NIMCET cheat sheet

Quadratic roots: sum = -b/a, product = c/a; discriminant D = b² - 4ac gives real and equal roots when D = 0, distinct real when D > 0, complex conjugates when D < 0.

Arithmetic mean ≥ geometric mean ≥ harmonic mean for positive numbers, with equality only when all numbers are equal; AM of a and b is (a+b)/2, GM is √(ab), HM is 2ab/(a+b).

Sum formulas: 1+2+…+n = n(n+1)/2; 1²+2²+…+n² = n(n+1)(2n+1)/6; 1³+2³+…+n³ = [n(n+1)/2]².

For an AP with first term a and common difference d: nth term = a + (n-1)d, sum of n terms = (n/2)[2a + (n-1)d]. For a GP: nth term = ar^(n-1), sum = a(rⁿ-1)/(r-1) for r ≠ 1.

Combinations and permutations: nPr = n!/(n-r)!, nCr = n!/[r!(n-r)!], nCr = nC(n-r), and nC0 + nC1 + … + nCn = 2ⁿ.

Binomial expansion: (x+y)ⁿ = Σ nCr x^(n-r) y^r; the general term is nCr x^(n-r) y^r, and the middle term exists only when n is even.

Probability: P(A∪B) = P(A) + P(B) - P(A∩B); for independent events P(A∩B) = P(A)P(B); conditional probability P(A|B) = P(A∩B)/P(B).

Coordinate geometry: distance = √[(x₂-x₁)² + (y₂-y₁)²]; slope m = (y₂-y₁)/(x₂-x₁); lines are parallel when m₁ = m₂ and perpendicular when m₁m₂ = -1.

Calculus: d/dx(xⁿ) = nx^(n-1), d/dx(sin x) = cos x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x; a function is increasing where its derivative is positive and decreasing where it is negative.

In a categorical syllogism, a conclusion follows only if it is forced by the premises; a term distributed in the conclusion must be distributed in the premise that contains it.

Contrapositive is logically equivalent to the original statement: "if P then Q" equals "if not Q then not P"; the converse ("if Q then P") is not equivalent.

For seating and ranking puzzles, fix the most constrained person or position first, then propagate each clue; eliminate any arrangement that contradicts a stated clue.

In blood-relation puzzles, draw a tree with vertical links for parent-child and horizontal links for siblings and spouses; never infer gender from a name alone.

Direction sense: track net displacement as vector components; two successive right turns equal a 180° reversal, and a left turn followed by a right turn returns the original heading.

Coding-decoding: check letter-position shifts (A=1 … Z=26) first, then reversal, then alternating or positional patterns; verify the rule on every given word before applying it.

In series completion, test differences, ratios, alternating sub-series, and squares or cubes of natural numbers; a single rule must explain all given terms.

Clock and calendar: the minute hand gains 5.5° per minute on the hour hand; hands coincide every 65 5/11 minutes, and an ordinary year has one odd day while a leap year has two.

Data interpretation: compute percentage change as (new - old)/old × 100, and read the axis scale and units before comparing any two bars or slices.

Binary to decimal: multiply each bit by 2 raised to its position index from the right, starting at 0; decimal to binary uses repeated division by 2, reading remainders bottom to top.

One byte equals 8 bits; storage units ascend as KB (2¹⁰ bytes), MB (2²⁰), GB (2³⁰), and TB (2⁴⁰) bytes.

Logic gates: AND outputs 1 only if all inputs are 1; OR outputs 1 if any input is 1; XOR outputs 1 when inputs differ; NAND and NOR are universal gates.

CPU components: the ALU performs arithmetic and logic operations, the control unit fetches and decodes instructions, and registers hold operands and intermediate results.

Memory hierarchy from fastest and smallest to slowest and largest: registers, cache, main memory (RAM), then secondary storage such as SSD and magnetic disk.

RAM is volatile and loses its contents on power-off; ROM is non-volatile and stores firmware such as the BIOS or boot loader.

In C, array indices start at 0, so an array of n elements has valid indices 0 to n-1; an index equal to n is out of bounds.

OSI reference model has seven layers, from physical and data link up through network, transport, session, presentation, and application; TCP is connection-oriented while UDP is connectionless.

HTML uses tags to mark up structure, CSS controls presentation, and JavaScript adds client-side behavior; a URL's scheme, host, and path identify how, where, and what to fetch.

A subject and its verb must agree in number; intervening phrases do not change the subject, so "the list of items is" is correct, not "are".

Collective nouns take a singular verb when acting as one unit and a plural verb when members act individually; "the team is winning" versus "the team are arguing".

Indefinite pronouns such as each, everyone, somebody, and neither take singular verbs; "none" may be singular or plural depending on the sense.

Use the comparative for two items and the superlative for three or more; "between" pairs with two and "among" with more than two.

Passive voice is formed with a form of "be" plus the past participle; the object of the active sentence becomes the subject of the passive one.

Reported speech shifts tense back one step, so present becomes past and past becomes past perfect; time words shift too, as "now" becomes "then" and "today" becomes "that day".

Idiomatic preposition pairs to memorise: "comply with", "capable of", "averse to", "insist on", "deprived of", "superior to", "prefer to".

In reading comprehension, the main idea is the claim the whole passage supports; a detail that appears once is usually a supporting fact, not the theme.

Commonly confused pairs: "affect" (verb, to influence) versus "effect" (noun, result); "principal" (main or head) versus "principle" (rule or belief).