JKSSB JE Civil cheat sheet

Concrete mix design by IS 10262 uses target mean strength f'ck = fck + 1.65·s, where s is the assumed standard deviation (4 N/mm² for M20–M25, 5 for M30–M50); water-cement ratio is chosen from the IS 456 durability table, not from strength alone.

Workability tests: slump test suits normal mixes (true slump 25–100 mm typical); compaction factor test is used for very dry or low-slump mixes; Vee-Bee consistometer measures remoulding effort in seconds for stiff mixes.

Bulk density and specific gravity: cement specific gravity ≈ 3.15, fine aggregate ≈ 2.65, coarse aggregate ≈ 2.70; 1 bag of cement = 50 kg = 0.0347 m³.

Standard brick size per IS 1077 is 190 × 90 × 90 mm (modular); with mortar the nominal size is 200 × 100 × 100 mm, giving 500 bricks per m³ of masonry.

Slope deflection and moment distribution are displacement methods; the three-moment (Clapeyron) theorem and unit-load (virtual work) method are force/flexibility methods. A fixed-end beam has 0° end rotation and carries fixed-end moments wL²/12 for UDL.

For a simply supported beam with central point load W and span L: max BM = WL/4, max deflection = WL³/48EI. For UDL w over L: max BM = wL²/8, max deflection = 5wL⁴/384EI.

Soil consistency limits (Atterberg): liquid limit, plastic limit, shrinkage limit. Plasticity index PI = LL − PL; a high PI (>30) indicates highly compressible clay. Standard Proctor gives OMC and MDD; higher compaction energy lowers OMC and raises MDD.

Surveying: 1° of latitude ≈ 111 km; Gunter's chain = 66 ft = 20.1168 m with 100 links; prismatic compass bears whole-circle bearing, surveyor's compass reads reduced bearing (0–90° in each quadrant).

Hydrology and fluids: rational formula Q = C·i·A (peak runoff); discharge Q = A·V; Bernoulli's equation p/γ + v²/2g + z = constant along a streamline for ideal, steady, incompressible flow.

Jammu & Kashmir: Union Territory since 31 October 2019 under the J&K Reorganisation Act; Jammu and Kashmir are the two divisions, with Srinagar (summer) and Jammu (winter) as capitals; Ladakh was carved out as a separate UT.

J&K statehood-era facts still asked: largest city by population is Srinagar; the Jhelum, Chenab, Ravi, Tawi and Indus are the major rivers; Wular Lake is the largest freshwater lake in India; Dal Lake is in Srinagar.

Indian Constitution: adopted 26 November 1949, came into force 26 January 1950; Fundamental Rights in Part III (Articles 12–35); Directive Principles in Part IV (Articles 36–51); Fundamental Duties in Article 51A added by the 42nd Amendment, 1976.

Constitutional offices: President elected by an electoral college (elected MPs + elected MLAs); Article 356 covers President's Rule; Article 370 (now abrogated) gave J&K special status; Article 35A dealt with permanent residents.

Indian geography superlatives: longest river Ganga; largest river island Majuli (Brahmaputra); highest peak Kanchenjunga (India); Thar Desert in Rajasthan; Deccan Plateau bounded by Western and Eastern Ghats.

Science basics: SI unit of force is newton, of pressure pascal, of power watt; speed of light ≈ 3 × 10⁸ m/s; pH below 7 is acidic; photosynthesis converts CO₂ and water to glucose using chlorophyll.

Indian economy and institutions: RBI established 1935, nationalised 1949; NITI Aayog replaced the Planning Commission in 2015; GST launched 1 July 2017; SEBI regulates securities markets.

Sports and awards: Bharat Ratna is India's highest civilian award; Dronacharya Award is for coaches; Ranji Trophy is cricket; Santosh Trophy is football; Khel Ratna is the top sporting honour.

Number series: check differences first, then ratios, then squares/cubes, then alternating (two interleaved) sequences; e.g. +3, +6, +12, +24 doubles.

Coding-decoding: letter positions A=1 … Z=26; opposite letters sum to 27 (A–Z, B–Y, C–X). Reverse alphabet position = 27 − forward position.

Blood relations: draw a family tree with gender symbols; 'mother's brother' = maternal uncle; 'father's sister's son' = cousin; never assume gender from a name or profession.

Direction sense: use a coordinate plane; after a left turn from North you face West, right turn from North you face East; net displacement uses Pythagoras on the N-S and E-W legs.

Syllogism: valid conclusions must hold in every possible Venn diagram; 'All A are B' does not imply 'All B are A'; 'Some A are B' is symmetric but not convertible to 'All'.

