TS ICET cheat sheet

Data sufficiency asks whether the given statements are enough to answer the question, not what the numeric answer is; options typically run: I alone, II alone, either alone, both together, or neither.

Decision rule: test statement I alone first with no help from II; if it yields a unique value, the answer cannot involve II. Then repeat for II alone.

Two linear equations in two unknowns give a unique solution only when the ratio of coefficients of x differs from the ratio of coefficients of y (not parallel, not identical).

A single equation such as 3x + 2y = 12 with x, y positive integers can still be sufficient if only one integer pair fits; always test integrality constraints.

For yes/no questions, sufficiency requires a definite yes or a definite no; a statement that sometimes gives yes and sometimes no is insufficient.

In geometry data sufficiency, a stated figure is not to scale; only explicitly given lengths, angles, or parallel/perpendicular markings may be used.

Beware redundant information: if statement I already fixes the quantity, statement II adds nothing, so the answer is 'I alone sufficient'.

When a question asks for a value and each statement independently gives that same unique value, the answer is 'either statement alone is sufficient'.

Number series: check successive differences, then ratios, then alternating or two-tier patterns (squares, cubes, primes, n²±1, ×2+1).

Coding-decoding: map letter positions A=1…Z=26; test shifts (+1, −1, +2), reversal (A↔Z), and position-sum codes before assuming a word-substitution code.

Blood relations: draw a family tree using + for male and − for female; 'mother's brother' = maternal uncle, 'father's sister's son' = cousin, 'son's wife' = daughter-in-law.

Seating: for a circle, facing centre means left is clockwise; facing outward means left is anticlockwise. For a row, left/right swap when the observer faces the row.

Syllogism: 'All A are B' does not imply 'All B are A'; 'Some A are B' converts to 'Some B are A'; 'No A is B' converts to 'No B is A'.

Clocks: minute hand gains 5.5° per minute on the hour hand; hands coincide every 65 5/11 minutes and are opposite every 32 8/11 minutes after 12:00.

Calendars: a normal year shifts the weekday by 1, a leap year by 2; 400 years contain 0 odd days, so the Gregorian calendar repeats every 400 years.

Cubes: a cube cut into n³ small cubes has 8 corner cubes with 3 painted faces, 12(n−2) with 2 faces, 6(n−2)² with 1 face, and (n−2)³ with none.

Percentage change: successive changes of a% then b% give net change a + b + ab/100 percent; a 20% rise then 20% fall leaves a net 4% loss.

Profit and loss are computed on cost price unless stated otherwise; selling price = CP × (100 + profit%)/100, and discount is computed on marked price.

Compound interest: A = P(1 + r/100)^n for annual compounding; for half-yearly use r/2 and 2n periods, for quarterly r/4 and 4n periods.

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.

Speed conversion: multiply km/h by 5/18 to get m/s; for a train of length L crossing a platform of length P, time = (L + P)/speed.

Average of n numbers is their sum divided by n; if every value is increased by k, the average rises by k, and if multiplied by k, the average is multiplied by k.

Arithmetic progression: nth term a + (n − 1)d, sum n/2 [2a + (n − 1)d]; geometric progression nth term ar^(n−1), sum a(rⁿ − 1)/(r − 1) for r ≠ 1.

Ratio: if a : b = 2 : 3 and b : c = 4 : 5, make b common by LCM, giving a : b : c = 8 : 12 : 15.

Identities: (a + b)² = a² + 2ab + b², a² − b² = (a − b)(a + b), a³ + b³ = (a + b)(a² − ab + b²), and x + 1/x squared equals x² + 1/x² + 2.

Quadratic ax² + bx + c = 0: roots sum to −b/a and multiply to c/a; discriminant b² − 4ac decides real, equal, or imaginary roots.

Mensuration: cylinder volume πr²h and curved surface 2πrh; cone volume ⅓πr²h with slant l = √(r² + h²); sphere volume 4/3 πr³ and surface 4πr².

Coordinate geometry: distance √((x₂ − x₁)² + (y₂ − y₁)²), midpoint average of coordinates, and slope (y₂ − y₁)/(x₂ − x₁); parallel lines share slope, perpendicular slopes multiply to −1.

