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Saudi Secondary Certificate Grade 12 Core Competencies

Essential formulas, digital technologies, physical principles, and analytical competencies from the Saudi Ministry of Education Grade 12 Specialized Tracks (STEM, Computer Science, and General).

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Limit Definition of the Derivative
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1
Limit Definition of the Derivative
f'(x) = lim (h -> 0) [f(x + h) - f(x)] / h, representing the instantaneous rate of change of a differentiable function f(x).
2
Chain Rule for Composite Functions
d/dx [f(g(x))] = f'(g(x)) * g'(x); the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
3
Fundamental Theorem of Calculus (Part 2)
If f is continuous on [a, b] and F is an antiderivative of f, then integral from a to b of f(x) dx = F(b) - F(a).
4
De Moivre's Theorem for Complex Numbers
[r (cos(theta) + i sin(theta))]^n = r^n (cos(n theta) + i sin(n theta)), where n is any integer.
5
Standard Dot Product and Angle Between Vectors
u . v = |u| |v| cos(theta) = u_x v_x + u_y v_y + u_z * v_z; vectors u and v are orthogonal if and only if u . v = 0.
6
Electromagnetic Wave Speed Equation
c = 1 / sqrt(mu_0 epsilon_0) = f lambda, where c is the speed of light in vacuum (approx 3.00 x 10^8 m/s), f is frequency, and lambda is wavelength.
7
Special Relativity Time Dilation Formula
Delta t = Delta t_0 / sqrt(1 - v^2/c^2) = gamma * Delta t_0, showing that time dilates (runs slower) for an observer in motion relative to a proper time observer.
8
de Broglie Wavelength of Matter Waves
lambda = h / p = h / (m * v), where h is Planck's constant and p is linear momentum, illustrating wave-particle duality.
9
First Law of Thermodynamics
Delta U = Q - W (or Delta U = Q + W_on), stating that change in internal energy equals heat added to the system minus work done by the system.
10
Nuclear Binding Energy and Mass Defect
E = Delta m c^2, where mass defect Delta m = [Z m_p + (A - Z) * m_n] - m_nucleus; energy required to disassemble a nucleus into constituent nucleons.
11
Radioactive Decay Law and Half-Life
N(t) = N_0 e^(-lambda t), and t_1/2 = ln(2) / lambda approx 0.693 / lambda, where lambda is the decay constant.
12
Standard Cell Potential of Electrochemical Cells
E_cell_std = E_reduction_std (cathode) - E_reduction_std (anode); a positive E_cell indicates a thermodynamically spontaneous reaction under standard conditions (Delta G < 0).
13
Collision Theory for Reaction Rates
Chemical reactions occur only when reacting particles collide with kinetic energy equal to or greater than the activation energy (E_a) and with proper spatial orientation.
14
Functional Group of Esters and Saponification
Esters contain the -COO- linkage; saponification is the base-catalyzed hydrolysis of esters (triglycerides) using NaOH/KOH to produce glycerol and fatty acid soap salts.
15
Time Complexity of Binary Search vs Linear Search
Binary search on a sorted array operates in O(log n) logarithmic time complexity; linear search requires O(n) linear comparisons in worst and average cases.
16
Stack vs Queue Data Structures
A Stack operates on Last-In, First-Out (LIFO) order with push/pop operations; a Queue operates on First-In, First-Out (FIFO) order with enqueue/dequeue operations.
17
Relational Database Primary Key and Foreign Key
A Primary Key uniquely identifies each tuple/row in a relation with no null values; a Foreign Key references the primary key of another table to maintain referential integrity.
18
SQL Structured Query Language Core Clauses
SELECT ... FROM ... WHERE ... GROUP BY ... HAVING ... ORDER BY; HAVING filters grouped aggregates, while WHERE filters individual rows before grouping.
19
Symmetric vs Asymmetric Encryption
Symmetric encryption (AES) uses a single shared secret key for encryption and decryption; Asymmetric encryption (RSA, ECC) uses a mathematically linked public/private key pair.
20
TCP/IP vs OSI Reference Model Layers
OSI has 7 layers: Physical, Data Link, Network, Transport, Session, Presentation, Application. TCP/IP condenses these into 4: Network Access, Internet, Transport, Application.
21
IPv4 vs IPv6 Addressing Capacity
IPv4 uses 32-bit addresses providing 2^32 (approx 4.3 billion) unique addresses; IPv6 uses 128-bit addresses providing 2^128 (approx 3.4 x 10^38) unique addresses.
22
Supervised vs Unsupervised Machine Learning
Supervised learning trains models on labeled input-output pairs (classification, regression); Unsupervised learning detects hidden patterns or groupings in unlabeled datasets (clustering, PCA).
23
Polymorphism and Inheritance in Object-Oriented Programming
Inheritance allows a subclass to derive attributes and methods from a superclass; Polymorphism allows different classes to implement methods of the same name with class-specific behaviors.
24
Normal Distribution (68-95-99.7 Empirical Rule)
In a Gaussian distribution, approximately 68.27% of observations lie within 1 standard deviation of the mean, 95.45% within 2 standard deviations, and 99.73% within 3 standard deviations.

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