CUSAT CAT B.Tech flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Discriminant of a quadratic

For ax² + bx + c = 0 with a ≠ 0, D = b² − 4ac. D > 0 means two distinct real roots, D = 0 means equal real roots, and D < 0 means complex conjugate roots.

Sum and product of roots

For ax² + bx + c = 0, the sum of the roots equals −b/a and the product equals c/a. These hold even when the roots are complex, provided the coefficients are real.

General term of binomial expansion

In the expansion of (a + b)^n, the (r + 1)th term is nCr · a^(n−r) · b^r for r from 0 to n. The expansion contains n + 1 terms in total.

Standard limit sin x over x

As x approaches 0 with x measured in radians, sin x / x approaches 1. The same limit holds for tan x / x, and (1 − cos x)/x² approaches 1/2.

Derivative of arcsin x

d/dx (arcsin x) = 1/√(1 − x²) for |x| < 1. The derivative of arccos x is the negative of this, and d/dx (arctan x) = 1/(1 + x²).

Integration by parts formula

∫u dv = uv − ∫v du. Choose u using the ILATE order: inverse trigonometric, logarithmic, algebraic, trigonometric, exponential. This order usually leaves a simpler remaining integral.

Eccentricity of a conic

Eccentricity is the ratio of distance to focus over distance to directrix. Ellipse e = √(1 − b²/a²) is below 1, hyperbola e = √(1 + b²/a²) exceeds 1, and a parabola has e = 1.

Binomial distribution mean and variance

For n independent Bernoulli trials each with success probability p, the number of successes has mean np and variance npq, where q = 1 − p. Its standard deviation is the square root of npq.

Dot and cross product

a·b = |a||b|cos θ is a scalar. a × b = |a||b|sin θ n̂ is a vector perpendicular to both, and its magnitude equals the area of the parallelogram they span.

De Moivre's theorem

For any integer n, (cos θ + i sin θ)^n = cos nθ + i sin nθ. It converts powers of complex numbers into multiple-angle form and yields the n distinct nth roots.

Determinant and matrix inverse

A square matrix is invertible only when its determinant is nonzero; then A⁻¹ = adj(A)/det(A). Determinants multiply: det(AB) = det A · det B, and det Aᵀ = det A.

Scalar triple product

The scalar triple product a·(b × c) equals the signed volume of the parallelepiped with edges a, b, c. It is zero exactly when the three vectors are coplanar.

Projectile range at given speed

For launch speed u at angle θ on level ground, range R = u²sin2θ/g, which is largest at 45°. Maximum height is u²sin²θ/(2g) and time of flight is 2u sinθ/g.

Parallel axis theorem

I = I_cm + Md², where I_cm is the moment of inertia about an axis through the centre of mass, M is the total mass, and d is the perpendicular distance between the two parallel axes.

Period of a simple pendulum

T = 2π√(L/g) applies for small oscillations, roughly below 10 degrees. The period does not depend on the bob mass but lengthens where g is smaller, as at high altitude.

Carnot engine efficiency

Efficiency equals 1 − T_c/T_h with both reservoir temperatures in kelvin. No heat engine working between those two reservoirs can exceed this value, which reaches 1 only if T_c is 0 K.

First law of thermodynamics

ΔU = Q − W, where Q is heat added to the system and W is work done by the system. For an adiabatic process Q = 0, so ΔU = −W and the gas cools on expansion.

Lens maker's formula

For a thin lens in air, 1/f = (n − 1)(1/R₁ − 1/R₂), with radii signed by convention. Combined with 1/f = 1/v − 1/u it locates images for any object distance.

Young double slit fringe width

Fringe width β = λD/d, where D is the slit-to-screen distance and d the slit separation. Width grows with wavelength, so red fringes are wider than violet ones in the same setup.

Photoelectric equation

hν = φ + K_max, where φ is the metal's work function. Emission stops below threshold frequency ν₀ = φ/h regardless of intensity, while intensity changes only the saturation current.

de Broglie wavelength

λ = h/p = h/(mv) for a particle of momentum p. An electron accelerated through V volts has λ about 12.27/√V ångström, small enough to produce observable electron diffraction.

Bohr hydrogen energy levels

E_n = −13.6/n² eV for hydrogen. The ground-state ionisation energy is 13.6 eV, and the Lyman series consists of transitions ending at n = 1, lying in the ultraviolet.

Molarity versus molality

Molarity is moles of solute per litre of solution and changes with temperature as volume expands. Molality is moles of solute per kilogram of solvent and is unaffected by temperature.

Kp and Kc relation

Kp = Kc(RT)^Δn, where Δn is the moles of gaseous products minus gaseous reactants. The two constants are numerically equal whenever Δn equals zero for the reaction.

Gibbs energy criterion

ΔG = ΔH − TΔS. At constant temperature and pressure a process is spontaneous when ΔG is negative, non-spontaneous when positive, and at equilibrium when ΔG equals zero.

Huckel aromaticity rule

A planar, cyclic, fully conjugated system with 4n + 2 pi electrons is aromatic; the same geometry with 4n pi electrons is antiaromatic. Benzene has six, cyclobutadiene four.

Markovnikov's rule

When HX adds to an unsymmetrical alkene, hydrogen goes to the carbon already carrying more hydrogens, forming the more stable carbocation. Peroxides reverse this to give the anti-Markovnikov product.

SN1 versus SN2 mechanism

SN1 is a two-step, first-order process favoured by tertiary halides in polar protic solvents and gives racemisation. SN2 is one-step, second order, favoured by primary halides and inverts configuration.

Nernst equation at 298 K

E = E° − (0.0591/n)log Q, where n is the number of electrons transferred and Q the reaction quotient. It gives the actual cell emf when concentrations or pressures differ from standard.

Faraday's first law of electrolysis

Mass deposited m = ZIt, where Z is the electrochemical equivalent, I the current and t the time. One faraday, about 96500 coulombs, deposits one equivalent weight of any substance.

Crystal field splitting energy

In an octahedral field the d orbitals split into lower t2g and higher eg sets by Δo. CFSE = (−0.4 n_t2g + 0.6 n_eg)Δo, and high-spin configurations dominate when Δo is below the pairing energy.

VSEPR shapes from electron pairs

Four electron pairs give tetrahedral geometry at about 109.5°. One lone pair narrows this to a trigonal pyramidal shape near 107°, and two lone pairs give a bent shape near 104.5°.