Trace the evolution of the 'Basic Structure Doctrine' of the Indian Constitution across Shankari Prasad (1951), Golaknath (1967), Kesavananda Bharati (1973), and Minerva Mills (1980).

1. Shankari Prasad (1951): SC held Parliament can amend Fundamental Rights under Art. 368 and a Constitutional Amendment is not 'law' under Art. 13(2). 2. Golaknath (1967): SC reversed itself, ruling Fundamental Rights are transcendental and cannot be abridged by Parliament. 3. Kesavananda Bharati v. State of Kerala (24 April 1973, 13-judge bench, 7:6 majority): Upheld the 24th Amendment confirming Parliament can amend any part of the Constitution (including FRs) under Art. 368, PROVIDED it does not alter or destroy the 'Basic Structure' of the Constitution. 4. Minerva Mills (1980): Struck down clauses of the 42nd Amendment, declaring judicial review and harmony between Part III (FRs) and Part IV (DPSPs) part of the Basic Structure.

In UPSC Ancient Indian History, match the major Indus Valley Civilization (Harappan) sites to their key archaeological findings and river locations.

1. Harappa (Ravi river, Daya Ram Sahni, 1921): Two rows of 6 granaries, coffin burial, red sandstone male torso. 2. Mohenjo-daro (Indus river, R.D. Banerji, 1922): Great Bath, Great Granary, Bronze Dancing Girl (lost-wax casting), Pashupati Seal, bearded priest. 3. Lothal (Bhogava river, Gujarat): Artificial brick dockyard (tidal port), double burial, rice husks, fire altars, bead-making factory. 4. Kalibangan (Ghaggar river, Rajasthan): Ploughed field surface, fire altars, camel bones, no street drainage system. 5. Dholavira (Rann of Kutch, UNESCO site): Divided into 3 parts (Citadel, Middle Town, Lower Town), giant rock-cut water reservoirs/dams, 10-sign signboard inscription. 6. Rakhigarhi (Haryana): Largest IVC site in India.

In UPSC Ancient History, identify the key Ashokan Major Rock Edicts (especially XIII, XII, and Maski/Gujarra) and Kautilya's Saptanga Theory of the Mauryan State.

Ashoka's inscriptions (deciphered by James Prinsep in 1837, mostly in Prakrit language and Brahmi script, plus Kharosthi in NW and Greek/Aramaic in Kandahar): 1. Major Rock Edict XIII: Describes the Kalinga War (261 BCE, 8th regnal year), Ashoka's remorse, conversion from Bherighosha to Dhammaghosha, and mentions 5 contemporary Hellenistic kings (Antiochus, Ptolemy, Antigonus, Magas, Alexander). 2. Minor Rock Edicts at Maski, Gujarra, Nittur, and Udegolam explicitly mention Ashoka by his personal name ('Devanampiya Piyadassi Ashoka'). 3. Kautilya's Arthashastra Saptanga Theory (7 limbs of state): Swami (King), Amatya (Minister), Janapada (Territory/People), Durga (Fort), Kosha (Treasury), Danda/Bala (Army), and Mitra (Ally).

Distinguish the five Constitutional Writs issued by the Supreme Court (Article 32) and High Courts (Article 226) in Indian Polity, and explain how the High Court's writ jurisdiction is wider than the Supreme Court's.

1. Habeas Corpus ('to have the body of'): Protects personal liberty against illegal detention (can be issued against both public authorities and private individuals!). 2. Mandamus ('we command'): Directs a public official/body to perform a mandatory public duty (cannot be issued against the President, Governor, or private individuals). 3. Prohibition ('to forbid'): Issued by a higher court to a lower court/tribunal to prevent it from exceeding its jurisdiction (preventive). 4. Certiorari ('to be certified'): Quashes an order already passed by a lower court/tribunal in excess of jurisdiction or error of law (curative). 5. Quo Warranto ('by what authority'): Challenges a person's legal claim to a public office. High Courts (Art. 226) can issue writs for Fundamental Rights AND 'for any other purpose' (ordinary legal rights), making their territorial scope narrower but subject-matter jurisdiction wider than Art. 32.

In JEE Rotational Mechanics, state the Parallel Axis Theorem, Perpendicular Axis Theorem, Moments of Inertia of standard bodies (ring, disc, hollow/solid sphere, rod), and pure rolling acceleration down an incline.

1. Parallel Axis Theorem: I = I_cm + M d²; Perpendicular Axis Theorem (for planar lamina): I_z = I_x + I_y. 2. Moments of Inertia about CM axis: Thin Ring = MR²; Circular Disc / Solid Cylinder = ½MR²; Hollow Sphere = ⅔MR²; Solid Sphere = ⅖MR²; Thin Rod (length L) = (1/12)ML² about center and (1/3)ML² about one end. 3. For a body with radius of gyration k rolling without slipping down an incline θ: acceleration a = g sin θ / (1 + k²/R²), and minimum coefficient of static friction required is μ_min = tan θ / (1 + R²/k²).

What were the subjects of the 101st, 102nd, 103rd, 104th, 105th, and 106th Constitutional Amendment Acts of India?

1. 101st Amendment (2016): Introduced Goods and Services Tax (GST) and Art. 246A, 269A, 279A. 2. 102nd Amendment (2018): Granted constitutional status to National Commission for Backward Classes (NCBC, Art. 338B). 3. 103rd Amendment (2019): Introduced 10% reservation for Economically Weaker Sections (EWS) under Art. 15(6) and 16(6) (upheld in Janhit Abhiyan case). 4. 104th Amendment (2020): Extended SC/ST reservation in Lok Sabha & State Assemblies by 10 years and ended the nomination of 2 Anglo-Indians (Art. 331/333). 5. 105th Amendment (2021): Restored State governments' power to identify and maintain their own SEBC/OBC lists. 6. 106th Amendment (2023, Nari Shakti Vandan Adhiniyam): Reserved one-third (33%) of seats for women in Lok Sabha (Art. 330A), State Legislative Assemblies (Art. 332A), and NCT Delhi Assembly.

In JEE Calculus, state Newton-Leibniz's Rule for differentiating a definite integral with variable limits, and the King's Property + Queen's Property of definite integrals.

1. Leibniz Rule: d/dx [∫_{g(x)}^{h(x)} f(t) dt] = f(h(x))·h'(x) − f(g(x))·g'(x). 2. King's Property: ∫_a^b f(x) dx = ∫_a^b f(a + b − x) dx (especially ∫_0^a f(x) dx = ∫_0^a f(a − x) dx). 3. Queen's Property: ∫_0^{2a} f(x) dx = ∫_0^a [f(x) + f(2a − x)] dx, which equals 2∫_0^a f(x) dx if f(2a − x) = f(x) and 0 if f(2a − x) = −f(x). Also, Wallis's Formula: ∫_0^{π/2} sin^n(x) dx = ∫_0^{π/2} cos^n(x) dx.

In JEE Simple Harmonic Motion (SHM) & Waves, give the velocity v(x), energy, simple/physical pendulum periods, Doppler effect formula, and organ pipe harmonics.

1. In SHM (x = A sin(ωt + φ)): v = ω√(A² − x²), a = −ω²x, Total Energy E = ½mω²A². 2. Simple Pendulum: T = 2π√(L/g); Physical (Compound) Pendulum: T = 2π√[I / (mgd)]. 3. Closed Organ Pipe (one end closed, length L): Only odd harmonics present, f_n = (2n − 1)v / (4L); Open Organ Pipe (both ends open): All harmonics present, f_n = n v / (2L).

In UPSC Polity, list the major Constitutional Bodies in India alongside their exact Constitutional Articles.