Section formula: a point dividing a join in ratio m : n internally has coordinates ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

Trigonometry: sin²θ + cos²θ = 1, tanθ = sinθ/cosθ; standard values sin30 = ½, cos30 = √3/2, tan45 = 1, sin60 = √3/2.

Angle of elevation problems use tan = height/distance from a right triangle; the angle of depression from a tower equals the angle of elevation from the ground point.

Area formulas: triangle ½ × base × height or √(s(s−a)(s−b)(s−c)); trapezium ½(a + b)h; rhombus ½ d₁d₂.

Mean of grouped data with frequencies: x̄ = Σfx/Σf; median class is where cumulative frequency first reaches N/2.

Mode is the most frequent value; for grouped data the modal class has the highest frequency, and empirical relation is Mode = 3 Median − 2 Mean.

Range = maximum − minimum; standard deviation is the square root of variance, and variance is the mean of squared deviations from the mean.

Coefficient of variation = (standard deviation / mean) × 100; a larger CV means greater relative variability.

In a table question, read the units in the header (thousands, lakhs, percent) before computing; a common error is mixing units across columns.

For pie charts, each sector angle = (component/total) × 360°, and percentage share = component/total × 100.

Percentage increase between two years = (new − old)/old × 100; average annual growth uses the number of intervals, not the number of years listed.

If every observation is increased by a constant, the mean and median rise by that constant but the standard deviation and range stay unchanged.

Synonym pairs to know: 'abundant' = plentiful, 'candid' = frank, 'lucid' = clear, 'obsolete' = outdated, 'tenacious' = persistent.

Antonym pairs to know: 'benevolent' ↔ malevolent, 'frugal' ↔ extravagant, 'transparent' ↔ opaque, 'scarce' ↔ plentiful.

Idioms: 'to let the cat out of the bag' = reveal a secret; 'a bolt from the blue' = a sudden shock; 'to bite the bullet' = endure pain bravely; 'once in a blue moon' = very rarely.

Subject-verb agreement: a singular subject takes a singular verb even with intervening phrases; 'each', 'every', 'either', 'neither' are singular.

Neither…nor and either…or take a verb agreeing with the nearer subject; 'one of the + plural noun' takes a singular verb.

Use 'fewer' with countable nouns and 'less' with uncountable; 'between' for two items, 'among' for more than two.

Fill-in-the-blank: check the preposition or verb form demanded by the surrounding words, e.g. 'comply with', 'consist of', 'prefer X to Y', 'deprived of'.

Sentence correction: watch for dangling modifiers, wrong tense sequence, double negatives, and misplaced 'only', which changes meaning by position.

Main-idea questions are answered by the thesis sentence, usually in the first or last paragraph; supporting details are distractors for such questions.

Inference questions accept only what the passage implies, not outside knowledge; an option that is true in general but unsupported by the text is wrong.

Tone and attitude words: critical, neutral, optimistic, sarcastic, objective; identify the author's stance from adjectives and qualifiers used.

Vocabulary in context: substitute each option into the sentence and keep the one that preserves the passage's meaning, not the dictionary's first sense.

Sentence ordering: find the opening sentence (general statement, no pronoun reference), then link sentences by pronouns, connectors such as 'however' and 'therefore', and repeated nouns.

Paragraph coherence: a sentence that introduces a new topic or breaks the connector chain is the odd one out.

Composition: a paragraph needs one controlling idea, supporting sentences, and a closing link; avoid shifting tense or person mid-paragraph.

For 'which of the following can be inferred' questions, eliminate options that overstate with words like 'always', 'never', or 'only'.

Computer basics: CPU executes instructions, RAM is volatile working memory, ROM is non-volatile, and secondary storage (SSD, HDD) retains data without power.

Units: 1 byte = 8 bits, 1 KB = 1024 bytes, 1 MB = 1024 KB, 1 GB = 1024 MB; a bit is a binary digit, 0 or 1.

Software categories: system software (operating systems, drivers) versus application software (word processors, browsers); open-source software publishes its source code.

