Kankor 14-Pass cheat sheet

Quadratic formula: for ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) / 2a; the discriminant b² − 4ac has two real roots if positive, one if zero, none if negative.

Pythagorean theorem: in a right triangle with legs a and b and hypotenuse c, a² + b² = c².

Area formulas: triangle = ½ × base × height; circle = πr²; trapezoid = ½(a + b)h.

Sum of interior angles of an n-sided polygon = (n − 2) × 180°; each interior angle of a regular n-gon = (n − 2) × 180° / n.

Arithmetic sequence: nth term aₙ = a₁ + (n − 1)d; sum Sₙ = n(a₁ + aₙ)/2. Geometric: aₙ = a₁r^(n−1).

sin²θ + cos²θ = 1, and tanθ = sinθ/cosθ; in a right triangle sinθ = opposite/hypotenuse.

log(ab) = log a + log b; log(a/b) = log a − log b; log(aⁿ) = n log a.

Probability of an event = favorable outcomes / total equally likely outcomes; P(not A) = 1 − P(A).

Volume: cylinder = πr²h; sphere = (4/3)πr³; cone = (1/3)πr²h.

Newton's second law: F = ma; weight W = mg, with g ≈ 9.8 m/s² near Earth's surface.

Speed = distance/time; acceleration = change in velocity/time; for free fall from rest, v = gt and d = ½gt².

Ohm's law: V = IR; electric power P = VI = I²R = V²/R.

Density = mass/volume; pressure = force/area; pressure in a liquid = ρgh.

pH < 7 is acidic, pH = 7 neutral, pH > 7 basic; each unit is a tenfold change in hydrogen-ion concentration.

Avogadro's number = 6.02 × 10²³ particles per mole; molar mass equals mass of one mole in grams.

Balanced chemical equations conserve atoms of each element on both sides; combustion of a hydrocarbon yields CO₂ and H₂O.

Photosynthesis: 6CO₂ + 6H₂O + light → C₆H₁₂O₆ + 6O₂, occurring in chloroplasts using chlorophyll.

Cell respiration releases energy from glucose; DNA carries genetic information and replicates before cell division.

Afghanistan lies in South-Central Asia, landlocked, bordered by Iran, Pakistan, Turkmenistan, Uzbekistan, Tajikistan, and China (via the Wakhan Corridor).

The Hindu Kush mountain range runs northeast to southwest across Afghanistan; the Amu Darya forms part of the northern border.

Afghanistan's independence in foreign affairs is dated to 1919 under King Amanullah Khan after the Third Anglo-Afghan War.

The Durand Line (1893) marks the boundary between Afghanistan and British India, now the Pakistan border.

Kabul is the capital; major rivers include the Helmand, Kabul, Amu Darya, and Hari Rud.

In civics, the three branches of government are legislative (makes laws), executive (implements laws), and judicial (interprets laws).

A constitution is the supreme law of a state; it defines state structure and citizens' fundamental rights and duties.

Latitude lines run east–west and measure distance north or south of the equator; longitude lines run north–south through the poles.

The Five Pillars of Islam: Shahadah (testimony), Salah (prayer), Zakah (almsgiving), Sawm (fasting in Ramadan), and Hajj (pilgrimage to Makkah).

The Qur'an has 114 surahs; the first surah is Al-Fatihah and the longest is Al-Baqarah.

Muslims perform five obligatory daily prayers: Fajr, Zuhr, Asr, Maghrib, and Isha.

Zakah is generally 2.5% of qualifying wealth held for one lunar year above the nisab threshold.

Hajj is performed in the month of Dhu al-Hijjah, with key rites at Arafah, Muzdalifah, and Mina, and circumambulation (tawaf) of the Ka'bah.

The Hijrah, the Prophet Muhammad's migration from Makkah to Madinah, occurred in 622 CE and marks year 1 of the Islamic calendar.

Fasting (Sawm) in Ramadan lasts from dawn (Fajr) to sunset (Maghrib), abstaining from food, drink, and other invalidating acts.

Dari is written in a modified Arabic script, read right to left; it uses additional letters such as پ, چ, ژ, گ.

Pashto is also written in a modified Arabic script and adds retroflex letters such as ټ, ډ, ړ, ږ, ښ, ڼ.

Dari grammar marks the definite direct object with the suffix را (rā); nouns have singular and plural forms.

Pashto nouns have two main cases (direct and oblique) and grammatical gender (masculine and feminine).