1. Election Commission of India (Art. 324); 2. Union & State Public Service Commissions — UPSC/SPSC (Art. 315–323); 3. Finance Commission (Art. 280 — constituted every 5 years by the President for tax devolution); 4. Goods and Services Tax (GST) Council (Art. 279A — added by 101st Amendment, 2016, chaired by Union Finance Minister); 5. Comptroller and Auditor General of India — CAG (Art. 148); 6. Attorney General of India (Art. 76) & Advocate General of State (Art. 165); 7. National Commissions for SCs (Art. 338), STs (Art. 338A — 89th Amendment), and Backward Classes — NCBC (Art. 338B — 102nd Amendment).

In JEE Vector & 3D Geometry, state the formulas for the shortest distance between two skew lines r = a_1 + λb_1 and r = a_2 + μb_2, the Vector Triple Product a × (b × c), and Lagrange's Identity.

1. Shortest Distance between two skew lines: d = |(a_2 − a_1) · (b_1 × b_2)| / |b_1 × b_2| (lines are coplanar/intersecting iff the scalar triple product [a_2 − a_1 b_1 b_2] = 0). For parallel lines (b_1 = b_2 = b): d = |(a_2 − a_1) × b| / |b|. 2. Vector Triple Product ('BAC-CAB' rule): a × (b × c) = (a · c)b − (a · b)c. 3. Lagrange's Identity: |a × b|² + (a · b)² = |a|² |b|².

In UPSC Indian Economy, distinguish Nominal GDP, Real GDP, GDP at Market Price vs Factor Cost, and Gross Value Added (GVA) at Basic Prices.

1. Nominal GDP values current year output at current market prices; Real GDP values current year output at constant base-year prices (2011–12 in India), with GDP Deflator = (Nominal GDP / Real GDP) × 100. 2. GNP = GDP + Net Factor Income from Abroad (NFIA); Net Domestic Product (NDP) = GDP − Depreciation. 3. Relation between GVA at Factor Cost, GVA at Basic Prices, and GDP at Market Prices: GVA at Basic Prices = GVA at Factor Cost + Production Taxes − Production Subsidies; GDP at Market Prices = Σ GVA at Basic Prices + Product Taxes (e.g., GST, customs, excise) − Product Subsidies.

In UPSC Indian Economy (Monetary Policy & Banking), explain the Monetary Policy Committee (MPC) structure and how RBI's quantitative instruments (Repo, Reverse Repo, SDF, MSF, CRR, SLR) regulate liquidity.

The statutory 6-member Monetary Policy Committee (MPC, under Section 45ZB of the RBI Act, 1934: 3 RBI members including the Governor + 3 Govt-appointed external members) targets CPI inflation at 4% ± 2% (2%–6% tolerance band). 1. Repo Rate: Rate at which RBI lends short-term funds to banks against government securities (raising Repo curbs inflation by contracting money supply). 2. Standing Deposit Facility (SDF = Repo − 0.25%): Uncollateralized floor of the LAF corridor for absorbing excess liquidity. 3. Marginal Standing Facility (MSF = Repo + 0.25%): Ceiling rate at which banks borrow overnight by dipping up to 2% into their SLR quota. 4. CRR (Cash Reserve Ratio): Fraction of NDTL banks must keep as non-interest-bearing cash with RBI; SLR (Statutory Liquidity Ratio): Fraction of NDTL banks keep in liquid assets (cash, gold, G-Secs).

In UPSC Modern Indian History, compare the Permanent Settlement (Zamindari), Ryotwari System, and Mahalwari System of British land revenue administration.

1. Permanent Settlement (Lord Cornwallis, 1793; planned by John Shore; covered ~19% of British India in Bengal, Bihar, Odisha, Varanasi): Recognised Zamindars as hereditary owners of land; fixed revenue permanently (10/11th to East India Company, 1/11th kept by Zamindar) with the Sunset Clause. 2. Ryotwari System (Thomas Munro & Alexander Read, 1820; covered ~51% of British India in Madras, Bombay, Assam, Coorg): Revenue settled directly with the individual cultivator (Ryot) with periodic reassessment every 20–30 years. 3. Mahalwari System (Holt Mackenzie, 1822, revised by William Bentinck, 1833; covered ~30% in NW Provinces, Central Provinces, Punjab): Revenue assessed collectively on the village estate ('Mahal') via the village headman (Lambardar).

In UPSC Indian Economy (Public Finance & Fiscal Policy), distinguish Revenue Deficit, Fiscal Deficit, Effective Revenue Deficit, and Primary Deficit.

1. Revenue Deficit = Revenue Expenditure − Revenue Receipts (signaling government dissaving for day-to-day consumption). 2. Effective Revenue Deficit = Revenue Deficit − Grants given to States/UTs for creation of Capital Assets. 3. Fiscal Deficit = Total Expenditure − (Revenue Receipts + Non-Debt Capital Receipts such as recovery of loans and disinvestment proceeds); Fiscal Deficit equals the government's total net borrowing requirement for the fiscal year. 4. Primary Deficit = Fiscal Deficit − Interest Payments on past debt (measuring current fiscal borrowing excluding legacy interest burden).

In UPSC Polity, compare the three types of Emergencies under Part XVIII of the Indian Constitution: Article 352, Article 356, and Article 360.

1. National Emergency (Art. 352 — war, external aggression, or armed rebellion): Must be approved by BOTH Houses of Parliament within 1 month by Special Majority; continues for 6 months at a time upon renewal; Lok Sabha can revoke it by a simple majority resolution. 2. President's Rule / State Emergency (Art. 356 — failure of constitutional machinery in a State; subject to judicial review per S.R. Bommai case, 1994): Approved by both Houses within 2 months by Simple Majority; max duration 3 years (beyond 1 year requires ECI certification + active National Emergency). 3. Financial Emergency (Art. 360): Approved within 2 months by Simple Majority; continues indefinitely until revoked (never proclaimed in India).

In UPSC Geography & Environment, summarize the key international environmental conventions: Ramsar (1971), CITES (1973), Vienna/Montreal/Kigali (Ozone & HFCs), Rio Earth Summit (1992), and CBD Protocols (Cartagena & Nagoya).

1. Ramsar Convention (1971, Iran): Conservation and wise use of wetlands; Montreux Record lists Ramsar sites facing ecological degradation (in India: Keoladeo National Park and Loktak Lake; Chilika Lake was removed after restoration). 2. CITES (Washington Convention, 1973): Regulates international trade in endangered wild flora and fauna (legally binding). 3. Vienna Convention (1985) & Montreal Protocol (1987): Phased out ozone-depleting substances (CFCs, halons); Kigali Amendment (2016) phases down hydrofluorocarbons (HFCs, potent greenhouse gases). 4. Rio Earth Summit (1992): Created three sister conventions—UNFCCC (climate: Kyoto Protocol 1997, Paris Agreement 2015), UNCBD (biodiversity), and UNCCD (desertification). 5. CBD Protocols: Cartagena Protocol (2000) governs biosafety of Living Modified Organisms (LMOs); Nagoya Protocol (2010) governs Access and Benefit-Sharing (ABS) of genetic resources.

In UPSC Polity, explain the origin and composition of Fundamental Duties (Part IVA, Article 51A) in the Indian Constitution.

Not in the original 1950 Constitution; incorporated by the 42nd Constitutional Amendment Act (1976) on the recommendation of the Sardar Swaran Singh Committee (inspired by the USSR Constitution). Originally 10 Fundamental Duties were added under Article 51A; the 11th Fundamental Duty—requiring every parent/guardian to provide opportunities for education to their child between ages 6 and 14—was added by the 86th Constitutional Amendment Act (2002).

In UPSC Modern Indian History, trace the chronological milestones of the Indian Freedom Struggle from 1905 to 1942.