Seating arrangement: fix the reference person, then place others clockwise/counter-clockwise; for linear rows facing North, a person's left is your left; facing South, left and right swap.

Clock and calendar: hands coincide 11 times in 12 hours (every 65 5/11 minutes); hands are opposite 11 times in 12 hours; an ordinary year shifts the day by 1, a leap year by 2.

Cubes and dice: a cube cut into n³ small cubes has 8 corner cubes (3 faces painted), 12(n−2) edge cubes (2 faces), 6(n−2)² face cubes (1 face), and (n−2)³ unpainted.

Percentage and profit: profit% = (SP − CP)/CP × 100; successive discounts of a% and b% give net = a + b − ab/100; a 20% increase followed by a 20% decrease leaves 96% of the original.

Simple and compound interest: SI = P·R·T/100; CI = P(1 + R/100)^T − P; for 2 years the CI − SI difference equals P(R/100)².

Time, speed, distance: distance = speed × time; km/h to m/s multiply by 5/18; average speed for equal distances = 2xy/(x + y); relative speed adds for opposite directions, subtracts for same direction.

Trains and boats: a train of length L crossing a pole takes L/speed; crossing a platform of length P takes (L + P)/speed; downstream speed = boat + stream, upstream = boat − stream.

Time and work: if A does a job in x days and B in y days, together they take xy/(x + y) days; use LCM of days as total work units to avoid fractions.

Ratio and mixtures: alligation rule gives ratio = (mean − cheaper) : (dearer − mean); if a vessel of volume V has x litres replaced by water n times, final milk = x(1 − x/V)^n.

Mensuration: circle area πr², circumference 2πr; sphere volume 4/3πr³, surface 4πr²; cylinder volume πr²h, curved surface 2πrh; cone volume 1/3πr²h, slant l = √(r² + h²).

Averages and numbers: average = sum/count; for consecutive terms the average is the middle term; HCF × LCM = product of two numbers; divisibility by 3 and 9 depends on digit sum.

Subject-verb agreement: singular subjects take singular verbs; 'each, every, either, neither, everyone' are singular; with 'either … or' the verb agrees with the nearer subject.

Articles: 'a/an' before singular countable nouns (an before vowel sound: an hour, a university); 'the' for unique things, superlatives, rivers, ranges, and previously mentioned nouns.

Prepositions: 'since' with a point of time, 'for' with a duration; 'on' for days/dates, 'in' for months/years; 'between' for two, 'among' for more than two.

Tenses: present perfect (has/have + past participle) for past actions with present relevance; past perfect (had + participle) for the earlier of two past actions; 'since' and 'for' pair with perfect tenses.

Voice: active to passive moves the object to subject position and uses be + past participle; only transitive verbs can be made passive; the agent is introduced by 'by'.

Direct to indirect speech: present tense changes to past, 'this' to 'that', 'now' to 'then', 'today' to 'that day'; questions become statements with 'if/whether' or the wh-word.

Common confusions: 'affect' (verb) vs 'effect' (noun); 'principal' (main/head) vs 'principle' (rule); 'lie/lay', 'its/it's', 'their/there/they're'.

Idioms and one-word substitutions: 'at the eleventh hour' = at the last moment; 'to beat about the bush' = to avoid the main point; 'philanthropist' = lover of mankind; 'inevitable' = cannot be avoided.

Modular ratio m = 280/(3σcbc) is used in working-stress design of reinforced concrete; the neutral axis depth factor k = mσcbc/(mσcbc + σst).

IS 456 cover requirements: 40 mm for footings, 30 mm for slabs/beams exposed to moderate conditions, 20 mm for mild exposure; minimum grade for RCC is M20.

Steel: Fe415 means characteristic yield strength 415 N/mm²; partial safety factor for steel in limit-state design is 1.15, for concrete 1.5.

Bending: for a rectangular section, moment of resistance Mu = 0.87 fy Ast d(1 − fy Ast/(fck b d)); lever arm z = d(1 − 0.42 xu/d).

Shear and bond: nominal shear stress τv = Vu/(b·d); maximum shear stress limited by IS 456 Table 20; development length Ld = φσs/(4τbd).

Soil bearing: ultimate bearing capacity from Terzaghi qu = cNc + γDfNq + 0.5γBNγ for a strip footing; safe bearing capacity = qu/FOS (usually 2.5–3).

Survey levelling: rise = backsight − foresight; reduced level of next point = RL of previous + rise (or − fall); check: ΣBS − ΣFS = last RL − first RL.

Irrigation and hydraulics: Duty (ha/cumec) × Delta (m) = 8.64; discharge over a rectangular weir Q = (2/3)·Cd·L·√(2g)·H^(3/2).