Networking terms: LAN is a local network, WAN spans wide areas, IP address identifies a device, HTTP/HTTPS are web protocols, and a URL locates a resource.

Business terms: GDP is the value of goods and services produced in a country in a period; inflation is a sustained rise in the general price level.

Commercial terminology: 'debit' and 'credit' in accounting, 'ledger' records transactions, 'invoice' requests payment, 'equity' is owners' claim on assets.

Market terms: IPO is a company's first public share sale, 'bull market' means rising prices, 'bear market' falling prices, and 'liquidity' is ease of converting assets to cash.

Current business developments: keep track of India's monetary policy repo rate set by the RBI, GST as an indirect tax, and major indices such as Sensex and Nifty.

Read the question stem first and classify it as a value question or a yes/no question before evaluating statements.

Eliminate options progressively: if I alone works, drop every option requiring II; if I fails, test II alone.

A statement giving a range or inequality rarely fixes a unique value; check whether the question asks for a maximum, minimum, or exact figure.

Do not carry data from statement I into statement II; each is evaluated independently against the stem.

For linear arrangements with fixed positions, fix the most constrained person first, then place others relative to that anchor.

In floor puzzles, list floors 1 to n and mark 'above/below' relations as inequalities before assigning names.

For syllogism possibility questions, a conclusion is valid only if it holds in every Venn diagram consistent with the premises.

In clock problems, the angle between hands at h hours m minutes is |30h − 5.5m| degrees; take the smaller angle.

Simple interest = P × R × T / 100; the amount is P plus that interest, and SI is the same each year.

If a sum doubles in t years at simple interest, the rate is 100/t percent per annum.

Boats: downstream speed = boat + stream, upstream = boat − stream; time ratio downstream : upstream is the inverse of the speed ratio.

Mixtures: replacement formula leaves P(1 − x/P)^n of the original component after n operations removing x litres.

Sum of interior angles of an n-sided polygon is (n − 2) × 180°; each exterior angle of a regular polygon is 360°/n.

Pythagoras: in a right triangle, hypotenuse² = sum of squares of the other two sides; triples 3-4-5, 5-12-13, 8-15-17 recur in problems.

Circle: area πr², circumference 2πr; the angle in a semicircle is 90°, and the tangent at a point is perpendicular to the radius there.

Logarithms: log(ab) = log a + log b, log(a/b) = log a − log b, log(aⁿ) = n log a.

Variance of the first n natural numbers is (n² − 1)/12, and their mean is (n + 1)/2.

Adding a new observation equal to the mean leaves the mean unchanged but lowers the standard deviation.

In bar graphs, compare heights only when the axis starts at zero; a truncated axis exaggerates differences.

For line graphs, slope between two points gives the rate of change, useful for growth-rate questions.

Use the past perfect for the earlier of two past actions: 'He had left before I arrived.'

After 'hardly', 'scarcely', or 'no sooner', use inversion: 'Hardly had he sat down when the phone rang.'

Collective nouns take a singular verb when acting as one unit and a plural verb when members act individually.

'Since' marks a point in time, 'for' a duration; 'besides' means in addition, 'beside' means next to.

Factual questions are answered directly in the passage; locate the keyword and read the surrounding two sentences.

'The author implies' questions require a logical step beyond the literal text but inside its evidence.

Purpose questions ask why a paragraph exists, answered by its function: to illustrate, to contrast, to define.

For odd-sentence-out items, check which sentence fails to share a subject, tense, or connector with the rest.

Generations: first-generation computers used vacuum tubes, later ones transistors, integrated circuits, and microprocessors.

The OS manages memory, processes, files, and devices; examples include Windows, Linux, and macOS.

Cyber hygiene: strong unique passwords, two-factor authentication, and caution with phishing emails that mimic legitimate senders.

Business structures: sole proprietorship, partnership, LLP, and company differ in liability, ownership transfer, and registration.

A statement that restates the question in different words adds no information and is insufficient.

When the question asks for a ratio, two statements giving the two absolute quantities separately can be sufficient together.