1. 1905: Partition of Bengal (Curzon) & Swadeshi Movement. 2. 1907: Surat Split (Moderates vs Extremists under Rash Behari Ghosh). 3. 1916: Lucknow Pact (INC-Muslim League unity & Moderate-Extremist reunion under A.C. Majumdar) + Home Rule Leagues (Tilak & Annie Besant). 4. 1917–1918: Gandhi's Champaran Satyagraha (1917, tinkathia indigo), Ahmedabad Mill Strike (1918, first hunger strike), Kheda Satyagraha (1918). 5. 1919: Rowlatt Act, Jallianwala Bagh Massacre (13 April 1919), Montagu-Chelmsford Reforms. 6. 1920–1922: Non-Cooperation & Khilafat Movement (called off after Chauri Chaura incident, Feb 1922). 7. 1929: Lahore Congress (Jawaharlal Nehru — Purna Swaraj resolution). 8. 1930–1931: Dandi Salt March (12 March–6 April 1930), Civil Disobedience, Gandhi-Irwin Pact, Second Round Table Conference. 9. 1932: Communal Award & Poona Pact (Gandhi & Ambedkar). 10. 1942: Cripps Mission & Quit India Movement (8 August 1942, 'Do or Die').

In UPSC Modern History, compare the key constitutional provisions of the Government of India Act 1909 (Morley-Minto), Government of India Act 1919 (Montagu-Chelmsford), and Government of India Act 1935.

1. Act of 1909 (Morley-Minto): Introduced Separate Electorates for Muslims (communal representation); Satyendra Prasad Sinha became the first Indian member of the Viceroy's Executive Council. 2. Act of 1919 (Montagu-Chelmsford): Introduced 'Dyarchy' in the Provinces (dividing provincial subjects into 'Transferred' and 'Reserved'); introduced Bicameralism and direct elections at the Centre; separated provincial budgets from the central budget. 3. Act of 1935: Abolished Dyarchy in the Provinces and introduced 'Provincial Autonomy' (while proposing Dyarchy at the Centre and an All-India Federation); divided powers into Federal, Provincial, and Concurrent lists (residuary powers with Viceroy); established the Reserve Bank of India (RBI) and Federal Court (1937).

In UPSC Polity, compare a Money Bill (Article 110), Financial Bill Category I (Article 117(1)), Financial Bill Category II (Article 117(3)), and a Constitutional Amendment Bill (Article 368).

1. Money Bill (Art. 110; Speaker's certification is final): Introduced ONLY in Lok Sabha with prior recommendation of the President; Rajya Sabha cannot amend or reject it (must return within 14 days); NO Joint Sitting (Art. 108); President can give or withhold assent but CANNOT return it for reconsideration. 2. Financial Bill Category I (Art. 117(1)): Contains Art. 110 matters plus general legislation; introduced ONLY in Lok Sabha with President's prior recommendation, but Rajya Sabha has full power to reject/amend and Joint Sitting IS allowed. 3. Financial Bill Category II (Art. 117(3)): Involves expenditure from the Consolidated Fund of India without Art. 110 matters; can be introduced in EITHER House without prior Presidential recommendation, but requires President's recommendation for consideration/passage; both Houses have equal powers and Joint Sitting IS allowed. 4. Constitutional Amendment Bill (Art. 368): Introduced in either House by a minister or private member WITHOUT prior Presidential recommendation; must pass EACH House separately by Special Majority (NO Joint Sitting!); President MUST give assent (24th Amendment, 1971).

In JEE Electrostatics & Capacitors, state Gauss's Law, electric field and potential of a uniformly charged spherical shell vs solid non-conducting sphere, and capacitor energy changes with a dielectric slab (K).

1. Gauss's Law: ∮ E · dA = Q_enclosed / ε_0. 2. Conducting / Hollow Spherical Shell (charge Q, radius R): Inside (r < R), E = 0 and V = kQ/R (constant!); Outside (r ≥ R), E = kQ/r² and V = kQ/r. 3. Uniformly Charged Non-Conducting Solid Sphere: Inside (r < R), E = (kQr)/R³ and V = [kQ/(2R)](3 − r²/R²) (at center V_c = 1.5 V_surface). 4. Inserting dielectric K into parallel-plate capacitor C_0 = ε_0 A/d: If battery is DISCONNECTED (Q constant): C' = KC_0, V' = V_0/K, E' = E_0/K, U' = U_0/K. If battery remains CONNECTED (V constant): C' = KC_0, Q' = KQ_0, E' = E_0, U' = KU_0.

In JEE Complex Numbers & Probability, state De Moivre's Theorem, the nth Roots of Unity properties, Bayes' Theorem, and Binomial/Poisson distributions.

1. De Moivre's Theorem: (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ) = e^(inθ). The n roots of z^n = 1 are 1, ω, ω², ..., ω^(n−1) where ω = e^(i2π/n); their sum 1 + ω + ω² + ... + ω^(n−1) = 0 and product = (−1)^(n−1). 2. Bayes' Theorem: P(E_i | A) = [P(E_i) P(A | E_i)] / [Σ_j P(E_j) P(A | E_j)]. 3. Binomial Distribution B(n, p): P(X = r) = nCr p^r q^(n−r), Mean = np, Variance = npq.

In JEE Differential Equations & Linear Algebra, state the Integrating Factor (I.F.) and solution of a first-order linear ODE, and Cayley-Hamilton Theorem + rank-nullity conditions for a system of linear equations AX = B.

1. For dy/dx + P(x)y = Q(x), the Integrating Factor is I.F. = e^(∫P(x)dx), and the general solution is y · (I.F.) = ∫ Q(x) · (I.F.) dx + C. 2. Cayley-Hamilton Theorem: Every square matrix A satisfies its own characteristic equation det(A − λI) = 0 (where Σ λ_i = Trace(A) and Π λ_i = det(A)). 3. Rouché-Capelli Theorem for AX = B (n variables): Consistent with a unique solution if rank(A) = rank([A|B]) = n; Consistent with infinitely many solutions if rank(A) = rank([A|B]) < n; Inconsistent (no solution) if rank(A) ≠ rank([A|B]).

In JEE Coordinate Geometry (Conic Sections), summarize the eccentricity (e), focus, directrix, latus rectum length, and slope-form tangent y = mx + c for a Parabola, Ellipse, and Hyperbola.

1. Parabola (y² = 4ax): e = 1; Focus (a, 0); Directrix x = −a; Latus Rectum = 4a; Tangent with slope m: y = mx + a/m (point of contact: (a/m², 2a/m)). 2. Ellipse (x²/a² + y²/b² = 1, a > b): e = √(1 − b²/a²) 1; Foci (±ae, 0); Directrices x = ±a/e; Latus Rectum = 2b²/a; Tangent: y = mx ± √(a²m² − b²).

In UPSC Polity, classify Directive Principles of State Policy (DPSP, Part IV, Articles 36–51) and list which DPSPs were added by the 42nd, 44th, 86th, and 97th Constitutional Amendments.

Borrowed from the Irish Constitution (non-justiciable under Art. 37): Key articles include Art. 39A (Free legal aid — 42nd Amendment), Art. 40 (Village Panchayats), Art. 44 (Uniform Civil Code), Art. 48A (Environment & wildlife protection — 42nd Amendment), Art. 50 (Separation of judiciary from executive), Art. 51 (International peace). Amendments: 1. 42nd Amendment (1976): Added Art. 39(f) (healthy development of children), 39A, 43A (workers' participation in management), and 48A. 2. 44th Amendment (1978): Added Art. 38(2) (minimize inequalities in income/status). 3. 86th Amendment (2002): Changed Art. 45 to early childhood care (0–6 yrs) and made elementary education (6–14 yrs) a Fundamental Right under Art. 21A. 4. 97th Amendment (2011): Added Art. 43B (co-operative societies).