Parliament: Lok Sabha maximum strength 552 (530 states, 20 UTs, 2 Anglo-Indians nominated - the nomination provision lapsed in 2020); Rajya Sabha maximum 250 (238 elected + 12 nominated).

Judiciary: Supreme Court established 26 January 1950; Article 32 gives the right to constitutional remedies (writs); High Court writs flow from Article 226, which is wider than Article 32.

Panchayati Raj: 73rd Amendment (1992) for panchayats, 74th for municipalities; three-tier system - gram panchayat, panchayat samiti, zila parishad.

J&K specifics: Jammu & Kashmir has 20 districts and 5 Lok Sabha seats; the Jammu–Srinagar National Highway passes through Banihal (Jawahar Tunnel) and the Chenani–Nashri tunnel.

National symbols and facts: national bird Indian peacock, animal Bengal tiger, flower lotus, river Ganga; ISRO headquarters Bengaluru; BARC at Trombay, Mumbai.

World organisations: UN founded 1945, HQ New York; WHO HQ Geneva; IMF and World Bank HQ Washington DC; WTO HQ Geneva, founded 1995.

Indian defence and institutions: Indian Military Academy at Dehradun; National Defence Academy at Khadakwasla, Pune; Border Security Force raised 1965, CRPF 1939.

Important days: Republic Day 26 January, Independence Day 15 August, Constitution Day 26 November, National Science Day 28 February, Engineers' Day 15 September (M. Visvesvaraya's birthday).

Alphabet series: note the gap pattern (e.g. +2, +4, +6); for mixed series, separate letters and numbers and solve each independently.

Analogy rule: identify the relation (whole–part, cause–effect, worker–workplace, synonym) and apply the same relation, not the same words, to the answer pair.

Odd-one-out: look for a property shared by all but one term - perfect squares, primes, multiples, or letter-position parity.

Ranking: total = (rank from top) + (rank from bottom) − 1; if two people swap places, the change in rank gives the number between them.

Statement and assumption: an assumption is something unstated but necessary for the statement to make sense; do not import outside facts or moral judgements.

Series with squares and cubes: memorise squares 1–30, cubes 1–15, and powers of 2 up to 2¹⁰ = 1024; many series are n² ± k or n³ ± k.

Matrix and figure reasoning: track rotation (90°/180°), number of elements added, and shading alternation; the answer must continue the same transformation.

Data sufficiency: decide whether the given statements are enough to answer the question; do not actually compute the value if sufficiency can be judged from the form.

Fractions to percentages: 1/8 = 12.5%, 1/6 = 16.67%, 1/7 = 14.28%, 1/9 = 11.11%, 1/11 = 9.09%, 1/12 = 8.33% - memorise these to convert quickly.

Partnership: profit share is proportional to capital × time; if capitals are in ratio a:b and periods in ratio x:y, profits are in ax:by.

Pipes and cisterns: a filling pipe is positive work, a draining pipe negative; net rate = sum of individual rates with signs, then time = 1/net rate.

Simple interest shortcut: SI for 2 years at R% equals 2R% of principal; the difference between CI and SI for 3 years = P(R/100)²(3 + R/100).

Ages: express present ages as variables and write the conditions as equations; the difference of ages stays constant over time.

Number system: sum of first n natural numbers = n(n+1)/2; sum of squares = n(n+1)(2n+1)/6; sum of cubes = [n(n+1)/2]².

Probability and permutations: P(E) = favourable/total; nPr = n!/(n−r)!; nCr = n!/(r!(n−r)!); for mutually exclusive events P(A or B) = P(A) + P(B).

Percentage change: a% increase then b% decrease gives net change = a − b − ab/100 percent; to restore an original value after a 25% rise, reduce by 20%.

Error spotting: check subject-verb agreement first, then tense consistency, then preposition/article, then pronoun reference - most errors fall in these four slots.

Comparatives: use 'than' with comparatives and 'to' with superior, inferior, senior, junior, prior, preferable; do not use double comparatives ('more better').

Conditionals: first conditional (if + present, will + base), second (if + past, would + base), third (if + past perfect, would have + participle).

Modals: 'should' for advice, 'must' for obligation, 'can' for ability, 'may' for permission/possibility, 'ought to' for moral duty; 'used to' for past habit.

Sentence improvement: prefer the concise, grammatically complete option; avoid options that change the meaning or introduce redundancy ('return back', 'discuss about').

Synonyms and antonyms: use word roots - 'bene' good, 'mal' bad, 'cred' believe, 'dict' say, 'chron' time; 'phil' love, 'phob' fear, 'anti' against.

Cloze and comprehension: read the whole passage first, then fill blanks using the surrounding grammar (tense, preposition, connector) as the deciding cue.