In age problems, a statement fixing the present ratio does not fix the actual ages without one absolute value.

If a statement produces two possible values, it is insufficient even if one seems more plausible.

Number of arrangements of n distinct items is n!; with repetition of a word's letters, divide by factorials of repeated letters.

Circular arrangements of n people are (n − 1)!; if clockwise and anticlockwise are the same, halve it.

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

Handshake problems among n people give n(n − 1)/2 handshakes.

If price rises by r%, consumption must fall by 100r/(100 + r) percent to keep expenditure unchanged.

Compound interest for two years equals simple interest plus interest on the first year's interest.

Difference between CI and SI for 2 years at r% is P(r/100)²; for 3 years it is P(r/100)²(3 + r/100).

A trader using false weights gains (true weight − false weight)/false weight × 100 percent.

Similar triangles have proportional sides and equal angles; the ratio of areas equals the square of the side ratio.

The centroid divides each median in the ratio 2 : 1 from the vertex.

Angle bisector theorem: the bisector of an angle of a triangle divides the opposite side in the ratio of the adjacent sides.

Area of an equilateral triangle of side a is (√3/4)a², and its altitude is (√3/2)a.

Approximate before computing: round large table values to estimate which option is near, then verify only that option.

For percentage-share questions, total the column first; the denominator is the column total, not the grand total, unless asked.

Ratio questions between two rows use the same column; cross-column ratios need explicit statement.

Check whether the chart shows absolute values or percentages; a percentage chart cannot give absolute totals without a base figure.

Prefixes: 'bene-' good, 'mal-' bad, 'ante-' before, 'post-' after, 'circum-' around, 'contra-' against.

Roots: 'cred' believe, 'dict' say, 'luc' light, 'path' feeling, 'chron' time, 'graph' write.

Suffix '-ous' means full of, '-ify' means to make, '-ism' denotes a practice or doctrine.

Phrasal verbs: 'give in' yield, 'give up' quit, 'put off' postpone, 'put up with' tolerate.

A topic sentence states the paragraph's claim; every following sentence must support, illustrate, or qualify it.

Transition words signal relationships: 'however' contrasts, 'therefore' concludes, 'for instance' illustrates, 'meanwhile' shifts time.

Parallel structure keeps listed items in the same grammatical form, improving readability and scoring.

Avoid sentence fragments and run-ons; join independent clauses with a semicolon or a conjunction.

The RBI sets India's repo rate and manages monetary policy; fiscal policy is the government's taxation and spending.

GST replaced multiple indirect taxes and is levied as CGST, SGST, and IGST on inter-state supply.

Primary market issues new securities; the secondary market trades existing ones through exchanges.

Balance of trade is exports minus imports of goods; a deficit means imports exceed exports.

Example: 'What is x?' with I: x² = 9 gives x = 3 or −3, insufficient; II: x > 0 alone insufficient; together x = 3, so both are needed.

Example: 'Is n even?' with I: n is divisible by 4 - always even, sufficient alone; note divisibility by 4 is stronger than needed.

A statement giving a unique answer even when the answer is 'no' is still sufficient.

Always ask whether the statement could yield a second consistent case; if yes, mark insufficient.

Odd days in a century: 100 years contain 5 odd days, 200 years 3, 300 years 1, 400 years 0.

To find the day of a date, add odd days of years elapsed since a known reference plus day-of-year offset.

A clock that gains uniformly: if it shows correct time at one instant, the gain accumulates at a fixed rate per hour.

Hands of a clock are at right angles 22 times in 12 hours and coincide 11 times in 12 hours.

If A is twice as efficient as B, A takes half the time of B, and their work rates are in ratio 2 : 1.

Pipes filling a tank: net rate is sum of inlet rates minus outlet rates; a negative net rate means the tank never fills.

If a train passes a pole in t seconds, its length equals speed × t; passing a platform adds the platform length.

Average speed for equal distances at speeds a and b is the harmonic mean 2ab/(a + b), not (a + b)/2.

If α and β are roots, the quadratic is x² − (α + β)x + αβ = 0.

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.