In UPSC Indian Economy (External Sector), distinguish the Current Account vs Capital Account of the Balance of Payments (BoP), and define NEER vs REER.

1. Current Account records non-asset flows: Balance of Trade in Visible Goods (merchandise exports minus imports) + Invisibles (Services trade like IT exports, Investment Income/profit/interest, and Unilateral Transfers/Remittances; India enjoys complete Current Account Convertibility since 1994). 2. Capital Account records asset/liability cross-border flows: Foreign Direct Investment (FDI), Foreign Portfolio Investment (FPI/FII), External Commercial Borrowings (ECBs), and NRI deposits (India has partial Capital Account Convertibility per Tarapore Committees). 3. NEER (Nominal Effective Exchange Rate) is the trade-weighted geometric average of the rupee's bilateral nominal exchange rates against a basket of 40 trading-partner currencies; REER (Real Effective Exchange Rate) adjusts NEER for relative domestic vs foreign inflation differentials (if REER > 100, the domestic currency is overvalued and exports lose price competitiveness).

Compare the 42nd Constitutional Amendment Act (1976, 'Mini-Constitution') and the 44th Constitutional Amendment Act (1978) in UPSC Polity.

1. 42nd Amendment (1976, Indira Gandhi govt): Added 'Socialist, Secular, and Integrity' to the Preamble; added Fundamental Duties (Part IVA); transferred 5 subjects from State List to Concurrent List (Education, Forests, Weights & Measures, Protection of Wild Animals & Birds, Administration of Justice); made Cabinet advice binding on the President (Art. 74). 2. 44th Amendment (1978, Morarji Desai govt): Repealed Right to Property from Fundamental Rights (Art. 19(1)(f) & 31) and made it a legal right under Art. 300A; replaced 'internal disturbance' with 'armed rebellion' for National Emergency (Art. 352); required written Cabinet recommendation for Emergency; made Art. 20 & 21 non-suspendable; allowed President to return Cabinet advice once for reconsideration.

In UPSC Modern Indian History, match the key Socio-Religious Reform Movements and organizations of the 19th century to their founders.

1. Atmiya Sabha (1815) & Brahmo Samaj (1828) + Abolition of Sati (1829 with Bentinck): Raja Rammohan Roy. 2. Tattvabodhini Sabha (1839): Debendranath Tagore. 3. Young Bengal Movement: Henry Vivian Derozio. 4. Widow Remarriage Act (1856): Ishwar Chandra Vidyasagar. 5. Prarthana Samaj (1867, Bombay): Atmaram Pandurang (with M.G. Ranade). 6. Satyashodhak Samaj (1873, Maharashtra, author of Gulamgiri): Jyotirao Phule. 7. Arya Samaj (1875, 'Go back to the Vedas', Shuddhi movement, Dayanand Anglo-Vedic schools): Swami Dayanand Saraswati. 8. Ramakrishna Mission (1897): Swami Vivekananda. 9. Aligarh Movement / Mohammedan Anglo-Oriental College (1875): Sir Syed Ahmed Khan.

In UPSC Indian Polity, summarize the six categories of Fundamental Rights (Part III, Articles 12–35) and identify which Fundamental Rights are available ONLY to Indian citizens (not foreigners) and which can NEVER be suspended during a National Emergency.

1. Right to Equality (Art. 14–18); 2. Right to Freedom (Art. 19–22); 3. Right against Exploitation (Art. 23–24); 4. Right to Freedom of Religion (Art. 25–28); 5. Cultural & Educational Rights (Art. 29–30); 6. Right to Constitutional Remedies (Art. 32 — called the 'heart and soul of the Constitution' by Dr. B.R. Ambedkar). Rights exclusive to Indian Citizens ONLY: Articles 15, 16, 19, 29, and 30. Under Article 359 (as amended by the 44th Amendment, 1978), Articles 20 and 21 can NEVER be suspended even during a National Emergency (Art. 352).

In JEE Magnetism & EMI, state the Biot-Savart Law, Ampere's Circuital Law, magnetic field at the center/axis of a circular loop and solenoid, Faraday-Lenz Law, and LCR resonance/Q-factor.

1. Biot-Savart Law: dB = (μ_0 / 4π) [I (dl × r̂) / r²]. 2. Circular Loop (radius R, N turns): On axis at distance x: B = (μ_0 N I R²) / [2(R² + x²)^(3/2)]; at center (x = 0): B = μ_0 N I / (2R). Ideal Solenoid: B = μ_0 n I. 3. Lorentz Force: F = q(E + v × B); Cyclotron radius r = mv / (qB), period T = 2πm / (qB). 4. Faraday-Lenz Law: ε = −dΦ_B/dt; Motional EMF = B l v. 5. Series LCR Circuit: Impedance Z = √[R² + (X_L − X_C)²]; Resonant frequency ω_0 = 1/√(LC); Quality Factor Q = ω_0 L / R = (1/R)√(L/C); Average Power P = V_rms I_rms cos φ (where Power Factor cos φ = R/Z).

In UPSC Ancient Indian History, summarize the four Buddhist Councils (venue, ruler/patron, president, and outcome) and distinguish Hinayana vs Mahayana Buddhism.

1. First Council (483 BCE, Sattapanni Cave, Rajagriha; Patron: Ajatashatru; President: Mahakassapa): Ananda compiled Sutta Pitaka (teachings) and Upali compiled Vinaya Pitaka (monastic rules). 2. Second Council (383 BCE, Vaishali; Patron: Kalasoka; President: Sabakami): Split into Sthaviravadins and Mahasanghikas. 3. Third Council (250 BCE, Pataliputra; Patron: Ashoka; President: Moggaliputta Tissa): Compiled Abhidhamma Pitaka (completing the Pali Tripitaka) and sent missionaries abroad. 4. Fourth Council (72 CE, Kundalvana, Kashmir; Patron: Kanishka; President: Vasumitra, Vice-President: Ashvaghosha): Composed Mahavibhasha Shastra in Sanskrit and formally divided Buddhism into Hinayana (lesser vehicle, Pali, symbolic worship) and Mahayana (greater vehicle, Sanskrit, idol worship of Buddha & Bodhisattvas).

In UPSC Indian Polity, list all 12 Schedules of the Indian Constitution using the 'TEARS OF OLD PM' framework and state which Constitutional Amendments added the 9th, 10th, 11th, and 12th Schedules.

1st (T): Territories & States; 2nd (E): Emoluments/salaries of constitutional offices; 3rd (A): Affirmations & Oaths; 4th (R): Rajya Sabha seat allocation; 5th (S): Scheduled Areas & Tribes administration; 6th (O): Other Tribal Areas (Assam, Meghalaya, Tripura, Mizoram — AMTM); 7th (F): Federal Lists (Union, State, Concurrent); 8th (O): Official Languages (22 languages); 9th (L): Land Reforms & Acts immune from initial judicial review (added by 1st Amendment, 1951; subject to judicial review post-24 April 1973 per I.R. Coelho case); 10th (D): Defection disqualification rules (52nd Amendment, 1985); 11th (P): Panchayats — 29 subjects (73rd Amendment, 1992); 12th (M): Municipalities — 18 subjects (74th Amendment, 1992).

UPSC Civil Services General Studies (Polity, History, Economy & Environment) & JEE Mathematics/Physics

Indian Constitution Articles, Schedules, Amendments (up to 106th), Ancient/Modern Indian History, RBI Monetary/Fiscal Economy, Environmental Conventions, and JEE Calculus, Conics & Physics.

