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EJU Mathematics, Physics and Chemistry Formulae

Flashcards with the key formulas for the EJU mathematics, physics and chemistry papers: quadratics, trigonometry, counting, calculus, mechanics, waves, electricity, gases and chemical calculations. Original study material.

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Vertex form of a quadratic

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Vertex form of a quadratic

y = a (x - p)^{2} + q

has its vertex at

(p, q)

Axis of symmetry of

y = a x^{2} + b x + c

x = - \frac{b}{2 a}

Discriminant of

a x^{2} + b x + c = 0

D = b^{2} - 4 a c

: two real roots if

D > 0

, one if

D = 0

, none if

D < 0

Sum and product of the roots

α + β = - \frac{b}{a}

α β = \frac{c}{a}

Law of cosines

a^{2} = b^{2} + c^{2} - 2 b c cos A

Law of sines

\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C} = 2 R

Area of a triangle from two sides and the included angle

\frac{1}{2} ab sin C

Permutations and combinations

_{n} P_{r} = \frac{n !}{( n - r )!}

_{n} C_{r} = \frac{n !}{r ! ( n - r )!}

Inclusion-exclusion for two sets

n (A \cup B) = n (A) + n (B) - n (A \cap B)

Variance of data

\frac{1}{n} \sum x^{2} - \overset{x}{ˉ}^{2}

Derivative of

x^{n}

and integral of

x^{n}

(x^{n})^{'} = n x^{n - 1}

;

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C

Double angle formulas

sin 2 θ = 2 sin θ cos θ

;

cos 2 θ = cos^{2} θ - sin^{2} θ = 1 - 2 sin^{2} θ

Logarithm laws

lo g_{a} M N = lo g_{a} M + lo g_{a} N

;

lo g_{a} M^{k} = k lo g_{a} M

;

lo g_{a} b = \frac{l o g _{c} b}{l o g _{c} a}

Sum of arithmetic and geometric series

S_{n} = \frac{n}{2} {2 a + (n - 1) d}

;

S_{n} = \frac{a ( r ^{n} - 1 )}{r - 1}

Kinematics with constant acceleration

v = v_{0} + a t

x = v_{0} t + \frac{1}{2} a t^{2}

v^{2} - v_{0}^{2} = 2 a x

Centripetal force

F = \frac{m v ^{2}}{r} = m r ω^{2}

Period of a spring-mass system

T = 2 π \frac{m}{k}

Doppler effect (moving source, still observer)

f^{'} = \frac{V}{V - v _{s}} f

for an approaching source.

Young's double-slit fringe spacing

Δ x = \frac{L λ}{d}

Energy stored in a capacitor

U = \frac{1}{2} C V^{2} = \frac{1}{2} Q V

Force on a current-carrying wire

F = B I L sin θ

First law of thermodynamics

Δ U = Q + W_{on gas} = Q - W_{by gas}

Ideal gas equation

p V = n R T

with

R = 8.3 \times 1 0^{3}

Pa L/(mol K).

Photon energy

E = h f = \frac{h c}{λ}

Molality and freezing point depression

Δ T = K_{f} m

, with

m

in mol per kg of solvent.

Faraday's law of electrolysis

Moles of electrons

= \frac{I t}{F}

with

F = 96500

C/mol.

Equilibrium constant for

a A + b B ⇌ c C

K_{c} = \frac{[ C ] ^{c}}{[ A ] ^{a} [ B ] ^{b}}

pH of a weak acid (small dissociation)

[H^{+}] = K_{a} C

Enthalpy change from bond energies

Δ H = \sum (bonds broken) - \sum (bonds formed)

Dilution and mixing

Dilution:

c_{1} V_{1} = c_{2} V_{2}

. Mixing:

c = \frac{c _{1} V _{1} + c _{2} V _{2}}{V _{1} + V _{2}}

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