Spelling: double letters often follow a short vowel (beginning, occurrence, accommodate, committee); '-ible' after roots like 'poss', 'respons', '-able' in 'comfortable', 'available'.

Earthwork by trapezoidal rule: volume between two sections = L × (A1 + A2)/2; prismoidal correction = L(A1 + A2 − 4Am)/6, subtracted from the trapezoidal volume.

Cement mortar 1:6 means 1 part cement to 6 parts sand by volume; dry volume = wet volume × 1.30–1.35 to allow for voids and shrinkage.

Plastering: 12 mm thick plaster is standard for internal walls, 15–20 mm for external; quantity of mortar = area × thickness × dry factor.

Rate analysis: lead and lift - lead is horizontal transport distance, lift is vertical height; both are paid separately in earthwork and masonry items.

Concrete curing: minimum 7 days for ordinary Portland cement, 10 days for OPC-blended, 14 days for concrete with mineral admixtures (IS 456).

Bricks classification: first class bricks are table-moulded, well burnt, sharp-edged, and give a clear ringing sound; water absorption should be under 20% (first class ≈ 15–20%).

Tests on cement: fineness by sieve (90 µm) or Blaine; soundness by Le Chatelier or autoclave; setting time - initial ≥ 30 min, final ≤ 600 min for OPC.

Tender and contract: item-rate contract pays per measured quantity; lump-sum contract fixes a total price; EPC and BOT are common modern delivery models.

Ancient India: Indus Valley Civilisation (c. 2500–1500 BCE) - Harappa, Mohenjo-daro, Lothal (dockyard), Kalibangan; Mauryan Empire founded by Chandragupta Maurya, Ashoka's edicts on rock and pillar.

Medieval India: Delhi Sultanate (1206–1526) - Slave, Khilji, Tughlaq, Sayyid, Lodi dynasties; Mughal Empire founded 1526 by Babur at the First Battle of Panipat.

Freedom movement: Revolt of 1857, INC founded 1885, Partition of Bengal 1905, Jallianwala Bagh 1919, Non-Cooperation 1920, Dandi March 1930, Quit India 1942.

Indian rivers: Ganga rises at Gangotri (Bhagirathi), Brahmaputra at Mansarovar (as Tsangpo), Indus in Tibet, Narmada and Tapi flow west into the Arabian Sea.

Climate and soils: monsoon from the southwest in June–September; black (regur) soil is best for cotton; alluvial soil covers the northern plains; laterite soil forms in heavy-rainfall areas.

Mountain passes: Zoji La (Srinagar–Leh), Banihal (Jammu–Srinagar), Nathu La (Sikkim), Shipki La (Himachal), Khyber and Bolan in the northwest.

Indian states: 28 states and 8 Union Territories; Rajasthan is the largest state by area, Goa the smallest; Uttar Pradesh has the largest population.

Industries and minerals: Jharia and Raniganj are coal fields; Bailadila (Chhattisgarh) iron ore; Kolar (Karnataka) gold; Panna (MP) diamonds; Digboi (Assam) oil.

Venn diagrams: 'All A are B' means A is inside B; 'No A is B' means disjoint circles; 'Some A are not B' means A has a part outside B.

Syllogism conversion rules: All A are B → Some B are A; Some A are B → Some B are A; No A is B → No B is A. All A are B does NOT convert to All B are A.

Statement and conclusion: a conclusion must follow from the statement alone; a statement that is generally true but not supported is rejected.

Cause and effect: identify whether one event is a plausible cause of the other, or whether both stem from a common third factor.

Series completion in figures: count the number of lines, sides, or dots; look for alternation (increase then decrease) and rotation by a fixed angle.

Paper folding and cutting: unfold step by step in reverse; each unfold mirrors the hole pattern across the fold line, doubling the number of holes.

Embedded figures: the answer must contain exactly the same lines as the original figure, with no extra or missing line.

Analogy with numbers: apply the same operation (×2 + 1, square − 3) to the given pair; check both directions before choosing.

Percentage of total in a pie chart: angle = (value/total) × 360°; if an angle is given, value = (angle/360) × total.

Average of a data set changes by (new value − old value)/n when one item is replaced; adding a value equal to the mean leaves the mean unchanged.

Successive percentage change in DI tables: multiply factors (1 + r1)(1 + r2) − 1 rather than adding percentages.

Cuboid and cube: volume lbh, surface 2(lb + bh + hl), diagonal √(l² + b² + h²); a cube of side a has volume a³ and diagonal a√3.

Triangle: area = ½ × base × height = √(s(s−a)(s−b)(s−c)) by Heron's formula; for an equilateral triangle of side a, area = (√3/4)a².