Level: advanced
Languages: en → en
Cards: 32
Markdown: UPSC Civil Services General Studies (Polity, History, Economy & Environment) & JEE Mathematics/Physics (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
1	Trace the evolution of the 'Basic Structure Doctrine' of the Indian Constitution across Shankari Prasad (1951), Golaknath (1967), Kesavananda Bharati (1973), and Minerva Mills (1980).	1. Shankari Prasad (1951): SC held Parliament can amend Fundamental Rights under Art. 368 and a Constitutional Amendment is not 'law' under Art. 13(2). 2. Golaknath (1967): SC reversed itself, ruling Fundamental Rights are transcendental and cannot be abridged by Parliament. 3. Kesavananda Bharati v. State of Kerala (24 April 1973, 13-judge bench, 7:6 majority): Upheld the 24th Amendment confirming Parliament can amend any part of the Constitution (including FRs) under Art. 368, PROVIDED it does not alter or destroy the 'Basic Structure' of the Constitution. 4. Minerva Mills (1980): Struck down clauses of the 42nd Amendment, declaring judicial review and harmony between Part III (FRs) and Part IV (DPSPs) part of the Basic Structure.
2	In UPSC Ancient Indian History, match the major Indus Valley Civilization (Harappan) sites to their key archaeological findings and river locations.	1. Harappa (Ravi river, Daya Ram Sahni, 1921): Two rows of 6 granaries, coffin burial, red sandstone male torso. 2. Mohenjo-daro (Indus river, R.D. Banerji, 1922): Great Bath, Great Granary, Bronze Dancing Girl (lost-wax casting), Pashupati Seal, bearded priest. 3. Lothal (Bhogava river, Gujarat): Artificial brick dockyard (tidal port), double burial, rice husks, fire altars, bead-making factory. 4. Kalibangan (Ghaggar river, Rajasthan): Ploughed field surface, fire altars, camel bones, no street drainage system. 5. Dholavira (Rann of Kutch, UNESCO site): Divided into 3 parts (Citadel, Middle Town, Lower Town), giant rock-cut water reservoirs/dams, 10-sign signboard inscription. 6. Rakhigarhi (Haryana): Largest IVC site in India.
3	In UPSC Ancient History, identify the key Ashokan Major Rock Edicts (especially XIII, XII, and Maski/Gujarra) and Kautilya's Saptanga Theory of the Mauryan State.	Ashoka's inscriptions (deciphered by James Prinsep in 1837, mostly in Prakrit language and Brahmi script, plus Kharosthi in NW and Greek/Aramaic in Kandahar): 1. Major Rock Edict XIII: Describes the Kalinga War (261 BCE, 8th regnal year), Ashoka's remorse, conversion from Bherighosha to Dhammaghosha, and mentions 5 contemporary Hellenistic kings (Antiochus, Ptolemy, Antigonus, Magas, Alexander). 2. Minor Rock Edicts at Maski, Gujarra, Nittur, and Udegolam explicitly mention Ashoka by his personal name ('Devanampiya Piyadassi Ashoka'). 3. Kautilya's Arthashastra Saptanga Theory (7 limbs of state): Swami (King), Amatya (Minister), Janapada (Territory/People), Durga (Fort), Kosha (Treasury), Danda/Bala (Army), and Mitra (Ally).
4	Distinguish the five Constitutional Writs issued by the Supreme Court (Article 32) and High Courts (Article 226) in Indian Polity, and explain how the High Court's writ jurisdiction is wider than the Supreme Court's.	1. Habeas Corpus ('to have the body of'): Protects personal liberty against illegal detention (can be issued against both public authorities and private individuals!). 2. Mandamus ('we command'): Directs a public official/body to perform a mandatory public duty (cannot be issued against the President, Governor, or private individuals). 3. Prohibition ('to forbid'): Issued by a higher court to a lower court/tribunal to prevent it from exceeding its jurisdiction (preventive). 4. Certiorari ('to be certified'): Quashes an order already passed by a lower court/tribunal in excess of jurisdiction or error of law (curative). 5. Quo Warranto ('by what authority'): Challenges a person's legal claim to a public office. High Courts (Art. 226) can issue writs for Fundamental Rights AND 'for any other purpose' (ordinary legal rights), making their territorial scope narrower but subject-matter jurisdiction wider than Art. 32.
5	In JEE Rotational Mechanics, state the Parallel Axis Theorem, Perpendicular Axis Theorem, Moments of Inertia of standard bodies (ring, disc, hollow/solid sphere, rod), and pure rolling acceleration down an incline.	1. Parallel Axis Theorem: I = I_cm + M d²; Perpendicular Axis Theorem (for planar lamina): I_z = I_x + I_y. 2. Moments of Inertia about CM axis: Thin Ring = MR²; Circular Disc / Solid Cylinder = ½MR²; Hollow Sphere = ⅔MR²; Solid Sphere = ⅖MR²; Thin Rod (length L) = (1/12)ML² about center and (1/3)ML² about one end. 3. For a body with radius of gyration k rolling without slipping down an incline θ: acceleration a = g sin θ / (1 + k²/R²), and minimum coefficient of static friction required is μ_min = tan θ / (1 + R²/k²).
6	What were the subjects of the 101st, 102nd, 103rd, 104th, 105th, and 106th Constitutional Amendment Acts of India?	1. 101st Amendment (2016): Introduced Goods and Services Tax (GST) and Art. 246A, 269A, 279A. 2. 102nd Amendment (2018): Granted constitutional status to National Commission for Backward Classes (NCBC, Art. 338B). 3. 103rd Amendment (2019): Introduced 10% reservation for Economically Weaker Sections (EWS) under Art. 15(6) and 16(6) (upheld in Janhit Abhiyan case). 4. 104th Amendment (2020): Extended SC/ST reservation in Lok Sabha & State Assemblies by 10 years and ended the nomination of 2 Anglo-Indians (Art. 331/333). 5. 105th Amendment (2021): Restored State governments' power to identify and maintain their own SEBC/OBC lists. 6. 106th Amendment (2023, Nari Shakti Vandan Adhiniyam): Reserved one-third (33%) of seats for women in Lok Sabha (Art. 330A), State Legislative Assemblies (Art. 332A), and NCT Delhi Assembly.
7	In JEE Calculus, state Newton-Leibniz's Rule for differentiating a definite integral with variable limits, and the King's Property + Queen's Property of definite integrals.	1. Leibniz Rule: d/dx [∫_{g(x)}^{h(x)} f(t) dt] = f(h(x))·h'(x) − f(g(x))·g'(x). 2. King's Property: ∫_a^b f(x) dx = ∫_a^b f(a + b − x) dx (especially ∫_0^a f(x) dx = ∫_0^a f(a − x) dx). 3. Queen's Property: ∫_0^{2a} f(x) dx = ∫_0^a [f(x) + f(2a − x)] dx, which equals 2∫_0^a f(x) dx if f(2a − x) = f(x) and 0 if f(2a − x) = −f(x). Also, Wallis's Formula: ∫_0^{π/2} sin^n(x) dx = ∫_0^{π/2} cos^n(x) dx.
8	In JEE Simple Harmonic Motion (SHM) & Waves, give the velocity v(x), energy, simple/physical pendulum periods, Doppler effect formula, and organ pipe harmonics.	1. In SHM (x = A sin(ωt + φ)): v = ω√(A² − x²), a = −ω²x, Total Energy E = ½mω²A². 2. Simple Pendulum: T = 2π√(L/g); Physical (Compound) Pendulum: T = 2π√[I / (mgd)]. 3. Closed Organ Pipe (one end closed, length L): Only odd harmonics present, f_n = (2n − 1)v / (4L); Open Organ Pipe (both ends open): All harmonics present, f_n = n v / (2L).
