Jharkhand Polytechnic (PECE) cheat sheet

Equations of motion: v = u + at; s = ut + ½at²; v² = u² + 2as (uniform acceleration only).

Newton's second law: F = ma; weight W = mg with g ≈ 9.8 m/s² near Earth's surface.

Work-energy theorem: W_net = ΔKE; kinetic energy KE = ½mv²; potential energy PE = mgh near Earth.

Conservation of momentum: for an isolated system, total momentum before = total momentum after any collision.

Ohm's law: V = IR; resistors in series add (R = R₁ + R₂); in parallel 1/R = 1/R₁ + 1/R₂.

Electrical power: P = VI = I²R = V²/R; energy consumed in kWh = power(kW) × time(h).

Snell's law: n₁sin θ₁ = n₂sin θ₂; refractive index n = c/v; total internal reflection occurs when the angle exceeds the critical angle sin θc = n₂/n₁.

Lens/mirror formula: 1/f = 1/v − 1/u (lens) and 1/f = 1/v + 1/u (mirror); power of a lens P = 1/f (in metres), unit dioptre.

Half-life: after n half-lives, remaining fraction = (1/2)^n; radioactive decay is exponential and random.

Mole concept: 1 mole = 6.022 × 10²³ particles = molar mass in grams = 22.4 L of any gas at STP.

Molarity M = moles of solute / volume of solution in litres; molality m = moles of solute / kg of solvent.

pH = −log[H⁺]; pH + pOH = 14 at 25 °C; pH < 7 acidic, pH = 7 neutral, pH > 7 basic.

Balancing: total atoms of each element and total charge must be equal on both sides of the equation.

Periodic trends: atomic radius decreases across a period and increases down a group; electronegativity increases across a period and decreases down a group.

Oxidation number rules: free element = 0; O usually −2; H usually +1; sum of oxidation numbers equals the ion's charge.

Le Chatelier's principle: adding reactant or removing product shifts equilibrium forward; increasing pressure favours the side with fewer gas moles.

Common strong acids: HCl, H₂SO₄, HNO₃; strong bases: NaOH, KOH; salts of strong acid + strong base are neutral.

Ideal gas equation: PV = nRT with R = 8.314 J mol⁻¹ K⁻¹; at constant T and P, equal volumes of gases hold equal moles (Avogadro).

Quadratic formula: for ax² + bx + c = 0, x = [−b ± √(b² − 4ac)] / 2a; discriminant D > 0 two real roots, D = 0 equal roots, D < 0 complex roots.

Sum and product of roots: α + β = −b/a and αβ = c/a.

Arithmetic progression: nth term aₙ = a + (n − 1)d; sum Sₙ = n/2 [2a + (n − 1)d].

Geometric progression: aₙ = ar^(n−1); Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1; infinite sum = a/(1 − r) when |r| < 1.

Trigonometric identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; sin 2θ = 2 sin θ cos θ.

Standard values: sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2, tan 45° = 1.

Distance formula √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint ((x₁ + x₂)/2, (y₁ + y₂)/2); slope m = (y₂ − y₁)/(x₂ − x₁).

Derivatives: d/dx(xⁿ) = nx^(n−1); d/dx(sin x) = cos x; d/dx(eˣ) = eˣ; d/dx(ln x) = 1/x.

Combinations and permutations: nPr = n!/(n − r)!; nCr = n!/[r!(n − r)!]; nCr = nC(n − r).

Jharkhand was carved out of Bihar on 15 November 2000, on the birth anniversary of Birsa Munda.

Capital of Jharkhand: Ranchi; the state shares borders with Bihar, Uttar Pradesh, Chhattisgarh, Odisha and West Bengal.

Jharkhand is known for mineral wealth - coal, iron ore, mica, bauxite and copper - and hosts major steel plants such as Jamshedpur (TISCO/Tata Steel).

Birsa Munda led the Munda rebellion (Ulgulan) against British rule in the late 19th century; he is honoured as Bhagwan Birsa Munda.

India's Constitution came into force on 26 January 1950; it was adopted by the Constituent Assembly on 26 November 1949.

Fundamental Rights are in Part III (Articles 12–35); the Right to Constitutional Remedies (Article 32) is called the heart and soul of the Constitution by Dr B. R. Ambedkar.

National symbols: national bird Indian peacock, national animal Bengal tiger, national flower lotus, national fruit mango.

ISRO is headquartered in Bengaluru; India's Mars mission was Mangalyaan (Mars Orbiter Mission), launched in 2013.

Number series: check successive differences, then ratios, then squares/cubes or alternating patterns before assuming a rule.

Alphabet positions: A = 1 … Z = 26; reverse position = 27 − forward position (e.g., A ↔ 26).

Odd-one-out: test each option for a shared property (prime, perfect square, divisibility) and pick the one that breaks it.

Coding-decoding: compute the shift between letters of the word and its code; the same shift usually applies to the new word.

Blood relations: draw a family tree with gender markers; 'maternal uncle' = mother's brother, 'paternal aunt' = father's sister.

Direction sense: net displacement uses Pythagoras when movements are perpendicular; opposite directions cancel.

Syllogism: a conclusion follows only if true in every possible diagram consistent with the statements; 'some' does not imply 'all'.

Clock: angle between hands = |30H − 5.5M| degrees; hands coincide 11 times in 12 hours.

Calendar: ordinary year shifts the weekday by 1, a leap year by 2; a century has 5 odd days for 100, 3 for 200, 1 for 300, 0 for 400 years.

Subject–verb agreement: singular subject takes singular verb; 'each, every, either, neither, everyone' are singular.

Articles: 'a/an' before singular countable nouns (an before vowel sounds); 'the' for specific or unique references.

Tenses: present perfect = has/have + past participle, used for past actions with present relevance; simple past for finished past time.

Active to passive: object becomes subject, verb becomes be + past participle, agent introduced by 'by'.

Direct to indirect speech: present tense in the reported clause shifts to past; 'now' → 'then', 'today' → 'that day', 'tomorrow' → 'the next day'.

Prepositions of time: 'in' for months/years, 'on' for days/dates, 'at' for clock times.

Synonyms/antonyms: match the word's part of speech and exact sense; e.g., 'benevolent' = kind, opposite 'malevolent'.

Common error: 'one of the + plural noun + singular verb' (One of the boys is present); 'discuss' and 'reach' take no 'about/at'.

Reading comprehension: locate the answer in the passage; the correct option must be supported by the text, not by outside knowledge.