Trapezium: area = ½ × (sum of parallel sides) × height; rhombus area = ½ × d1 × d2.

Speed and average in DI: average speed over two equal distances = 2v1v2/(v1 + v2), not the arithmetic mean of the speeds.

Ratio to percentage and fraction: convert all data to one form (fraction or decimal) before comparing; compare cross-products for two ratios a/b and c/d.

One-word substitutions: 'ephemeral' = short-lived, 'ubiquitous' = present everywhere, 'meticulous' = very careful, 'ambiguous' = open to more than one meaning.

Idioms: 'a blessing in disguise' = apparent misfortune that turns out well; 'to let the cat out of the bag' = to reveal a secret; 'once in a blue moon' = very rarely.

Antonym pairs to memorise: abundant/scarce, candid/evasive, frugal/extravagant, obstinate/flexible, transient/permanent.

Active vocabulary for letters and essays: use formal connectors - 'furthermore', 'consequently', 'nevertheless', 'in addition' - and avoid contractions in formal writing.

Punctuation: use a comma before a coordinating conjunction joining two independent clauses; use a semicolon between closely related independent clauses without a conjunction.

Paragraph and precis writing: keep one central idea per paragraph; a precis should be about one-third of the original length and in your own words.

Commonly misspelt words: separate, necessary, accommodate, occurrence, receive, definitely, maintenance, questionnaire.

Formal letter structure: sender's address, date, receiver's address, subject line, salutation, body in three paragraphs, complimentary close and signature.

Water demand: per-capita demand for Indian cities is typically 135–150 lpcd for domestic use; peak factor 1.5–2 for design of distribution mains.

Water treatment sequence: screening → aeration → coagulation-flocculation (alum) → sedimentation → filtration (rapid sand) → disinfection (chlorination).

BOD and COD: BOD measures oxygen consumed by microbes over 5 days at 20 °C; COD is a faster chemical measure and is always higher than BOD.

Sewer design: self-cleansing velocity ≈ 0.6–0.9 m/s; minimum diameter for public sewers is usually 150 mm; manholes spaced 30–50 m in straight runs.

Highway geometrics: camber for flexible pavement 2–3% (heavy rainfall up to 4%); superelevation e = v²/(127R) with design speed in km/h and R in metres.

Pavement layers: subgrade → sub-base → base → wearing course for flexible pavements; rigid pavements carry load by slab action and need joints (expansion, contraction, construction).

Railways: broad gauge 1676 mm, metre gauge 1000 mm, narrow gauge 762 mm; sleeper density is expressed as number of sleepers per rail length.

Surveying instruments: total station measures horizontal and vertical angles plus distance electronically; GPS gives 3-D position; theodolite measures angles only.

Five-Year Plans ran from 1951 to 2017 (12th Plan ended 2017); NITI Aayog now issues three-year action agendas and seven-year strategy documents.

Banking: RBI is the central bank and regulator; repo rate is the rate at which RBI lends to banks, reverse repo is the opposite; CRR and SLR are reserve requirements.

Inflation: CPI measures retail price change, WPI measures wholesale; the RBI inflation target is 4% with a band of ±2%.

Space and defence: Chandrayaan-3 landed near the lunar south pole in 2023; Mangalyaan entered Mars orbit in 2014; INS Vikrant is India's first indigenous aircraft carrier.

Health and science: WHO declared COVID-19 a pandemic in March 2020; vaccines include Covishield and Covaxin; India's UPI is a real-time payment system run by NPCI.

Constitutional bodies: Election Commission (Article 324), Finance Commission (Article 280), UPSC (Article 315), CAG (Article 148), Attorney General (Article 76).

Environment: Project Tiger launched 1973; Chipko movement 1973; Ramsar sites are wetlands of international importance; Kaziranga is famous for the one-horned rhinoceros.

Awards and honours: Nobel Prize categories are Physics, Chemistry, Medicine, Literature, Peace, and Economics (1969 onward); Jnanpith is for literature in Indian languages.

Clock angle: angle = |30H − 5.5M| degrees, where H is the hour and M the minutes; take the smaller angle if the result exceeds 180°.

Calendar: for a date, the day depends on the odd days from a reference; 100 years have 5 odd days, 200 years 3, 300 years 1, 400 years 0.

Blood relation shorthand: use 'M + F' for siblings, 'M − F' for mother-daughter, and always draw the tree before answering.

Direction and distance: after moving N, E, S, W, net displacement = √(x² + y²); the final facing direction depends on the net turn (right = +90°, left = −90°).

Seating arrangement: in a circle facing the centre, clockwise means the person to your right is counter-clockwise; facing outward reverses this.

