10.1–10.2.

Formulas:

  • Length of tangent: √(Distance from center² - Radius²)
  • Tangents from a point: PA = PB

Example:
NCERT Exercise 10.2, Q1: Find tangent length from (5, 4) to circle with center (2, 1), radius 3. Distance OP = √18, tangent = √(18 - 9) = 3 units.


Areas Related to Circles

This topic covers areas of circles, sectors, and segments. Study steps include memorizing formulas, solving shaded region problems, and practicing NCERT exercises 11.1–11.3, with RD Sharma for complex figures.

Formulas:

  • Area of circle: πr²
  • Sector area: (θ/360) × πr²
  • Segment area: Sector area - Triangle area

Example:
NCERT Example 3: Find area of sector with radius 6 cm, angle 60°. Area = (60/360) × π × 6² = 6π cm².



Surface Areas and Volumes

This chapter involves surface area and volume of solids like cones, cylinders, and frustums. Study steps include memorizing formulas, solving conversion problems, and practicing NCERT exercises 12.1–12.2.

Formulas:

  • Volume of cone: ⅓ πr²h
  • Surface area of sphere: 4πr²
  • Frustum volume: ⅓ πh(R² + r² + Rr)

Example:
NCERT Example 4: Find volume of cone with radius 7 cm, height 24 cm. Volume = ⅓ π × 7² × 24 = 1232 cm³.


Statistics

Statistics involves mean, median, mode, and ogives for data analysis. Study steps include learning formulas, drawing ogives, and solving NCERT exercises 13.1–13.4, with RS Aggarwal for graphical problems.

Formulas:

  • Mean (step-deviation): a + (Σfidi/Σfi) × h
  • Median: l + [(n/2 - cf)/f] × h
  • Mode: l + [(f₁ - f₀)/(2f₁ - f₀ - f₂)] × h

Example:
NCERT Example 3: Find mean of grouped data using step-deviation. Σfidi/Σfi = 35, mean = 50 + (35 × 10)/100 = 52.5.



Probability

Probability measures event likelihood: P(E) = favorable outcomes/total outcomes. Study steps include understanding outcomes, solving problems on coins, dice, and cards, and practicing NCERT exercise 14.1.

Formulas:

  • Probability: P(E) = Number of favorable outcomes/Total outcomes
  • Complementary event: P(not E) = 1 - P(E)

Example:
NCERT Example 2: Find probability of drawing a red ball from a bag with 5 red, 8 green balls. P = 5/(5+8) = 5/13.


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