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Tawjihiya Bahrain Core Formulas and Scientific Principles

Essential formulas, mathematical identities, physical laws, and chemical principles for the Bahrain Secondary School General Certificate examinations.

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Fundamental Theorem of Calculus (Evaluation Part)
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1
Fundamental Theorem of Calculus (Evaluation Part)
Integral from a to b of f(x) dx = F(b) - F(a), where F'(x) = f(x) on the closed interval [a, b].
2
Derivatives of Basic Trigonometric Functions
d/dx [sin(x)] = cos(x); d/dx [cos(x)] = -sin(x); d/dx [tan(x)] = sec^2(x).
3
Product and Quotient Rules for Differentiation
Product: (u v)' = u' v + u v'. Quotient: (u / v)' = (u' v - u * v') / v^2.
4
Chain Rule for Differentiating Composite Functions
d/dx [f(g(x))] = f'(g(x)) g'(x); or dy/dx = (dy/du) (du/dx).
5
Integration by Parts Formula
Integral of u dv = u * v - integral of v du, derived from the product rule of differentiation.
6
Area Under a Curve y = f(x) Above the X-Axis
Area = integral from a to b of f(x) dx, where f(x) >= 0 on the interval [a, b].
7
Volume of Revolution (Disk Method)
V = pi * integral from a to b of [f(x)]^2 dx for rotation about the x-axis.
8
Standard Equation of a Circle with Center (h, k)
(x - h)^2 + (y - k)^2 = r^2, where r is the radius and (h, k) are the coordinates of the center.
9
Dot Product of Two Vectors (Geometric and Algebraic)
u . v = |u| |v| cos(theta) = u_x v_x + u_y v_y + u_z * v_z.
10
Cross Product Magnitude of Two Vectors
|u x v| = |u| |v| sin(theta), giving the area of the parallelogram formed by vectors u and v.
11
Newton's Second Law of Motion
Sigma F = m * a = dp/dt; net external force equals mass times acceleration or rate of change of momentum.
12
Work-Kinetic Energy Theorem
W_net = Delta KE = 0.5 m v_f^2 - 0.5 m v_i^2; net work done on an object equals its change in kinetic energy.
13
Law of Conservation of Linear Momentum
In an isolated system with no net external force: Sigma p_initial = Sigma p_final.
14
Gravitational Potential Energy Near Earth's Surface
PE_grav = m g h, where m is mass, g is gravitational acceleration (9.8 m/s^2), and h is height.
15
Snell's Law of Optical Refraction
n_1 sin(theta_1) = n_2 sin(theta_2), where n is the refractive index and theta is the angle from the normal.
16
Photoelectric Effect Energy Equation (Einstein)
E_photon = h * f = Phi + KE_max, where h is Planck's constant, f is frequency, and Phi is the work function.
17
Coulomb's Law of Electrostatic Force
F = k_e |q_1 q_2| / r^2, where k_e is Coulomb's constant (8.99 x 10^9 N m^2 / C^2) and r is separation.
18
Ohm's Law and Electrical Power Formula
V = I R; Power P = V I = I^2 * R = V^2 / R, where V is voltage, I is current, and R is resistance.
19
Ideal Gas Law Equation of State
P V = n R * T, where P is pressure, V is volume, n is moles, R is universal gas constant, and T is Kelvin temperature.
20
Le Chatelier's Principle in Chemical Equilibrium
If an equilibrium system experiences a change in concentration, temperature, or pressure, it shifts to counteract the disturbance.
21
Arrhenius Definition of Acids and Bases
Acid: produces hydrogen ions (H+ or H3O+) in aqueous solution. Base: produces hydroxide ions (OH-) in aqueous solution.
22
Calculation of pH and pOH from Ion Concentrations
pH = -log10[H3O+]; pOH = -log10[OH-]; at 25 degrees Celsius: pH + pOH = 14.0.
23
Standard Enthalpy of Reaction Formula (Hess's Law Application)
Delta H_reaction = Sigma [n Delta H_f(products)] - Sigma [m Delta H_f(reactants)].
24
First-Order Radioactive Decay Integrated Rate Law
ln(N_t / N_0) = -k * t; half-life t_(1/2) = ln(2) / k = 0.693 / k.
25
Galvanic Electrochemical Cell Potential Calculation
E_cell = E_cathode - E_anode (reduction potentials); standard cell potential must be positive for a spontaneous reaction.

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