Coding by position: if the code of 'CAT' is 'DBU', each letter is shifted +1; if 'ACT' → 'ZDS', letters alternate +1 and −1.

Ranking with ties: if two people share a rank, the next rank is skipped; total count = rank from top + rank from bottom − number of ties.

Logical deduction: 'either/or' conclusions in syllogism apply only when both conclusions are individually possible but not simultaneously true.

Work equivalence: if A is twice as efficient as B, A takes half the time; efficiency is inversely proportional to time.

Work with wages: wages are shared in the ratio of work done, which equals the ratio of 1/time for each worker.

Pipes with a leak: if a pipe fills in x hours and a leak empties in y hours, net time = xy/(y − x) when y > x.

Train crossing another train: same direction time = (L1 + L2)/(v1 − v2); opposite direction time = (L1 + L2)/(v1 + v2).

Boats: time downstream = D/(b + s), upstream = D/(b − s); if downstream and upstream times are equal, the stream speed is zero.

Relative speed in km/h: convert to m/s by multiplying by 5/18 before dividing lengths in metres.

Work in alternate days: if A works on odd days and B on even days, compute two-day work = 1/x + 1/y, then find the number of two-day cycles.

Circular track: two runners meet at the starting point after LCM of their individual lap times; they meet anywhere after LCM of (circumference/relative speed).

Parts of speech: a noun names, a pronoun replaces, an adjective qualifies a noun, an adverb modifies a verb/adjective/adverb, a conjunction joins, a preposition relates.

Phrases vs clauses: a clause has a subject and a finite verb; a phrase does not. 'Because it rained' is a clause; 'after the rain' is a phrase.

Types of sentences: declarative (statement), interrogative (question), imperative (command/request), exclamatory (strong feeling); identify by the verb form and punctuation.

Transformation: 'too … to' → 'so … that … not'; 'no sooner … than' → 'as soon as'; 'unless' → 'if … not'.

Reading comprehension: answer only from the passage; eliminate options that are true in general but not stated; the main idea is usually in the first or last paragraph.

Para jumbles: find the opening sentence (introduces the topic, no pronouns referring back), then link sentences by pronouns, connectors, and repeated nouns.

Figures of speech: simile uses 'like/as', metaphor states one thing is another, personification gives human qualities to non-human things, hyperbole is exaggeration.

Common error patterns: 'one of the + plural noun + singular verb'; 'the number of + plural noun + singular verb'; 'a number of + plural noun + plural verb'.

Hydrostatic pressure: p = ρgh; total force on a vertical rectangular gate = ½ρgH² × width, acting at H/3 from the base.

Continuity: A1V1 = A2V2 for incompressible flow; velocity increases where the cross-section narrows.

Bernoulli with losses: p1/γ + V1²/2g + z1 = p2/γ + V2²/2g + z2 + hL, where hL includes friction (Darcy–Weisbach hf = fLV²/2gD) and minor losses.

Orifice and mouthpiece: discharge Q = Cd·A·√(2gH); Cd ≈ 0.61–0.62 for a sharp-edged orifice, Cv ≈ 0.98, Cc ≈ 0.62.

Notches and weirs: rectangular weir Q = (2/3)Cd·L·√(2g)·H^1.5; triangular (V-notch) Q = (8/15)Cd·√(2g)·tan(θ/2)·H^2.5.

Flow measurement: venturimeter uses the pressure difference between throat and inlet; orifice meter has a lower Cd but is cheaper; pitot tube measures velocity at a point.

Open channel flow: Manning's equation V = (1/n)R^(2/3)S^(1/2); hydraulic radius R = A/P; critical flow occurs at Froude number = 1.

Pumps and turbines: centrifugal pump converts kinetic energy to pressure; Pelton wheel is an impulse turbine for high head, Francis and Kaplan are reaction turbines for medium and low head.

J&K districts include Jammu, Samba, Kathua, Udhampur, Reasi, Rajouri, Poonch, Doda, Kishtwar, Ramban, Srinagar, Ganderbal, Budgam, Pulwama, Anantnag, Kulgam, Shopian, Baramulla, Kupwara, Bandipora.

The Jhelum originates at Verinag in Anantnag and flows through Srinagar; the Tawi flows through Jammu city; the Chenab enters via Kishtwar and Doda.

Passes and routes: Banihal Pass connects Jammu with the Kashmir Valley; Zoji La connects Srinagar with Kargil and Leh; Mughal Road links Poonch with Shopian.

Tourism and heritage: Gulmarg, Pahalgam, Sonamarg are hill stations; Vaishno Devi shrine is near Katra; Amarnath cave yatra is in the Himalayas of south Kashmir.

