NDA Study Notes – Maths, English & GAT (Basic Course)

1. Mathematics – High-Value Formulas & Ideas

Target: Formulas yaad + basic intuition samajh lena. Details already chapters me diye gaye hain.

1.1 Algebra – Identities & Equations

• Standard identities: (a + b)² = a² + 2ab + b² (a – b)² = a² – 2ab + b² a² – b² = (a – b)(a + b)
• (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
• Quadratic equation: ax² + bx + c = 0 Roots: x = (–b ± √(b² – 4ac)) / 2a
• Relation of roots: If roots are α, β then α + β = –b/a,   αβ = c/a
• Tip (Hindi): हमेशा पहले discriminant Δ = b² – 4ac निकालिए. अगर Δ negative है तो real roots नहीं होंगे.

1.2 Trigonometry – Ratios & Identities

• Basic ratios: sin θ = Perpendicular / Hypotenuse, cos θ = Base / Hypotenuse, tan θ = Perpendicular / Base
• Important values (0°, 30°, 45°, 60°, 90°) – इन्हें table की तरह याद रखें (exam में सीधा use होता है).
• Key identities:
  • sin²θ + cos²θ = 1
  • 1 + tan²θ = sec²θ
  • 1 + cot²θ = cosec²θ
• Complementary angles: sin(90° – θ) = cos θ, cos(90° – θ) = sin θ, tan(90° – θ) = cot θ
• Tip: Exam में multiple identities combine करके सवाल पूछा जाता है. हमेशा एक identity से दूसरे तक simplify कीजिए.

1.3 Mensuration – Area & Volume

• Rectangle: Area = l × b, Perimeter = 2(l + b)
• Square: Area = a², Perimeter = 4a
• Triangle: Area = ½ × base × height Heron’s formula: Area = √{s(s – a)(s – b)(s – c)}, where s = (a + b + c)/2
• Circle: Circumference = 2πr, Area = πr²
• Cylinder: Curved surface area (CSA) = 2πrh, Total surface area (TSA) = 2πr(r + h), Volume = πr²h
• Cone: Slant height l = √(h² + r²); Volume = ⅓πr²h
• Tip: NDA me mostly direct formula based but conceptual questions bhi aate hain (e.g., painting area, fencing length).

1.4 Number System & Basic Arithmetic

• Types: Natural, Whole, Integers, Rational, Irrational, Real numbers.
• Divisibility basics – 2, 3, 5, 9, 11 rules strongly yaad rakho.
• LCM & HCF by prime factorization.
• Fractions & decimals – conversion, ratio & proportion, percentages.
• Percentage–Profit–Loss–Discount–Simple Interest–Compound Interest ke basic relationships yaad rakho.
• Tip: Speed ke liye basic fraction–percentage pairs याद करें (1/2 = 50%, 1/3 ≈ 33.33%, 1/4 = 25%, 3/4 = 75% etc.).

1.5 Coordinate Geometry & Statistics (Basic)

• Distance between two points: d = √{(x₂ – x₁)² + (y₂ – y₁)²}
• Midpoint of line joining (x₁, y₁), (x₂, y₂): = ((x₁ + x₂)/2, (y₁ + y₂)/2)
• Slope of line: m = (y₂ – y₁) / (x₂ – x₁)
• Mean (Average): Sum of observations / Total number.
• Median: Middle value in arranged data.
• Mode: Most frequent value.

2. English – Grammar & Usage Essentials

NDA English me mostly error detection, vocabulary, comprehension & usage based questions hote hain.

2.1 Tenses – Basic Framework

• Three main tenses: Present, Past, Future. Har tense ke 4 forms: Simple, Continuous, Perfect, Perfect Continuous.
• Simple Present: habitual / general truths – “He goes to school”, “The sun rises in the east.”
• Present Continuous: action happening now – “He is going to school.”
• Simple Past: finished action in past – “He went to school yesterday.”
• Present Perfect: action completed with link to present – “He has finished his homework.”
• Tip (Hindi): Ek table bana ke sare tense forms ko example ke saath likh ke yaad kariye. NDA me mostly sense of tense check hota hai, rare hi complicated structure aata hai.

