VITEEE cheat sheet

For ax² + bx + c = 0, the roots are x = [−b ± √(b² − 4ac)] / (2a); the discriminant D = b² − 4ac gives D > 0 two real roots, D = 0 equal roots, D < 0 complex conjugate roots.

Standard derivatives: d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x; the product rule is (uv)′ = u′v + uv′.

Integration by parts: ∫u dv = uv − ∫v du. The LIATE order (logarithmic, inverse trig, algebraic, trig, exponential) indicates a suitable choice of u.

Probability addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For independent events, P(A ∩ B) = P(A)P(B); conditional probability is P(A|B) = P(A ∩ B)/P(B).

For a 2×2 matrix with entries a, b, c, d, the determinant is ad − bc; the inverse exists only when ad − bc ≠ 0 and equals (1/(ad − bc)) times [[d, −b], [−c, a]].

Mendel's dihybrid cross of two heterozygotes gives a 9:3:3:1 phenotype ratio when genes assort independently; a monohybrid cross gives 3:1, and a test cross of a heterozygote gives 1:1.

In photosynthesis, light reactions occur in thylakoid membranes and produce ATP, NADPH, and oxygen from water; the Calvin cycle occurs in the stroma and fixes CO₂ using RuBisCO.

In the nephron, most glucose, amino acids, and sodium are reabsorbed in the proximal convoluted tubule; the loop of Henle builds a medullary osmotic gradient used to concentrate urine.

Mitochondria carry out oxidative phosphorylation and produce most cellular ATP; ribosomes synthesise proteins; lysosomes contain hydrolytic enzymes that digest macromolecules and worn-out organelles.

For uniformly accelerated motion: v = u + at, s = ut + ½at², and v² = u² + 2as, where u is initial velocity, v final velocity, a acceleration, and s displacement.

Newton's second law gives F = ma; limiting friction is f = μN, where μ is the coefficient of friction and N the normal reaction.

Work-energy theorem: net work done equals change in kinetic energy, W = ΔKE = ½mv² − ½mu²; power is P = W/t = Fv for constant force.

For SHM, a spring-mass system has T = 2π√(m/k); a simple pendulum has T = 2π√(l/g) for small oscillations, independent of bob mass.

Coulomb's law: F = kq₁q₂/r² with k ≈ 9 × 10⁹ N·m²/C². A parallel-plate capacitor has C = ε₀A/d, and energy stored is ½CV².

Ohm's law: V = IR. Resistors in series add (R = R₁ + R₂ + …); in parallel, 1/R = 1/R₁ + 1/R₂ + …; electrical power is P = VI = I²R.

Magnetic force on a moving charge is F = qvB sin θ; on a straight current-carrying wire it is F = BIL sin θ, with direction given by Fleming's left-hand rule.

Thin lens formula: 1/f = 1/v − 1/u using Cartesian sign convention; Snell's law is n₁ sin θ₁ = n₂ sin θ₂, and total internal reflection requires travel from denser to rarer medium beyond the critical angle.

Photoelectric effect: Kmax = hf − φ. de Broglie relation: λ = h/p; for an electron accelerated through V volts, λ ≈ 1.227/√V nanometres.

Mole concept: n = m/M; one mole contains 6.022 × 10²³ particles, and one mole of an ideal gas occupies 22.4 L at STP.

Periodic trends: across a period, atomic radius decreases and ionisation energy increases; down a group, atomic radius increases and ionisation energy decreases.

Equilibrium relation: Kp = Kc(RT)^Δn, where Δn is the change in moles of gas. Le Chatelier's principle predicts the direction of shift after a disturbance.

Nernst equation at 298 K: E = E° − (0.0591/n) log Q, where n is the number of electrons transferred and Q is the reaction quotient.

Gibbs free energy: ΔG = ΔH − TΔS. A process is spontaneous when ΔG < 0; at equilibrium ΔG = 0.

pH = −log₁₀[H⁺]. A buffer of weak acid and its salt follows the Henderson–Hasselbalch equation: pH = pKa + log([salt]/[acid]).

Markovnikov's rule: addition of HX to an unsymmetrical alkene places H on the carbon with more hydrogens and X on the more substituted carbon.

First-order kinetics: t½ = 0.693/k, independent of initial concentration; the integrated form is ln([A]₀/[A]) = kt.

Aldehyde tests: Tollens' reagent (ammoniacal AgNO₃) gives a silver mirror, and Fehling's solution gives a red precipitate of Cu₂O with aldehydes but not ketones.

Successive percentage changes of a% and b% give a net change of a + b + ab/100 percent; a 20% rise followed by a 10% fall gives 1.2 × 0.9 = 1.08, an 8% net rise.

If a : b = c : d, then ad = bc. To divide a quantity Q in the ratio a : b : c, the shares are Qa/(a+b+c), Qb/(a+b+c), and Qc/(a+b+c).

Average = sum of observations ÷ number of observations. Adding a constant k to every value raises the average by k; multiplying every value by k multiplies the average by k.

Speed = distance ÷ time. For opposite-direction motion, relative speed is the sum of speeds; for same-direction motion, it is the difference.

Number series: check differences first for arithmetic patterns, ratios for geometric patterns, and alternating or prime-square patterns when simple differences fail.

Letter coding: A = 1 through Z = 26; the reverse position is 27 minus the forward position. Many codes shift each letter by a fixed number of places.

Direction sense: track net displacement using perpendicular components; the shortest distance from the start is found by Pythagoras when movements are at right angles.

Syllogism: draw Venn diagrams for the statements. A conclusion is valid only if it must be true in every diagram consistent with the premises.

Blood relations: decode each symbol before combining relations; for example, father's sister is aunt, mother's brother is uncle, and brother's son is nephew.

Subject-verb agreement: a singular subject takes a singular verb and a plural subject takes a plural verb. Phrases such as 'along with' or 'as well as' do not change the verb number.

Present perfect (has/have + past participle) is used with 'since' for a starting point and 'for' for a duration: She has lived here since 2019.

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

Articles: 'a' precedes consonant sounds, 'an' precedes vowel sounds, and 'the' is used for specific or previously mentioned nouns.

Active to passive: move the object to subject position, use the correct form of 'be' plus the past participle, and add 'by' plus the original subject when needed.

First conditional: if + present simple, will + base verb, for real future possibilities. Second conditional: if + past simple, would + base verb, for unreal present situations.

Synonyms are words with similar meanings and antonyms have opposite meanings; the correct choice must fit the sentence's context and grammatical form.

Common idioms: 'a blessing in disguise' means an apparent misfortune that turns out beneficial; 'to beat around the bush' means to avoid the main point.

Modals: 'can' indicates ability, 'may' permission or possibility, 'must' strong obligation, and 'should' advice; modal verbs are followed by the base form.