Lakes: Wular (largest freshwater lake in India, in Bandipora), Dal (Srinagar), Manasbal, Nigeen, and Pangong in Ladakh.

Economy: horticulture (apple, saffron, walnut) is central; the Kashmir saffron has GI status; handicrafts include pashmina, carpets, and papier-mâché.

Institutions: University of Jammu and University of Kashmir are the two main universities; Sher-e-Kashmir University of Agricultural Sciences has campuses in both divisions.

Rivers and dams: Salal, Baglihar, and Dulhasti are hydroelectric projects on the Chenab; Uri and Kishanganga are on the Jhelum system.

Scheduling puzzles: list the fixed constraints first (A before B, C on Wednesday), then place the most restricted item and propagate.

Grouping puzzles: if 'X and Y cannot be together' and 'X must be selected', then Y is out; use elimination on the options.

Arrangement with floors: floor 1 is the lowest; 'above' means higher floor number; 'immediately above' means consecutive numbers.

Selection with conditions: 'at least one of A and B' means not both absent; combine with other constraints to fix the group size.

Ranking and order: if A is taller than B but shorter than C, the order is C > A > B; convert every statement into a chain before answering.

Decision making: choose the option that is lawful, ethical, and practical; reject options that are illegal, biased, or that shift blame.

Data arrangement tables: for 5–7 items, draw a grid with categories as rows and positions as columns; fill certainties first.

Input-output and machine puzzles: track the pattern of rearrangement step by step; the rule is usually sorting or moving the smallest/largest to an end.

Quadratic equation: roots from ax² + bx + c = 0 are x = [−b ± √(b² − 4ac)]/2a; sum of roots = −b/a, product = c/a.

Algebraic identities: (a + b)² = a² + 2ab + b²; a² − b² = (a + b)(a − b); (a + b)³ = a³ + b³ + 3ab(a + b).

If x + 1/x = k, then x² + 1/x² = k² − 2 and x³ + 1/x³ = k³ − 3k.

HCF and LCM: for two numbers, HCF × LCM = product; HCF of fractions = HCF of numerators/LCM of denominators; LCM of fractions = LCM of numerators/HCF of denominators.

Divisibility rules: by 4 - last two digits divisible by 4; by 8 - last three digits; by 11 - alternating sum of digits is 0 or a multiple of 11.

Remainder theorem: the remainder of p(x) divided by (x − a) is p(a); for numbers, use modular arithmetic to simplify large powers.

Number of factors: if N = a^p · b^q, the number of divisors is (p + 1)(q + 1); the number of ways to express N as a product of two factors is half that count (rounded up).

Logarithms: log(ab) = log a + log b; log(a/b) = log a − log b; log(a^n) = n log a; log_b a = log a/log b.

Error type 1 - subject-verb: 'The list of items are long' is wrong; the verb agrees with 'list', so use 'is'.

Error type 2 - pronoun: 'Each student must bring their book' is now accepted in informal usage, but exam keys often prefer 'his or her' for strict agreement.

Error type 3 - tense: 'He has gone yesterday' is wrong; use simple past 'went' with a definite past time marker.

Error type 4 - preposition: 'discuss about', 'married with', 'comprise of', 'superior than' are wrong; use 'discuss', 'married to', 'comprise', 'superior to'.

Error type 5 - article: 'He is best student' is wrong; use 'the best student' for superlatives.

Sentence improvement: prefer the option that keeps the original meaning and fixes only the error; reject options that are grammatically correct but change the sense.

Parallelism: items in a list must share the same form - 'reading, writing, and swimming', not 'reading, writing, and to swim'.

Redundancy: 'return back', 'repeat again', 'free gift', 'final outcome', 'very unique' are redundant; the shorter correct form is the answer.

Limit state of collapse in flexure: the section is under-reinforced when xu/d < 0.48 for Fe415, 0.46 for Fe500; over-reinforced sections are avoided because they fail without warning.

Minimum tension steel in a beam = 0.85bd/fy; maximum = 0.04bD; minimum shear reinforcement is provided by stirrups at spacing not exceeding 0.75d.

Slab design: one-way slabs have Ly/Lx > 2; two-way slabs need design for both directions; minimum thickness for a simply supported slab is span/30 for mild steel.

Column: short column when both slenderness ratios (lex/D and ley/b) are under 12; minimum longitudinal steel 0.8% and maximum 6% of gross area.

Footing: one-way shear and two-way (punching) shear are checked; punching shear perimeter is at d/2 from the column face.

Steel connections: bolt value in shear = (π/4)d²·fub/√3 divided by the partial safety factor; weld strength depends on throat thickness = 0.7 × size.