2.2 Subject–Verb Agreement

• Singular subject → singular verb; plural subject → plural verb. “He goes”, “They go”.
• Words like each, every, everyone, neither, either are usually singular. “Each of the boys is ready.”
• With “as well as / along with / together with” – verb agrees with first subject. “The captain, along with his players, was honoured.”
• Collective nouns (team, jury, family) – usually singular unless each member is acting separately.

2.3 Articles, Prepositions & Common Errors

• a / an – first mention, general sense; the – specific / already known / unique things (the sun, the earth).
• Prepositions: in, on, at – time & place ke saath usage yaad rakhiye. “In the morning”, “at night”, “on Monday”, “at 5 pm”.
• Common error: double negatives – “I don’t want nothing” ❌; correct – “I don’t want anything” ✔.

2.4 Vocabulary & Comprehension

• Synonyms & Antonyms – roz 5–10 words copy me likh ke revise kariye.
• Idioms & Phrases – important NDA list se daily revise.
• Reading Comprehension tip: Pehle question read karo, phir passage; ya passage fast skim karke underline keywords.
• Tip: English improve karne ke liye daily 10–15 minutes English newspaper ka small article ya story padhiye – meaning Hindi me samajh kar.

3. General Awareness (History, Geography, Polity, Economics, Science)

GS me focus hona chahiye – basics + important facts. Depth UPSC civil services jitni nahi chahiye, but NDA ke level ka strong foundation.

3.1 History – High-Yield Pointers

• Ancient: Indus Valley Civilization (town planning, drainage), Vedic age (Rig, Sama, Yajur, Atharva), Mahajanapadas, Mauryan empire (Ashoka – Dhamma, edicts).
• Medieval: Delhi Sultanate (Slave, Khilji, Tughlaq, Sayyid, Lodi), Mughal rulers (Babar, Akbar, Jahangir, Shah Jahan, Aurangzeb) – unke major policies & architecture.
• Modern: British expansion, 1857 revolt, INC formation (1885), Gandhian movements (Non-Cooperation, Civil Disobedience, Quit India), freedom fighters & key dates.
• Tip: Ek timeline page banao jisme important years + events likhe ho. Exam se pehle ek baar fast revise karo.

3.2 Geography – India + World Basics

• Earth basics – latitude, longitude, time zones, important lines (Equator, Tropic of Cancer/Capricorn, Arctic/Antarctic Circle).
• India’s physical features – Himalayas, Northern plains, Peninsular plateau, Coastal plains, Islands.
• Rivers – Indus, Ganga, Brahmaputra systems; major peninsular rivers (Godavari, Krishna, Kaveri, Narmada, Tapi).
• Climate of India – monsoon pattern, seasons, rainfall distribution.
• Resources – coal, petroleum, natural gas, iron ore, important agricultural crops.

3.3 Polity – Constitution & Governance

• Framing of Constitution, Constituent Assembly, adoption date (26 Nov 1949), enforcement (26 Jan 1950).
• Key features – Sovereign, Socialist, Secular, Democratic, Republic.
• Fundamental Rights, Directive Principles of State Policy (DPSP), Fundamental Duties.
• Parliament – Lok Sabha, Rajya Sabha, President; basic functions & powers.
• Prime Minister & Council of Ministers, Governor & Chief Minister basics.
• Tip: Ek small notebook me important Articles ka short list bana ke rakhiye (e.g. 14, 19, 21, 32, 368 etc.).

3.4 Economics – Simple Concepts

• Basic terms – GDP, GNP, inflation, deflation, budget deficit, direct & indirect taxes.
• Indian banking system – RBI, commercial banks, basic functions (CRR, SLR concept level).
• Five-Year Plans (old framework), NITI Aayog ka basic idea.
• Government schemes overview (just awareness level) – rural development, education, health.

3.5 General Science – Physics, Chemistry, Biology Highlights

• Physics: Motion, force, work–energy–power, basic electricity (Ohm’s law, series/parallel circuits), heat & temperature, optics basics.
• Chemistry: Matter & its states, atomic structure, periodic table idea, chemical reactions (combination, decomposition, displacement), acids–bases–salts, metals & non-metals.
• Biology: Cell structure, human body systems (digestive, respiratory, circulatory, nervous), health & diseases, nutrition, basic genetics idea (not very deep).
• Tip: Science ke liye class 9–10 NCERT base book jaisa revision kaafi helpful hai. Notes se headings dekh ke NCERT se cross-check kar sakte ho.