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Thanawiya Amma Core Formulas and Scientific Principles

24 flashcards to revise the key facts for the Thanawiya Amma. Study material written for practice, not the official exam.

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Binomial General Term Formula (nCr)
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1
Binomial General Term Formula (nCr)
T_(r+1) = nCr a^(n-r) b^r in the expansion of (a + b)^n.
2
Sum and Product of Roots of Quadratic ax^2 + bx + c = 0
Sum = -b / a, and Product = c / a (Vieta's formulas).
3
Derivative of Exponential Function e^(g(x))
d/dx [e^(g(x))] = g'(x) * e^(g(x)).
4
Derivative of Natural Logarithm ln(g(x))
d/dx [ln(g(x))] = g'(x) / g(x) for g(x) > 0.
5
Euler's Complex Exponential Form
z = r e^(i theta) = r (cos theta + i sin theta), where r is modulus and theta is argument.
6
Complex Cube Roots of Unity (omega)
1 + omega + omega^2 = 0, and omega^3 = 1.
7
Vector Dot Product (Scalar Product)
A . B = |A| |B| cos(theta) = A_xB_x + A_yB_y + A_z*B_z. If A . B = 0, vectors are orthogonal.
8
Vector Cross Product Magnitude
|A x B| = |A| |B| sin(theta), representing the area of the parallelogram spanned by A and B.
9
Equation of a Plane in 3D Space
A(x - x0) + B(y - y0) + C(z - z0) = 0, where n = (A, B, C) is the normal vector.
10
Standard Equation of a Sphere in 3D
(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2 with center (a, b, c) and radius r.
11
Integration by Parts Formula
integral(u dv) = u * v - integral(v du).
12
Volume of Solid of Revolution (Disk Method)
V = pi * integral_a^b [f(x)]^2 dx for revolution about the x-axis.
13
Faraday's Law of Electromagnetic Induction
EMF = -N * (delta Phi / delta t), where negative sign denotes Lenz's law.
14
Transformer Voltage and Turns Ratio
Vs / Vp = Ns / Np = Ip / Is (for ideal 100% efficient transformer).
15
de Broglie Wavelength Equation
lambda = h / p = h / (m * v), where h is Planck's constant (6.626 x 10^-34 J s).
16
Einstein's Photoelectric Equation
K_max = h f - W = h (f - f_0), where W is the work function and f_0 is threshold frequency.
17
Resonant Frequency in Series RLC Circuit
f_0 = 1 / (2 pi sqrt(L * C)), occurring when X_L = X_C and impedance Z = R.
18
Faraday's Law of Electrolysis (Charge to Mass)
m = (M I t) / (z * F), where F is Faraday's constant (approx 96,500 C/mol).
19
pH and pOH Relationship at 25 C
pH = -log10[H+], pOH = -log10[OH-], and pH + pOH = 14 (Kw = 1.0 x 10^-14).
20
Le Chatelier's Principle (Exothermic Reaction)
Increasing temperature shifts equilibrium in the reverse (endothermic) direction, reducing product yield and Kc.
21
Haber-Bosch Ammonia Synthesis Reaction
N2(g) + 3H2(g) <=> 2NH3(g) + Heat, catalyzed by finely divided iron at 450-500 C and 200 atm.
22
Contact Process for Sulfuric Acid Catalyst
Vanadium pentoxide (V2O5) catalyzes 2SO2 + O2 <=> 2SO3 at 450 C.
23
Markovnikov's Rule in Organic Addition
In addition of HX to an unsymmetrical alkene, hydrogen bonds to the carbon with more hydrogen atoms.
24
Tautomerism in Ethyne Catalytic Hydration
Ethyne + H2O (H2SO4/HgSO4) yields vinyl alcohol (unstable enol), which tautomerizes into acetaldehyde (ethanal).

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