Deflection control: span/effective depth limits - 20 for simply supported, 26 for continuous beams, 7 for cantilever (for spans up to 10 m).

Durability: minimum cement content and maximum w/c ratio depend on exposure class (mild, moderate, severe, very severe, extreme) per IS 456 Table 5.

Capitals: Australia - Canberra, Canada - Ottawa, Switzerland - Bern, Turkey - Ankara, Brazil - Brasília, Myanmar - Naypyidaw.

Currencies: Japan - yen, UK - pound sterling, Bangladesh - taka, Nepal - Nepalese rupee, Sri Lanka - Sri Lankan rupee, Russia - rouble.

First in India: first Governor-General of independent India - Lord Mountbatten; first President - Dr Rajendra Prasad; first woman Prime Minister - Indira Gandhi.

National parks: Jim Corbett (Uttarakhand) is India's oldest; Gir (Gujarat) for Asiatic lions; Sundarbans (West Bengal) for Royal Bengal tigers; Periyar (Kerala) for elephants.

Dances: Bharatanatyam (Tamil Nadu), Kathak (Uttar Pradesh), Kathakali (Kerala), Odissi (Odisha), Kuchipudi (Andhra Pradesh), Manipuri (Manipur), Ghoomar (Rajasthan).

Famous monuments: Taj Mahal (Agra, UNESCO), Qutub Minar (Delhi), Red Fort (Delhi), Sun Temple (Konark), Hampi (Karnataka), Sanchi Stupa (MP).

Books and authors: 'Discovery of India' - Jawaharlal Nehru; 'Wings of Fire' - A.P.J. Abdul Kalam; 'Gitanjali' - Rabindranath Tagore.

Abbreviations: ISRO, DRDO, CSIR, BARC, NTPC, ONGC, SAIL - know the full forms and their headquarters for one-mark questions.

Word formation: check whether the given word can be formed using only the letters in the source word, respecting letter counts.

Letter series with alternating case: separate uppercase and lowercase sequences; the pattern often alternates +1, +3, +5.

Number analogy: if 4:64, the rule is cube (4³); if 5:25, it is square; test the rule on the second pair before answering.

Odd pair: in a list of word pairs, the odd pair breaks the relation - e.g. all pairs are synonyms except one antonym pair.

Classification: group items by a common property (all capitals, all planets, all metals); the odd item lacks that property.

Statement and course of action: a valid action must be practical, address the problem directly, and be within the authority of the person mentioned.

Cause and effect: if two statements are given, check whether one can plausibly cause the other; if both are effects of a common cause, mark accordingly.

Symbol substitution: replace each symbol with the given operation and apply BODMAS strictly; do not assume the usual meaning of the symbol.

Percentage of a number: x% of N = (x/100) × N; to find what percent A is of B, compute (A/B) × 100.

Ratio simplification: divide all terms by the HCF; to compare ratios a/b and c/d, compare ad and bc.

Proportion: if a:b = c:d, then ad = bc; for continued proportion a:b = b:c, b² = ac.

Average with additions: if the average of n items is A and a new item x is added, the new average is (nA + x)/(n + 1).

Weighted average: average = (n1A1 + n2A2)/(n1 + n2); the result always lies between A1 and A2.

Percentage error: error% = (|true − measured|/true) × 100; if a value is misread as 10% more, the computed result is off by about 9.09%.

Population growth and depreciation: P after n years = P0(1 + r)^n for growth and P0(1 − r)^n for depreciation.

Ratio of ages: if the present ratio is a:b and after n years it is c:d, set (a k + n)/(b k + n) = c/d and solve for k.

Articles with countable/uncountable nouns: 'a/an' only with singular countable nouns; 'the' with specific or unique items; no article with general plurals and abstract nouns.

Prepositions of time: 'at' for clock times, 'on' for days and dates, 'in' for months, years, seasons, and longer periods.

Gerund vs infinitive: verbs like enjoy, avoid, suggest, mind take the gerund; verbs like want, decide, hope, agree take the infinitive.

Reported speech time shift: 'now' → 'then', 'today' → 'that day', 'tomorrow' → 'the next day', 'yesterday' → 'the previous day', 'here' → 'there'.

Conditional inversion: 'Had I known' = 'If I had known'; 'Were I rich' = 'If I were rich'; 'Should you need help' = 'If you should need help'.

Collective nouns: 'team, jury, committee' take a singular verb when acting as one unit and a plural verb when members act individually.

Question tags: positive statement takes a negative tag and vice versa; 'I am' takes 'aren't I'; 'Let's' takes 'shall we'.

Punctuation in lists: use commas for items, a semicolon when items themselves contain commas, and a colon before an explanation or list.