For an AP, the sum of terms equidistant from the ends is constant and equals the sum of the first and last terms.

Logarithm base change: log_b a = log a / log b, useful for comparing growth.

For an even number of ordered observations, the median is the average of the two middle values.

The mean is affected by extreme values; the median and mode are more robust to outliers.

In a symmetric distribution, mean, median, and mode coincide; in a right-skewed distribution, mean > median > mode.

Weighted mean = Σ(w × x)/Σw, used when observations carry different importance.

'One of my friends is' not 'are'; the relative clause after 'one of the' often takes a plural verb depending on the intended subject.

Do not use double comparatives: 'more better' is wrong; use 'better'.

'Amount' for uncountable, 'number' for countable; 'much' for uncountable, 'many' for countable.

Avoid redundant pairs such as 'return back', 'discuss about', and 'repeat again'.

Skim the questions before reading the passage so you know which details to mark while reading.

Underline names, dates, numbers, and contrast words; most factual questions target these.

For 'except' questions, three options are supported and one is not; find the odd one.

Do not let personal opinion override the passage's stated view in tone questions.

A compiler translates an entire program before execution; an interpreter translates and runs line by line.

The internet is a network of networks; the World Wide Web is a service of hyperlinked documents on it.

Databases store structured records; SQL is the standard query language for relational databases.

Cloud computing delivers storage and processing over the internet, billed on usage.

For 'what is the area of the triangle?' a statement giving base and height is sufficient; one giving only two sides is not, unless the included angle is known.

A statement that a quadrilateral is a rhombus fixes equal sides but not the angles; area still needs a diagonal or angle.

Two angles of a triangle fix the third, so a statement giving two angles is sufficient for the third angle.

Stating that two lines are parallel plus one transversal angle fixes all eight angles.

In ranking puzzles, 'A is 5th from the top and 8th from the bottom' gives total = 5 + 8 − 1 = 12.

To find a person's position from the other end, total − position + 1.

In blood relations, 'A is the son of B's sister' makes A the nephew of B, not the son.

'My father's only son' refers to the speaker if male; such phrases often collapse to 'me'.

Divisibility by 3 or 9 depends on digit sum; by 4 on the last two digits; by 8 on the last three digits; by 11 on alternating digit sums.

HCF × LCM of two numbers equals their product; HCF divides the difference of the numbers.

A number is prime if it has no factor up to its square root; 1 is neither prime nor composite.

Remainder theorem: the remainder of p(x) divided by (x − a) is p(a).

Cuboid: volume lbh, total surface 2(lb + bh + hl), diagonal √(l² + b² + h²).

Cube: volume a³, total surface 6a², diagonal a√3.

Hemisphere: curved surface 2πr², total surface 3πr², volume ⅔πr³.

When a solid is melted and recast, volume is conserved; equate initial and final volumes to find the new dimension.

Identify whether the table gives totals or subtotals; summing subtotals double-counts.

For 'by what percent did X change', use the earlier year as the denominator.

If a row total is missing, derive it from column totals before answering share questions.

Cross-check an option's order of magnitude; a mismatch signals a unit or decimal error.

'To turn a deaf ear' = refuse to listen; 'to take to task' = reprimand; 'to make both ends meet' = live within means.

'A white elephant' = costly but useless possession; 'a red-letter day' = an important day.

'To smell a rat' = suspect something wrong; 'to show the white flag' = surrender.

'In the nick of time' = just in time; 'at the eleventh hour' = at the last moment.

Pronouns such as 'he', 'this', and 'such' cannot open a paragraph because they need a prior referent.

Connectors like 'thus' and 'consequently' mark later sentences; 'first' and 'initially' mark early ones.

Chronological or cause-effect chains usually fix the order; identify the cause sentence before the effect sentence.

The final sentence often restates the idea or gives a conclusion and can be identified by summary words.

Assets = liabilities + equity is the accounting equation underlying the balance sheet.

Depreciation spreads the cost of a fixed asset over its useful life; it is a non-cash expense.

Working capital = current assets − current liabilities, measuring short-term financial health.

A company's market capitalisation = share price × number of shares outstanding.