Class 10 Mathematics — Complete Notes and Solved PYQs

Chapter 1: Real Numbers

Euclid's Division Lemma

For any two positive integers a and b: a = bq + r (0 ≤ r < b)

HCF using Euclid's Algorithm: Example: HCF(72, 48)

• 72 = 48 × 1 + 24
• 48 = 24 × 2 + 0
• HCF = 24

Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of primes in exactly one way (ignoring order).

HCF × LCM = Product of two numbers

Irrational Numbers

√2, √3, √5, π, e are irrational. Proof by contradiction: Assume √2 = p/q (in lowest terms) → 2 = p²/q² → p² = 2q² → p is even → q is also even → contradicts lowest terms assumption.

Decimal Expansions

• Terminating: denominator has only factors 2 and 5. Example: 7/8 = 0.875
• Non-terminating recurring: other denominators. Example: 1/3 = 0.333...
• Non-terminating non-recurring: irrational numbers

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Chapter 2: Polynomials

Degree: Highest power of variable.

• Linear: ax + b (degree 1) — 1 zero
• Quadratic: ax² + bx + c (degree 2) — 2 zeros
• Cubic: ax³ + bx² + cx + d (degree 3) — 3 zeros

Relationship between zeros and coefficients:

For quadratic ax² + bx + c with zeros α and β:

• Sum of zeros: α + β = -b/a
• Product of zeros: αβ = c/a

For cubic ax³ + bx² + cx + d with zeros α, β, γ:

• α + β + γ = -b/a
• αβ + βγ + αγ = c/a
• αβγ = -d/a

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Chapter 4: Quadratic Equations

Standard form: ax² + bx + c = 0 (a ≠ 0)

Methods of Solution:

1. Factoring: Find two numbers p,q where p+q=b, p×q=ac
2. Completing the Square: Make perfect square trinomial
3. Quadratic Formula: x = (-b ± √(b²-4ac)) / 2a

Discriminant D = b² - 4ac:

| D | Nature of Roots | |---|---| | D > 0 | Two distinct real roots | | D = 0 | Two equal real roots | | D < 0 | No real roots (complex) |

Solved Example: Solve 2x² - 5x + 3 = 0

• D = 25 - 24 = 1 > 0
• x = (5 ± 1) / 4
• x = 3/2 or x = 1

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Chapter 6: Triangles

Similarity of Triangles

Two triangles are similar if their corresponding angles are equal and sides are proportional.

Criteria:

• AA (Angle-Angle): Two angles equal
• SSS: All three sides proportional
• SAS: Two sides proportional and included angle equal

Basic Proportionality Theorem (Thales' Theorem): If a line is parallel to one side of a triangle and intersects the other two sides, it divides them in the same ratio. DE || BC → AD/DB = AE/EC

Pythagoras Theorem: In right triangle: AC² = AB² + BC² (hypotenuse² = sum of squares of other two sides)

Converse: If AC² = AB² + BC², then angle B = 90°.

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Chapter 8: Introduction to Trigonometry

Trigonometric Ratios (for angle θ):

• sin θ = Opposite/Hypotenuse
• cos θ = Adjacent/Hypotenuse
• tan θ = Opposite/Adjacent = sin θ/cos θ
• cosec θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ

Standard Values:

| Angle | 0° | 30° | 45° | 60° | 90° | |---|---|---|---|---|---| | sin | 0 | 1/2 | 1/√2 | √3/2 | 1 | | cos | 1 | √3/2 | 1/√2 | 1/2 | 0 | | tan | 0 | 1/√3 | 1 | √3 | undefined |

Trigonometric Identities:

1. sin²θ + cos²θ = 1
2. 1 + tan²θ = sec²θ
3. 1 + cot²θ = cosec²θ

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Chapter 7: Coordinate Geometry

Distance Formula: d = √[(x₂-x₁)² + (y₂-y₁)²]

Section Formula: Point dividing (x₁,y₁) to (x₂,y₂) in ratio m:n

• x = (mx₂ + nx₁)/(m+n)
• y = (my₂ + ny₁)/(m+n)

Midpoint Formula: ((x₁+x₂)/2, (y₁+y₂)/2)

Area of Triangle: ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|

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Chapter 13: Surface Areas and Volumes

| Shape | Curved SA | Total SA | Volume | |---|---|---|---| | Cube (side a) | — | 6a² | a³ | | Cuboid | — | 2(lb+bh+lh) | lbh | | Cylinder (r,h) | 2πrh | 2πr(r+h) | πr²h | | Cone (r,l,h) | πrl | πr(r+l) | ⅓πr²h | | Sphere (r) | 4πr² | 4πr² | ⁴⁄₃πr³ | | Hemisphere | 2πr² | 3πr² | ⅔πr³ |

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Chapter 14: Statistics

Mean:

• Direct method: x̄ = Σfᵢxᵢ / Σfᵢ
• Assumed mean: x̄ = a + Σfᵢdᵢ / Σfᵢ

Median: Value at (n/2)th position For grouped data: M = l + [(n/2 - cf)/f] × h

Mode: Most frequent value For grouped data: Mode = l + [(f₁-f₀)/(2f₁-f₀-f₂)] × h

Relation: Mode = 3(Median) - 2(Mean)

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Solved CBSE PYQs

Q1 (2023): Find HCF of 96 and 404 using Euclid's Algorithm. 404 = 96×4 + 20 → 96 = 20×4 + 16 → 20 = 16×1 + 4 → 16 = 4×4 + 0 HCF = 4

Q2 (2023): If one zero of polynomial 3x²+kx-4 is 4/3, find k. 3(4/3)²+k(4/3)-4=0 → 3(16/9)+4k/3-4=0 → 16/3+4k/3=4 → 4k/3=-4/3 → k=-1

Q3 (2022): Prove sin⁴θ - cos⁴θ = 2sin²θ - 1 LHS = (sin²θ + cos²θ)(sin²θ - cos²θ) = 1×(sin²θ - cos²θ) = sin²θ - (1-sin²θ) = 2sin²θ - 1 = RHS ✓

Q4 (2024): Two poles of heights 6m and 11m are 12m apart. Find the distance between tops. Using Pythagoras: d = √(12² + 5²) = √(144+25) = √169 = 13m

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Quick Revision Checklist

• Euclid's algorithm: a = bq + r
• HCF × LCM = Product of numbers
• Quadratic formula and discriminant conditions
• Sum and product of roots: -b/a and c/a
• Thales' theorem: DE||BC → AD/DB = AE/EC
• Pythagoras theorem and its converse
• Trig values for 0°, 30°, 45°, 60°, 90°
• Three trig identities (sin²+cos²=1 etc.)
• Distance, section, midpoint formulas
• Mean, median, mode formulas for grouped data