9	In UPSC Polity, list the major Constitutional Bodies in India alongside their exact Constitutional Articles.	1. Election Commission of India (Art. 324); 2. Union & State Public Service Commissions — UPSC/SPSC (Art. 315–323); 3. Finance Commission (Art. 280 — constituted every 5 years by the President for tax devolution); 4. Goods and Services Tax (GST) Council (Art. 279A — added by 101st Amendment, 2016, chaired by Union Finance Minister); 5. Comptroller and Auditor General of India — CAG (Art. 148); 6. Attorney General of India (Art. 76) & Advocate General of State (Art. 165); 7. National Commissions for SCs (Art. 338), STs (Art. 338A — 89th Amendment), and Backward Classes — NCBC (Art. 338B — 102nd Amendment).
10	In JEE Vector & 3D Geometry, state the formulas for the shortest distance between two skew lines r = a_1 + λb_1 and r = a_2 + μb_2, the Vector Triple Product a × (b × c), and Lagrange's Identity.	1. Shortest Distance between two skew lines: d = \|(a_2 − a_1) · (b_1 × b_2)\| / \|b_1 × b_2\| (lines are coplanar/intersecting iff the scalar triple product [a_2 − a_1 b_1 b_2] = 0). For parallel lines (b_1 = b_2 = b): d = \|(a_2 − a_1) × b\| / \|b\|. 2. Vector Triple Product ('BAC-CAB' rule): a × (b × c) = (a · c)b − (a · b)c. 3. Lagrange's Identity: \|a × b\|² + (a · b)² = \|a\|² \|b\|².
11	In UPSC Indian Economy, distinguish Nominal GDP, Real GDP, GDP at Market Price vs Factor Cost, and Gross Value Added (GVA) at Basic Prices.	1. Nominal GDP values current year output at current market prices; Real GDP values current year output at constant base-year prices (2011–12 in India), with GDP Deflator = (Nominal GDP / Real GDP) × 100. 2. GNP = GDP + Net Factor Income from Abroad (NFIA); Net Domestic Product (NDP) = GDP − Depreciation. 3. Relation between GVA at Factor Cost, GVA at Basic Prices, and GDP at Market Prices: GVA at Basic Prices = GVA at Factor Cost + Production Taxes − Production Subsidies; GDP at Market Prices = Σ GVA at Basic Prices + Product Taxes (e.g., GST, customs, excise) − Product Subsidies.
12	In UPSC Indian Economy (Monetary Policy & Banking), explain the Monetary Policy Committee (MPC) structure and how RBI's quantitative instruments (Repo, Reverse Repo, SDF, MSF, CRR, SLR) regulate liquidity.	The statutory 6-member Monetary Policy Committee (MPC, under Section 45ZB of the RBI Act, 1934: 3 RBI members including the Governor + 3 Govt-appointed external members) targets CPI inflation at 4% ± 2% (2%–6% tolerance band). 1. Repo Rate: Rate at which RBI lends short-term funds to banks against government securities (raising Repo curbs inflation by contracting money supply). 2. Standing Deposit Facility (SDF = Repo − 0.25%): Uncollateralized floor of the LAF corridor for absorbing excess liquidity. 3. Marginal Standing Facility (MSF = Repo + 0.25%): Ceiling rate at which banks borrow overnight by dipping up to 2% into their SLR quota. 4. CRR (Cash Reserve Ratio): Fraction of NDTL banks must keep as non-interest-bearing cash with RBI; SLR (Statutory Liquidity Ratio): Fraction of NDTL banks keep in liquid assets (cash, gold, G-Secs).
13	In UPSC Modern Indian History, compare the Permanent Settlement (Zamindari), Ryotwari System, and Mahalwari System of British land revenue administration.	1. Permanent Settlement (Lord Cornwallis, 1793; planned by John Shore; covered ~19% of British India in Bengal, Bihar, Odisha, Varanasi): Recognised Zamindars as hereditary owners of land; fixed revenue permanently (10/11th to East India Company, 1/11th kept by Zamindar) with the Sunset Clause. 2. Ryotwari System (Thomas Munro & Alexander Read, 1820; covered ~51% of British India in Madras, Bombay, Assam, Coorg): Revenue settled directly with the individual cultivator (Ryot) with periodic reassessment every 20–30 years. 3. Mahalwari System (Holt Mackenzie, 1822, revised by William Bentinck, 1833; covered ~30% in NW Provinces, Central Provinces, Punjab): Revenue assessed collectively on the village estate ('Mahal') via the village headman (Lambardar).
14	In UPSC Indian Economy (Public Finance & Fiscal Policy), distinguish Revenue Deficit, Fiscal Deficit, Effective Revenue Deficit, and Primary Deficit.	1. Revenue Deficit = Revenue Expenditure − Revenue Receipts (signaling government dissaving for day-to-day consumption). 2. Effective Revenue Deficit = Revenue Deficit − Grants given to States/UTs for creation of Capital Assets. 3. Fiscal Deficit = Total Expenditure − (Revenue Receipts + Non-Debt Capital Receipts such as recovery of loans and disinvestment proceeds); Fiscal Deficit equals the government's total net borrowing requirement for the fiscal year. 4. Primary Deficit = Fiscal Deficit − Interest Payments on past debt (measuring current fiscal borrowing excluding legacy interest burden).
15	In UPSC Polity, compare the three types of Emergencies under Part XVIII of the Indian Constitution: Article 352, Article 356, and Article 360.	1. National Emergency (Art. 352 — war, external aggression, or armed rebellion): Must be approved by BOTH Houses of Parliament within 1 month by Special Majority; continues for 6 months at a time upon renewal; Lok Sabha can revoke it by a simple majority resolution. 2. President's Rule / State Emergency (Art. 356 — failure of constitutional machinery in a State; subject to judicial review per S.R. Bommai case, 1994): Approved by both Houses within 2 months by Simple Majority; max duration 3 years (beyond 1 year requires ECI certification + active National Emergency). 3. Financial Emergency (Art. 360): Approved within 2 months by Simple Majority; continues indefinitely until revoked (never proclaimed in India).
16	In UPSC Geography & Environment, summarize the key international environmental conventions: Ramsar (1971), CITES (1973), Vienna/Montreal/Kigali (Ozone & HFCs), Rio Earth Summit (1992), and CBD Protocols (Cartagena & Nagoya).	1. Ramsar Convention (1971, Iran): Conservation and wise use of wetlands; Montreux Record lists Ramsar sites facing ecological degradation (in India: Keoladeo National Park and Loktak Lake; Chilika Lake was removed after restoration). 2. CITES (Washington Convention, 1973): Regulates international trade in endangered wild flora and fauna (legally binding). 3. Vienna Convention (1985) & Montreal Protocol (1987): Phased out ozone-depleting substances (CFCs, halons); Kigali Amendment (2016) phases down hydrofluorocarbons (HFCs, potent greenhouse gases). 4. Rio Earth Summit (1992): Created three sister conventions—UNFCCC (climate: Kyoto Protocol 1997, Paris Agreement 2015), UNCBD (biodiversity), and UNCCD (desertification). 5. CBD Protocols: Cartagena Protocol (2000) governs biosafety of Living Modified Organisms (LMOs); Nagoya Protocol (2010) governs Access and Benefit-Sharing (ABS) of genetic resources.