In both languages the verb generally agrees with the subject in person and number; Pashto also shows ergative agreement in past transitive clauses.

Classical Dari literature includes works by Rumi, Saadi, Hafiz, and Jami; the Ghaznavid and Timurid courts patronized Persian literature.

Khushal Khan Khattak and Rahman Baba are central classical Pashto poets; Rahman Baba is known for mystical, lyrical verse.

The Bayazid Roshan and later Pashto writers contributed to a distinct Pashto literary tradition alongside Dari's Persianate canon.

Percent change = (new − old)/old × 100%; simple interest I = P × r × t.

The mean of a data set = sum of values / number of values; the median is the middle value when ordered.

Slope of a line through (x₁,y₁) and (x₂,y₂) = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes.

Distance between two points = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Binomial expansion: (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; a² − b² = (a − b)(a + b).

Circumference of a circle = 2πr; arc length = (θ/360°) × 2πr for central angle θ.

nPr = n!/(n − r)! and nCr = n!/(r!(n − r)!) for permutations and combinations.

Kinetic energy KE = ½mv²; gravitational potential energy PE = mgh; work = force × distance in the direction of force.

Newton's third law: every action has an equal and opposite reaction; momentum p = mv is conserved in collisions.

The three common states of matter are solid, liquid, and gas; sublimation is solid-to-gas directly, and condensation is gas-to-liquid.

An atom contains protons and neutrons in the nucleus and electrons in shells; atomic number = number of protons.

Covalent bonds share electrons between nonmetals; ionic bonds transfer electrons between metals and nonmetals.

Blood is pumped by the heart through arteries away from the heart and veins toward it; capillaries allow exchange of gases and nutrients.

Human blood groups follow ABO and Rh systems; the universal red-cell donor is O-negative and the universal recipient is AB-positive.

Enzymes are biological catalysts that speed reactions by lowering activation energy and are usually specific to their substrates.

The Earth rotates on its axis once about every 24 hours, causing day and night; it revolves around the Sun in about 365.25 days, causing seasons.

The Tropic of Cancer (23.5° N) and Tropic of Capricorn (23.5° S) mark the limits of the tropics.

The Ring of Fire is a zone of frequent earthquakes and volcanoes around the Pacific Ocean.

The Silk Road passed through Afghanistan, linking China with Persia, Arabia, and Europe via cities such as Balkh and Herat.

A democracy typically features free elections, rule of law, separation of powers, and protection of minority rights.

Taxes are compulsory payments to the state used to fund public goods and services; a budget balances revenue and expenditure.

The United Nations was founded in 1945; its principal organs include the General Assembly and Security Council.

Climate differs from weather: climate is the long-term average pattern of a region, while weather is short-term atmospheric conditions.

The Qur'an was revealed to the Prophet Muhammad over about 23 years; the first revealed verses are the opening of Surah Al-Alaq.

Surah Al-Fatihah has seven verses and is recited in every unit (rak'ah) of the daily prayers.

The two main sources of Islamic law are the Qur'an and the Sunnah (sayings and practices of the Prophet).

Eid al-Fitr follows Ramadan (1 Shawwal); Eid al-Adha falls on 10 Dhu al-Hijjah and commemorates Prophet Ibrahim's willingness to sacrifice.

Wudu (ablution) is required before prayer and is invalidated by acts such as using the toilet or deep sleep.

The Qiblah is the direction of the Ka'bah in Makkah, toward which Muslims face during prayer.

The Night of Power (Laylat al-Qadr) falls in the last ten nights of Ramadan and is considered the holiest night.

Dari and Pashto both use the ezafe construction (e.g., Dari کتابِ خوب, 'a good book') to link a noun to a following modifier.

In Dari, the present perfect is formed with the past participle plus the auxiliary, e.g., رفته است ('has gone').

Pashto verbs distinguish present and past stems; the past stem often differs from the present stem, e.g., ځي (goes) vs. لاړ (went).

Metaphor (استعاره) and simile (تشبیه) are core figures of speech in both traditions; a simile uses an explicit comparison word like مانند in Dari.

The ghazal and masnavi are major poetic forms in Dari; the landay (a two-line folk couplet) is a distinctive Pashto form.

Proverbs (مثل in Dari, متل in Pashto) carry moral or practical lessons and are common in both oral traditions.

Correct spelling and the use of the correct script letters (e.g., ی vs. ې in Pashto) are tested in language sections.

