Bihar Polytechnic cheat sheet

Newton's second law: F = ma; the net force equals mass times acceleration, and force is measured in newtons (kg·m/s²).

Equations of motion for uniform acceleration: v = u + at, s = ut + ½at², v² = u² + 2as.

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

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

Lens and mirror formula: 1/v − 1/u = 1/f for lenses and 1/v + 1/u = 1/f for mirrors; magnification m = v/u.

Snell's law: n₁ sin θ₁ = n₂ sin θ₂; refractive index n = c/v, and total internal reflection occurs beyond the critical angle sin θc = 1/n.

First law of thermodynamics: ΔQ = ΔU + ΔW; isothermal work by an ideal gas is W = nRT ln(V₂/V₁).

de Broglie relation λ = h/p links particle momentum to wavelength; photon energy E = hν = hc/λ.

Mole concept: 1 mole = 6.022 × 10²³ particles and equals molar mass in grams; molarity M = moles of solute / litres of solution.

Ideal gas equation PV = nRT with R = 8.314 J/(mol·K) or 0.0821 L·atm/(mol·K).

pH = −log[H⁺]; at 25 °C pH + pOH = 14, and a buffer follows the Henderson–Hasselbalch equation pH = pKa + log([A⁻]/[HA]).

Oxidation is loss of electrons and increase in oxidation number; reduction is gain of electrons and decrease in oxidation number.

Le Chatelier's principle: raising pressure favours the side with fewer gas moles; raising temperature favours the endothermic direction.

Hybridisation from steric number: 2 = sp (linear), 3 = sp² (trigonal planar), 4 = sp³ (tetrahedral).

Balancing redox in acidic medium: balance atoms, add H₂O to balance oxygen, add H⁺ to balance hydrogen, then balance charge with electrons.

Bond order = (bonding electrons − antibonding electrons)/2; higher bond order means shorter, stronger bonds.

Quadratic roots: for ax² + bx + c = 0, sum = −b/a and product = c/a; discriminant D = b² − 4ac decides real, equal, or complex roots.

Arithmetic progression: nth term aₙ = a + (n−1)d and sum Sₙ = n/2[2a + (n−1)d]; geometric sum Sₙ = a(rⁿ−1)/(r−1) for r ≠ 1.

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

Derivative rules: d/dx(xⁿ) = nxⁿ⁻¹, product rule (uv)' = u'v + uv', quotient rule (u/v)' = (u'v − uv')/v².

Definite integral property ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx; also ∫ f'(x)/f(x) dx = ln|f(x)| + C.

Straight line: slope m = (y₂ − y₁)/(x₂ − x₁); point-slope form y − y₁ = m(x − x₁); perpendicular slopes satisfy m₁m₂ = −1.

Circle with centre (h, k) and radius r: (x − h)² + (y − k)² = r²; general form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f).

Probability of an event P(E) = favourable outcomes / total equally likely outcomes, with 0 ≤ P(E) ≤ 1 and P(E') = 1 − P(E).

Percentage change = (new − old)/old × 100; successive changes of a% and b% give a net change of a + b + ab/100 percent.

Simple interest SI = PRT/100; compound interest amount A = P(1 + R/100)ⁿ, so CI = A − P.

Speed, time, distance: distance = speed × time; for equal distances at speeds x and y, average speed = 2xy/(x + y).

Work: if A finishes a job in x days and B in y days, together they take xy/(x + y) days.

Ratio and mixture: alligation rule gives ratio of quantities as (mean − cheaper) : (dearer − mean).

Permutation and combination: nPr = n!/(n−r)! and nCr = n!/[r!(n−r)!], with nCr = nC(n−r).

Number series and coding-decoding questions: check differences, ratios, or squares/cubes first; in letter coding, note the fixed shift in alphabet position.

Logical reasoning: in syllogisms, a conclusion follows only if true in every possible Venn diagram of the premises; converse of a statement is not automatically valid.