17	In UPSC Polity, explain the origin and composition of Fundamental Duties (Part IVA, Article 51A) in the Indian Constitution.	Not in the original 1950 Constitution; incorporated by the 42nd Constitutional Amendment Act (1976) on the recommendation of the Sardar Swaran Singh Committee (inspired by the USSR Constitution). Originally 10 Fundamental Duties were added under Article 51A; the 11th Fundamental Duty—requiring every parent/guardian to provide opportunities for education to their child between ages 6 and 14—was added by the 86th Constitutional Amendment Act (2002).
18	In UPSC Modern Indian History, trace the chronological milestones of the Indian Freedom Struggle from 1905 to 1942.	1. 1905: Partition of Bengal (Curzon) & Swadeshi Movement. 2. 1907: Surat Split (Moderates vs Extremists under Rash Behari Ghosh). 3. 1916: Lucknow Pact (INC-Muslim League unity & Moderate-Extremist reunion under A.C. Majumdar) + Home Rule Leagues (Tilak & Annie Besant). 4. 1917–1918: Gandhi's Champaran Satyagraha (1917, tinkathia indigo), Ahmedabad Mill Strike (1918, first hunger strike), Kheda Satyagraha (1918). 5. 1919: Rowlatt Act, Jallianwala Bagh Massacre (13 April 1919), Montagu-Chelmsford Reforms. 6. 1920–1922: Non-Cooperation & Khilafat Movement (called off after Chauri Chaura incident, Feb 1922). 7. 1929: Lahore Congress (Jawaharlal Nehru — Purna Swaraj resolution). 8. 1930–1931: Dandi Salt March (12 March–6 April 1930), Civil Disobedience, Gandhi-Irwin Pact, Second Round Table Conference. 9. 1932: Communal Award & Poona Pact (Gandhi & Ambedkar). 10. 1942: Cripps Mission & Quit India Movement (8 August 1942, 'Do or Die').
19	In UPSC Modern History, compare the key constitutional provisions of the Government of India Act 1909 (Morley-Minto), Government of India Act 1919 (Montagu-Chelmsford), and Government of India Act 1935.	1. Act of 1909 (Morley-Minto): Introduced Separate Electorates for Muslims (communal representation); Satyendra Prasad Sinha became the first Indian member of the Viceroy's Executive Council. 2. Act of 1919 (Montagu-Chelmsford): Introduced 'Dyarchy' in the Provinces (dividing provincial subjects into 'Transferred' and 'Reserved'); introduced Bicameralism and direct elections at the Centre; separated provincial budgets from the central budget. 3. Act of 1935: Abolished Dyarchy in the Provinces and introduced 'Provincial Autonomy' (while proposing Dyarchy at the Centre and an All-India Federation); divided powers into Federal, Provincial, and Concurrent lists (residuary powers with Viceroy); established the Reserve Bank of India (RBI) and Federal Court (1937).
20	In UPSC Polity, compare a Money Bill (Article 110), Financial Bill Category I (Article 117(1)), Financial Bill Category II (Article 117(3)), and a Constitutional Amendment Bill (Article 368).	1. Money Bill (Art. 110; Speaker's certification is final): Introduced ONLY in Lok Sabha with prior recommendation of the President; Rajya Sabha cannot amend or reject it (must return within 14 days); NO Joint Sitting (Art. 108); President can give or withhold assent but CANNOT return it for reconsideration. 2. Financial Bill Category I (Art. 117(1)): Contains Art. 110 matters plus general legislation; introduced ONLY in Lok Sabha with President's prior recommendation, but Rajya Sabha has full power to reject/amend and Joint Sitting IS allowed. 3. Financial Bill Category II (Art. 117(3)): Involves expenditure from the Consolidated Fund of India without Art. 110 matters; can be introduced in EITHER House without prior Presidential recommendation, but requires President's recommendation for consideration/passage; both Houses have equal powers and Joint Sitting IS allowed. 4. Constitutional Amendment Bill (Art. 368): Introduced in either House by a minister or private member WITHOUT prior Presidential recommendation; must pass EACH House separately by Special Majority (NO Joint Sitting!); President MUST give assent (24th Amendment, 1971).
21	In JEE Electrostatics & Capacitors, state Gauss's Law, electric field and potential of a uniformly charged spherical shell vs solid non-conducting sphere, and capacitor energy changes with a dielectric slab (K).	1. Gauss's Law: ∮ E · dA = Q_enclosed / ε_0. 2. Conducting / Hollow Spherical Shell (charge Q, radius R): Inside (r < R), E = 0 and V = kQ/R (constant!); Outside (r ≥ R), E = kQ/r² and V = kQ/r. 3. Uniformly Charged Non-Conducting Solid Sphere: Inside (r < R), E = (kQr)/R³ and V = [kQ/(2R)](3 − r²/R²) (at center V_c = 1.5 V_surface). 4. Inserting dielectric K into parallel-plate capacitor C_0 = ε_0 A/d: If battery is DISCONNECTED (Q constant): C' = KC_0, V' = V_0/K, E' = E_0/K, U' = U_0/K. If battery remains CONNECTED (V constant): C' = KC_0, Q' = KQ_0, E' = E_0, U' = KU_0.
22	In JEE Complex Numbers & Probability, state De Moivre's Theorem, the nth Roots of Unity properties, Bayes' Theorem, and Binomial/Poisson distributions.	1. De Moivre's Theorem: (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ) = e^(inθ). The n roots of z^n = 1 are 1, ω, ω², ..., ω^(n−1) where ω = e^(i2π/n); their sum 1 + ω + ω² + ... + ω^(n−1) = 0 and product = (−1)^(n−1). 2. Bayes' Theorem: P(E_i \| A) = [P(E_i) P(A \| E_i)] / [Σ_j P(E_j) P(A \| E_j)]. 3. Binomial Distribution B(n, p): P(X = r) = nCr p^r q^(n−r), Mean = np, Variance = npq.
23	In JEE Differential Equations & Linear Algebra, state the Integrating Factor (I.F.) and solution of a first-order linear ODE, and Cayley-Hamilton Theorem + rank-nullity conditions for a system of linear equations AX = B.	1. For dy/dx + P(x)y = Q(x), the Integrating Factor is I.F. = e^(∫P(x)dx), and the general solution is y · (I.F.) = ∫ Q(x) · (I.F.) dx + C. 2. Cayley-Hamilton Theorem: Every square matrix A satisfies its own characteristic equation det(A − λI) = 0 (where Σ λ_i = Trace(A) and Π λ_i = det(A)). 3. Rouché-Capelli Theorem for AX = B (n variables): Consistent with a unique solution if rank(A) = rank([A\|B]) = n; Consistent with infinitely many solutions if rank(A) = rank([A\|B]) < n; Inconsistent (no solution) if rank(A) ≠ rank([A\|B]).
24	In JEE Coordinate Geometry (Conic Sections), summarize the eccentricity (e), focus, directrix, latus rectum length, and slope-form tangent y = mx + c for a Parabola, Ellipse, and Hyperbola.	1. Parabola (y² = 4ax): e = 1; Focus (a, 0); Directrix x = −a; Latus Rectum = 4a; Tangent with slope m: y = mx + a/m (point of contact: (a/m², 2a/m)). 2. Ellipse (x²/a² + y²/b² = 1, a > b): e = √(1 − b²/a²) < 1; Foci (±ae, 0); Directrices x = ±a/e; Latus Rectum = 2b²/a; Tangent: y = mx ± √(a²m² + b²). 3. Hyperbola (x²/a² − y²/b² = 1): e = √(1 + b²/a²) > 1; Foci (±ae, 0); Directrices x = ±a/e; Latus Rectum = 2b²/a; Tangent: y = mx ± √(a²m² − b²).
