Class 10 — Notes 📙

Key points, formulas and digest answers — SSC Maharashtra Board & CBSE Board

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Mathematics

Chapter 1 — Real Numbers

📌 Key Points
  • Euclid's Division Lemma: For any two integers a and b, a = bq + r where 0 ≤ r < b
  • HCF: found using Euclid's division algorithm — apply lemma repeatedly till remainder = 0
  • Fundamental Theorem of Arithmetic: every composite number can be expressed as a product of primes in a unique way
  • HCF × LCM = Product of two numbers
  • If p/q is rational and in lowest terms, then denominator q = 2ⁿ × 5ᵐ for terminating decimal
  • √2, √3, √5 are all irrational numbers — proved by contradiction
✅ Digest Answers
Q. 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
Q. Prove √2 is irrational.
Assume √2 = p/q (lowest terms). Then 2 = p²/q² → p² = 2q² → p² is even → p is even → p = 2k → 4k² = 2q² → q² = 2k² → q is even. But p and q both even contradicts our assumption that p/q is in lowest terms. So √2 is irrational.
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Mathematics

Chapter 3 — Pair of Linear Equations

📌 Key Points
  • General form: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
  • Unique solution (consistent): a₁/a₂ ≠ b₁/b₂ — lines intersect
  • Infinite solutions: a₁/a₂ = b₁/b₂ = c₁/c₂ — lines coincide
  • No solution (inconsistent): a₁/a₂ = b₁/b₂ ≠ c₁/c₂ — lines parallel
  • Methods: Substitution, Elimination, Cross-multiplication, Graphical
✅ Digest Answers
Q. Solve by elimination: x + y = 5 and x − y = 1
Adding both: 2x = 6 → x = 3. Substituting: 3 + y = 5 → y = 2. Solution: x = 3, y = 2
Q. When does a pair of linear equations have no solution?
When the lines are parallel — they never meet. Condition: a₁/a₂ = b₁/b₂ ≠ c₁/c₂. The system is called inconsistent.
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Mathematics

Chapter 8 — Introduction to Trigonometry

📌 Key Formulas
  • sin θ = Opposite/Hypotenuse, cos θ = Adjacent/Hypotenuse, tan θ = Opposite/Adjacent
  • cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
  • Identity 1: sin²θ + cos²θ = 1
  • Identity 2: 1 + tan²θ = sec²θ
  • Identity 3: 1 + cot²θ = cosec²θ
  • sin 0°=0, sin 30°=1/2, sin 45°=1/√2, sin 60°=√3/2, sin 90°=1
  • cos 0°=1, cos 30°=√3/2, cos 45°=1/√2, cos 60°=1/2, cos 90°=0
  • tan 0°=0, tan 30°=1/√3, tan 45°=1, tan 60°=√3, tan 90°=undefined
✅ Digest Answers
Q. Prove: sin²θ + cos²θ = 1
In right triangle ABC, AB² + BC² = AC² (Pythagoras). Dividing by AC²: (AB/AC)² + (BC/AC)² = 1 → sin²θ + cos²θ = 1 ✓
Q. Find sin 60° × cos 30° + cos 60° × sin 30°
= (√3/2)(√3/2) + (1/2)(1/2) = 3/4 + 1/4 = 1 (= sin 90°)
⚗️
Science — Chemistry

Chapter 1 — Chemical Reactions and Equations

📌 Key Points
  • Signs of chemical reaction: change in colour, temperature, state, smell, gas evolved, precipitate formed
  • Combination: A + B → AB. Example: C + O₂ → CO₂
  • Decomposition: AB → A + B. Example: 2H₂O → 2H₂ + O₂
  • Displacement: A + BC → AC + B. Example: Zn + CuSO₄ → ZnSO₄ + Cu
  • Double displacement: AB + CD → AD + CB. Example: AgNO₃ + NaCl → AgCl + NaNO₃
  • Oxidation: gain of oxygen or loss of hydrogen or loss of electrons
  • Reduction: loss of oxygen or gain of hydrogen or gain of electrons
  • Redox reaction: oxidation and reduction occur simultaneously
  • Rancidity: oxidation of oils and fats giving bad smell. Prevented by antioxidants, vacuum packing, N₂ flushing
✅ Digest Answers
Q. What is a displacement reaction? Give an example.
A reaction where a more reactive element displaces a less reactive element from its compound. Example: Fe + CuSO₄ → FeSO₄ + Cu. Iron displaces copper because iron is more reactive than copper.
Q. Balance: Fe + H₂O → Fe₃O₄ + H₂
3Fe + 4H₂O → Fe₃O₄ + 4H₂
Q. What is corrosion? How to prevent it?
Corrosion is the slow oxidation of metals in presence of air and moisture. Example: rusting of iron (Fe → Fe₂O₃). Prevention: painting, galvanizing (zinc coating), oiling, alloying.
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Science — Physics