Solve linear equations by isolating the variable: for 3x + 5 = 20, x = 5.

Rules of exponents: aᵐ × aⁿ = a^(m+n); aᵐ / aⁿ = a^(m−n); (aᵐ)ⁿ = a^(mn); a⁰ = 1 for a ≠ 0.

The sum of angles in a triangle is 180°; in an isosceles triangle the base angles are equal.

Area of a parallelogram = base × height; area of a rhombus = ½ × d₁ × d₂.

For a right triangle, tanθ = opposite/adjacent; cosθ = adjacent/hypotenuse.

A function f(x) assigns exactly one output to each input; the domain is the set of valid inputs.

The derivative of xⁿ is n·x^(n−1); the derivative of a constant is 0.

Frequency × wavelength = wave speed (v = fλ); sound travels faster in solids than in gases.

Light travels at about 3 × 10⁸ m/s in a vacuum; reflection angle equals incidence angle.

Heat transfer occurs by conduction (solids), convection (fluids), and radiation (no medium needed).

The periodic table is arranged by increasing atomic number; elements in the same group have similar chemical properties.

Acids react with bases in neutralization to form salt and water: HCl + NaOH → NaCl + H₂O.

The human body has 206 bones in adults; the skeleton supports the body, protects organs, and enables movement.

The circulatory and respiratory systems cooperate: lungs oxygenate blood, and the heart circulates it to tissues.

Mitosis produces two genetically identical daughter cells; meiosis produces four gametes with half the chromosome number.

Afghanistan's population is ethnically diverse, including Pashtun, Tajik, Hazara, Uzbek, and others; Dari and Pashto are the official languages.

The Ahmad Shah Durrani empire (founded 1747) is often cited as the beginning of modern Afghan statehood.

The Anglo-Afghan Wars (1839–42, 1878–80, 1919) shaped Afghanistan's modern borders and independence.

The Earth's atmosphere is mostly nitrogen (~78%) and oxygen (~21%), with trace gases such as argon and carbon dioxide.

A map scale converts map distance to real distance; a large-scale map shows a small area in greater detail.

The three types of government power are legislative, executive, and judicial; checks and balances prevent concentration of power.

Human rights include civil, political, economic, social, and cultural rights; the Universal Declaration was adopted in 1948.

The Prophet Muhammad was born in Makkah around 570 CE and received the first revelation at about age 40 in the cave of Hira.

The migration (Hijrah) to Madinah in 622 CE established the first Muslim community and the Islamic calendar.

The Battle of Badr (2 AH / 624 CE) was the first major battle between the Muslims and the Quraysh.

The Farewell Pilgrimage and Sermon of the Prophet occurred in 10 AH, emphasizing equality and the sanctity of life and property.

The first four caliphs (Rashidun) were Abu Bakr, Umar, Uthman, and Ali.

The Qur'an is divided into 30 juz' and 114 surahs; surahs revealed in Makkah are generally shorter and emphasize faith.

Sadaqah is voluntary charity distinct from the obligatory Zakah.

In Dari, the past continuous is formed with the auxiliary 'داشت' plus the past participle, e.g., می‌رفت ('was going').

Pashto word order is generally subject–object–verb, similar to Dari, with the verb placed at the end of the clause.

Dari uses the suffix ها for plural nouns and the suffix های for adjectives in the ezafe construction.

The Pashto alphabet has about 45 letters, including retroflex consonants and the letters ښ and ږ.

Literary devices in both languages include tashbih (simile), isti'ara (metaphor), and jinās (wordplay).

Rumi's Masnavi and Saadi's Gulistan and Bustan are foundational texts of Dari/Persian literature studied in schools.

Pashto folk literature includes the landay (two-line couplet) and the tappa, often expressing love, grief, or social themes.

Solve quadratic equations by factoring: x² − 5x + 6 = 0 factors as (x − 2)(x − 3) = 0, giving x = 2 or 3.

The sum of the roots of ax² + bx + c = 0 is −b/a; the product is c/a.

The area of a sector = (θ/360°) × πr² for central angle θ in degrees.

The volume of a rectangular prism = length × width × height; surface area of a cube = 6s².

If two triangles are similar, corresponding sides are proportional and corresponding angles are equal.

The equation of a circle centered at (h, k) with radius r is (x − h)² + (y − k)² = r².

The mean of a frequency distribution = Σ(f × x) / Σf.