IOE Entrance Nepal flashcards

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

Quadratic formula

For ax²+bx+c=0 with a≠0, the roots are x=(−b±√(b²−4ac))/(2a). The discriminant b²−4ac decides whether roots are real distinct, equal, or complex.

Sum and product of roots

For ax²+bx+c=0, the sum of roots equals −b/a and the product equals c/a. These let you build an equation from its roots.

AP nth term and sum

Arithmetic progression with first term a and common difference d: nth term is a+(n−1)d; sum of n terms is n/2 times (2a+(n−1)d).

Infinite geometric series

A geometric series with first term a and ratio r converges only when |r|<1, and its sum is a/(1−r). Otherwise the sum diverges to infinity.

Binomial general term

In (a+b)ⁿ the term containing b^k is C(n,k)·a^(n−k)·b^k. The expansion has n+1 terms and coefficients are symmetric.

Double angle identities

sin2θ=2sinθcosθ; cos2θ=cos²θ−sin²θ=1−2sin²θ; tan2θ=2tanθ/(1−tan²θ). These follow from the sum formulas with A=B=θ.

Chain rule

The derivative of f(g(x)) is f′(g(x))·g′(x). Differentiate the outer function, keep the inner unchanged, then multiply by the inner derivative.

Perpendicular line slopes

Two non-vertical lines are perpendicular exactly when the product of their slopes is −1, so one slope is the negative reciprocal of the other.

Equations of motion

For constant acceleration: v=u+at, s=ut+½at², and v²=u²+2as. Each links initial velocity u, final velocity v, acceleration a, time t, displacement s.

Projectile range and height

For launch speed u at angle θ on level ground: range R=u²sin2θ/g and maximum height H=u²sin²θ/(2g). Range is greatest at 45°.

Work–energy theorem

The net work done on a body equals its change in kinetic energy. Work by gravity is mgh; a spring stores elastic potential energy ½kx².

Circular motion acceleration

An object moving in a circle of radius r at speed v has centripetal acceleration v²/r, directed toward the centre. Angular speed ω relates by a=ω²r.

Escape velocity

The minimum speed to leave a planet's gravity without further propulsion is √(2GM/R), equal to √(2gR). For Earth it is about 11.2 km/s.

Simple pendulum period

A simple pendulum of length L has period T=2π√(L/g) for small swings. Period is independent of bob mass and depends on local gravity.

Carnot efficiency

The maximum efficiency of a heat engine between hot and cold reservoirs is 1−T_c/T_h, with both temperatures in kelvin. Real engines are always less efficient.

Resistors in parallel

For resistors in parallel, the reciprocal of total resistance equals the sum of reciprocals: 1/R=1/R₁+1/R₂. Total resistance is less than the smallest branch.

Mole and molar volume

One mole contains 6.022×10²³ particles and equals the molar mass in grams. At STP one mole of an ideal gas occupies about 22.4 litres.

Molarity versus molality

Molarity is moles of solute per litre of solution and varies with temperature. Molality is moles per kilogram of solvent and is temperature-independent.

pH scale

pH equals −log[H⁺]. At 25 °C, pH+pOH=14, so pH below 7 is acidic, 7 neutral, above 7 basic. Each unit is a tenfold change in [H⁺].

Ideal gas equation

PV=nRT relates pressure, volume, moles, and absolute temperature. Use kelvin; R is 0.0821 L·atm/(mol·K) or 8.314 J/(mol·K).

Periodic trend directions

Across a period left to right, atomic radius falls while electronegativity and ionisation energy rise. Down a group, radius rises and the other two fall.

Oxidation number rules

Free elements are 0; oxygen is usually −2; hydrogen is +1. Oxidation numbers in a neutral compound sum to zero, and in an ion to its charge.

Subject–verb agreement rule

A singular subject takes a singular verb and a plural subject a plural verb. With 'either…or' or 'neither…nor', the verb agrees with the nearer subject.

Present perfect tense

Formed with have or has plus the past participle. It describes an action completed at an unspecified time or one continuing relevance to the present.

Third conditional

Structure: if + past perfect, then would have + past participle. It describes an unreal past situation and its imaginary result.

Active to passive voice

Passive uses a form of 'be' plus the past participle. The active object becomes the subject; only transitive verbs can be made passive.

Prepositions of time

Use 'in' with months, years, and seasons; 'on' with days and dates; 'at' with clock times and with night or noon.

Fewer versus less

Use 'fewer' with countable plural nouns, as in fewer students. Use 'less' with uncountable nouns, as in less water or less time.

SI base units

Seven base units: metre, kilogram, second, ampere, kelvin, mole, and candela. All other SI units are derived from these by multiplication or division.

Derived SI units

Newton is kg·m/s²; joule is N·m; watt is J/s; pascal is N/m²; coulomb is A·s; volt is W/A; ohm is V/A; hertz is s⁻¹.

Metric prefix powers

kilo 10³, mega 10⁶, giga 10⁹, milli 10⁻³, micro 10⁻⁶, nano 10⁻⁹, pico 10⁻¹². Convert by moving the decimal point the matching number of places.

Contrapositive in logic

The contrapositive of 'if P then Q' is 'if not Q then not P', which is logically equivalent. The converse and inverse are not equivalent to the original.