Chapter 12 — Electricity

📌 Key Formulas
  • Electric current I = Q/t. Unit: Ampere (A)
  • Ohm's Law: V = IR where V = voltage, I = current, R = resistance
  • Resistance R = ρL/A where ρ = resistivity, L = length, A = area
  • Resistors in series: R = R₁ + R₂ + R₃
  • Resistors in parallel: 1/R = 1/R₁ + 1/R₂ + 1/R₃
  • Power P = VI = I²R = V²/R. Unit: Watt (W)
  • Electric energy E = P × t = VIt. Unit: Joule or kWh
  • 1 kWh = 3.6 × 10⁶ J (one unit of electricity)
  • Heating effect of current: H = I²Rt (Joule's law)
✅ Digest Answers
Q. State Ohm's Law.
At constant temperature, the current flowing through a conductor is directly proportional to the potential difference across it. V = IR. The ratio V/I is constant and is called resistance.
Q. Why are household appliances connected in parallel?
In parallel: 1) Each appliance gets the same voltage (220V). 2) Each appliance works independently — if one fails others keep working. 3) Total resistance decreases allowing more current.
Q. A bulb has resistance 100Ω. Voltage = 200V. Find power.
P = V²/R = 200²/100 = 40000/100 = 400 W
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History

Rise of Nationalism in Europe

📌 Key Points
  • Nationalism: feeling of pride and loyalty towards one's nation
  • French Revolution (1789): first major event that spread ideas of nationalism
  • Napoleon: spread revolutionary ideas across Europe through conquests
  • Unification of Germany: led by Otto von Bismarck using blood and iron policy, completed 1871
  • Unification of Italy: led by Garibaldi, Mazzini, Cavour, completed 1861
  • Zollverein: German customs union (1834) that helped economic unity
  • Balkans: region of conflict due to nationalism and fall of Ottoman Empire
  • Romanticism: cultural movement that promoted national identity through art, poetry, music
✅ Digest Answers
Q. What role did Napoleon play in spreading nationalism?
Napoleon's conquests spread ideas of liberty, equality and fraternity from French Revolution to Europe. He introduced the Civil Code (1804) abolishing feudal privileges. Though he was a conqueror, his administrative reforms inspired nationalism in conquered territories.
Q. Who were the three architects of Italian unification?
1. Giuseppe Mazzini — gave vision and ideology of unified Italy. 2. Count Cavour — used diplomacy and war. 3. Giuseppe Garibaldi — led armed movement in southern Italy with his Red Shirts.
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English

Letter Writing & Grammar

📌 Key Points — Formal Letter
  • Formal letter format: Sender's address → Date → Receiver's address → Subject → Salutation → Body → Closing → Signature
  • Salutation for formal: Dear Sir/Madam
  • Closing for formal: Yours faithfully / Yours sincerely
  • Subject line should be short and clear
  • Active voice: Subject does the action — "Ram ate the mango"
  • Passive voice: Subject receives the action — "The mango was eaten by Ram"
  • Passive voice formation: Object + is/am/are/was/were + past participle + by + subject
✅ Digest Answers
Q. Change to passive: "The teacher teaches the students."
The students are taught by the teacher.
Q. What is the difference between "Yours faithfully" and "Yours sincerely"?
"Yours faithfully" is used when you do NOT know the name of the person (Dear Sir/Madam). "Yours sincerely" is used when you know and use the name of the person (Dear Mr. Sharma).

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