25	In UPSC Polity, classify Directive Principles of State Policy (DPSP, Part IV, Articles 36–51) and list which DPSPs were added by the 42nd, 44th, 86th, and 97th Constitutional Amendments.	Borrowed from the Irish Constitution (non-justiciable under Art. 37): Key articles include Art. 39A (Free legal aid — 42nd Amendment), Art. 40 (Village Panchayats), Art. 44 (Uniform Civil Code), Art. 48A (Environment & wildlife protection — 42nd Amendment), Art. 50 (Separation of judiciary from executive), Art. 51 (International peace). Amendments: 1. 42nd Amendment (1976): Added Art. 39(f) (healthy development of children), 39A, 43A (workers' participation in management), and 48A. 2. 44th Amendment (1978): Added Art. 38(2) (minimize inequalities in income/status). 3. 86th Amendment (2002): Changed Art. 45 to early childhood care (0–6 yrs) and made elementary education (6–14 yrs) a Fundamental Right under Art. 21A. 4. 97th Amendment (2011): Added Art. 43B (co-operative societies).
26	In UPSC Indian Economy (External Sector), distinguish the Current Account vs Capital Account of the Balance of Payments (BoP), and define NEER vs REER.	1. Current Account records non-asset flows: Balance of Trade in Visible Goods (merchandise exports minus imports) + Invisibles (Services trade like IT exports, Investment Income/profit/interest, and Unilateral Transfers/Remittances; India enjoys complete Current Account Convertibility since 1994). 2. Capital Account records asset/liability cross-border flows: Foreign Direct Investment (FDI), Foreign Portfolio Investment (FPI/FII), External Commercial Borrowings (ECBs), and NRI deposits (India has partial Capital Account Convertibility per Tarapore Committees). 3. NEER (Nominal Effective Exchange Rate) is the trade-weighted geometric average of the rupee's bilateral nominal exchange rates against a basket of 40 trading-partner currencies; REER (Real Effective Exchange Rate) adjusts NEER for relative domestic vs foreign inflation differentials (if REER > 100, the domestic currency is overvalued and exports lose price competitiveness).
27	Compare the 42nd Constitutional Amendment Act (1976, 'Mini-Constitution') and the 44th Constitutional Amendment Act (1978) in UPSC Polity.	1. 42nd Amendment (1976, Indira Gandhi govt): Added 'Socialist, Secular, and Integrity' to the Preamble; added Fundamental Duties (Part IVA); transferred 5 subjects from State List to Concurrent List (Education, Forests, Weights & Measures, Protection of Wild Animals & Birds, Administration of Justice); made Cabinet advice binding on the President (Art. 74). 2. 44th Amendment (1978, Morarji Desai govt): Repealed Right to Property from Fundamental Rights (Art. 19(1)(f) & 31) and made it a legal right under Art. 300A; replaced 'internal disturbance' with 'armed rebellion' for National Emergency (Art. 352); required written Cabinet recommendation for Emergency; made Art. 20 & 21 non-suspendable; allowed President to return Cabinet advice once for reconsideration.
28	In UPSC Modern Indian History, match the key Socio-Religious Reform Movements and organizations of the 19th century to their founders.	1. Atmiya Sabha (1815) & Brahmo Samaj (1828) + Abolition of Sati (1829 with Bentinck): Raja Rammohan Roy. 2. Tattvabodhini Sabha (1839): Debendranath Tagore. 3. Young Bengal Movement: Henry Vivian Derozio. 4. Widow Remarriage Act (1856): Ishwar Chandra Vidyasagar. 5. Prarthana Samaj (1867, Bombay): Atmaram Pandurang (with M.G. Ranade). 6. Satyashodhak Samaj (1873, Maharashtra, author of Gulamgiri): Jyotirao Phule. 7. Arya Samaj (1875, 'Go back to the Vedas', Shuddhi movement, Dayanand Anglo-Vedic schools): Swami Dayanand Saraswati. 8. Ramakrishna Mission (1897): Swami Vivekananda. 9. Aligarh Movement / Mohammedan Anglo-Oriental College (1875): Sir Syed Ahmed Khan.
29	In UPSC Indian Polity, summarize the six categories of Fundamental Rights (Part III, Articles 12–35) and identify which Fundamental Rights are available ONLY to Indian citizens (not foreigners) and which can NEVER be suspended during a National Emergency.	1. Right to Equality (Art. 14–18); 2. Right to Freedom (Art. 19–22); 3. Right against Exploitation (Art. 23–24); 4. Right to Freedom of Religion (Art. 25–28); 5. Cultural & Educational Rights (Art. 29–30); 6. Right to Constitutional Remedies (Art. 32 — called the 'heart and soul of the Constitution' by Dr. B.R. Ambedkar). Rights exclusive to Indian Citizens ONLY: Articles 15, 16, 19, 29, and 30. Under Article 359 (as amended by the 44th Amendment, 1978), Articles 20 and 21 can NEVER be suspended even during a National Emergency (Art. 352).
30	In JEE Magnetism & EMI, state the Biot-Savart Law, Ampere's Circuital Law, magnetic field at the center/axis of a circular loop and solenoid, Faraday-Lenz Law, and LCR resonance/Q-factor.	1. Biot-Savart Law: dB = (μ_0 / 4π) [I (dl × r̂) / r²]. 2. Circular Loop (radius R, N turns): On axis at distance x: B = (μ_0 N I R²) / [2(R² + x²)^(3/2)]; at center (x = 0): B = μ_0 N I / (2R). Ideal Solenoid: B = μ_0 n I. 3. Lorentz Force: F = q(E + v × B); Cyclotron radius r = mv / (qB), period T = 2πm / (qB). 4. Faraday-Lenz Law: ε = −dΦ_B/dt; Motional EMF = B l v. 5. Series LCR Circuit: Impedance Z = √[R² + (X_L − X_C)²]; Resonant frequency ω_0 = 1/√(LC); Quality Factor Q = ω_0 L / R = (1/R)√(L/C); Average Power P = V_rms I_rms cos φ (where Power Factor cos φ = R/Z).
31	In UPSC Ancient Indian History, summarize the four Buddhist Councils (venue, ruler/patron, president, and outcome) and distinguish Hinayana vs Mahayana Buddhism.	1. First Council (483 BCE, Sattapanni Cave, Rajagriha; Patron: Ajatashatru; President: Mahakassapa): Ananda compiled Sutta Pitaka (teachings) and Upali compiled Vinaya Pitaka (monastic rules). 2. Second Council (383 BCE, Vaishali; Patron: Kalasoka; President: Sabakami): Split into Sthaviravadins and Mahasanghikas. 3. Third Council (250 BCE, Pataliputra; Patron: Ashoka; President: Moggaliputta Tissa): Compiled Abhidhamma Pitaka (completing the Pali Tripitaka) and sent missionaries abroad. 4. Fourth Council (72 CE, Kundalvana, Kashmir; Patron: Kanishka; President: Vasumitra, Vice-President: Ashvaghosha): Composed Mahavibhasha Shastra in Sanskrit and formally divided Buddhism into Hinayana (lesser vehicle, Pali, symbolic worship) and Mahayana (greater vehicle, Sanskrit, idol worship of Buddha & Bodhisattvas).
32	In UPSC Indian Polity, list all 12 Schedules of the Indian Constitution using the 'TEARS OF OLD PM' framework and state which Constitutional Amendments added the 9th, 10th, 11th, and 12th Schedules.	1st (T): Territories & States; 2nd (E): Emoluments/salaries of constitutional offices; 3rd (A): Affirmations & Oaths; 4th (R): Rajya Sabha seat allocation; 5th (S): Scheduled Areas & Tribes administration; 6th (O): Other Tribal Areas (Assam, Meghalaya, Tripura, Mizoram — AMTM); 7th (F): Federal Lists (Union, State, Concurrent); 8th (O): Official Languages (22 languages); 9th (L): Land Reforms & Acts immune from initial judicial review (added by 1st Amendment, 1951; subject to judicial review post-24 April 1973 per I.R. Coelho case); 10th (D): Defection disqualification rules (52nd Amendment, 1985); 11th (P): Panchayats — 29 subjects (73rd Amendment, 1992); 12th (M): Municipalities — 18 subjects (74th Amendment, 1992).