Afghanistan Grade 12 Graduation cheat sheet

Quadratic formula: for ax^2 + bx + c = 0 with a ≠ 0, the roots are x = (-b ± √(b^2 - 4ac)) / (2a).

Discriminant rule: b^2 - 4ac > 0 gives two distinct real roots, = 0 gives one repeated root, < 0 gives no real roots.

Arithmetic sequence: nth term a_n = a_1 + (n - 1)d; sum S_n = n(a_1 + a_n)/2.

Geometric sequence: nth term a_n = a_1 · r^(n-1); sum S_n = a_1(1 - r^n)/(1 - r) for r ≠ 1.

Pythagorean theorem: in a right triangle with legs a, b and hypotenuse c, a^2 + b^2 = c^2.

Sine rule: a/sin A = b/sin B = c/sin C; cosine rule: c^2 = a^2 + b^2 - 2ab·cos C.

Derivative power rule: d/dx (x^n) = n·x^(n-1); derivative of a constant is 0.

Logarithm rules: log(xy) = log x + log y; log(x/y) = log x - log y; log(x^n) = n·log x.

Circle measures: area = πr^2, circumference = 2πr; volume of a sphere = (4/3)πr^3.

Dari is written right to left in a modified Arabic script; the alphabet has 32 letters, with short vowels usually unwritten.

The ezafe (-e) links a noun to a following modifier: ketāb-e sabz = 'the green book'.

Sentence order is subject–object–verb: 'Ali the book read' = Ali ketāb rā khānd.

The object marker rā follows a definite direct object and does not appear after indefinite objects.

Classical poetic forms: ghazal (rhymed couplets with a radif), qasida (panegyric), masnavi (rhyming couplets), ruba'i (quatrain).

Bayt = two hemistichs (mesra'); a ghazal typically runs 5–15 bayts and ends with the poet's takhallos (pen name).

Persian meters follow the 'aruz system of quantitative syllables, not stress-based rhythm.

Verb prefixes mark aspect and negation: mi- marks progressive/habitual, be- marks subjunctive or imperative, na- marks negation.

Pashto is written right to left in a modified Arabic script; its alphabet has about 45 letters, adding retroflex and additional consonants.

Pashto nouns have two genders (masculine, feminine) and inflect for two main cases (direct and oblique) plus plural.

Word order is subject–object–verb; adjectives normally follow the noun they modify.

Verbs agree with the subject in person, number, and gender (gender agreement appears in past tenses).

Ergative-like split: in past transitive constructions the verb may agree with the object rather than the subject.

The two main dialect groups are soft (southern, e.g. Kandahar) and hard (northern/eastern, e.g. Peshawar), differing in pronunciation of certain consonants.

Classical Pashto poetry uses the ghazal form; Khushal Khan Khattak and Rahman Baba are central figures of the tradition.

Landay is a folk couplet form: two lines of roughly nine and thirteen syllables, often on love or war.

Five pillars: shahada (testimony), salat (prayer), zakat (almsgiving), sawm (fasting Ramadan), hajj (pilgrimage).

The Qur'an contains 114 suras; the longest is al-Baqara and the shortest is al-Kawthar.

Salat is performed five times daily: fajr, zuhr, asr, maghrib, isha.

Zakat is normally 2.5 percent of qualifying wealth held for one lunar year (nisab applies).

Hajj rites occur in Dhu al-Hijja and include ihram, tawaf of the Ka'ba, sa'i, and standing at Arafat on 9 Dhu al-Hijja.

Wudu (ablution) is required before salat; tayammum with clean earth substitutes when water is unavailable or harmful.

The two main branches of Islam are Sunni and Shi'a; they differ chiefly on succession to leadership after the Prophet.

Sources of Islamic law: Qur'an, Sunna (hadith), ijma (consensus), and qiyas (analogical reasoning).

Newton's second law: F = ma; weight is W = mg, with g ≈ 9.8 m/s^2 near Earth's surface.

Kinematics: v = u + at; s = ut + (1/2)at^2; v^2 = u^2 + 2as.

Ohm's law: V = IR; electric power P = VI = I^2R = V^2/R.

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

Mole concept: one mole contains Avogadro's number (6.02 × 10^23) of particles; molar mass is expressed in g/mol.

Acids release H+ and bases release OH- in water; pH = -log[H+], with pH 7 neutral at 25 °C.

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

Cell respiration releases energy from glucose; aerobic respiration yields far more ATP than anaerobic.

DNA bases pair A–T and G–C; the human somatic cell has 46 chromosomes (23 pairs).

Afghanistan is landlocked, bordered by Iran, Pakistan, Turkmenistan, Uzbekistan, Tajikistan, and China (via the Wakhan Corridor).

The Hindu Kush range runs southwest–northeast and divides the country into northern and southern drainage zones.

Major rivers include the Amu Darya (northern border), Helmand, Kabul, and Harirud; most flow to inland basins or neighboring countries.

The Amu Darya forms most of Afghanistan's northern boundary with Central Asia.

Kabul has been the capital of modern Afghanistan; the country's modern borders were largely fixed by 19th-century agreements.

Afghanistan's climate is mostly arid to semi-arid continental, with cold winters and hot summers and large diurnal temperature ranges.

Traditional economy rests on agriculture and pastoralism; wheat, fruits, and nuts are key crops, and irrigation relies on karez channels.

The Durrani Empire (from 1747, Ahmad Shah Durrani) is conventionally treated as the foundation of the modern Afghan state.

Present perfect: have/has + past participle, used for past actions with present relevance or unspecified time.

Passive voice: be + past participle; the object becomes the subject (e.g., 'The letter was written by Ali').

Conditionals: first (if + present, will + verb), second (if + past, would + verb), third (if + past perfect, would have + participle).

Comparative and superlative: short adjectives take -er/-est; longer ones take more/most.

Reported speech shifts tense back one step: 'I am tired' becomes 'He said he was tired'.

Relative clauses: use who for people, which for things, where for places.

Countable nouns take many/few; uncountable nouns take much/little.

Prepositions of time: in for months/years, on for days/dates, at for clock times.

Citizenship carries both rights (expression, education, fair trial) and duties (obeying law, paying taxes, defending the country).

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

Rule of law means all persons and institutions, including the state, are accountable to publicly known laws.

Separation of powers prevents concentration of authority by assigning distinct functions to each branch.

Human rights are universal, inalienable, and indivisible; the Universal Declaration of Human Rights was adopted in 1948.

National symbols include the flag, national anthem, and national language(s); Pashto and Dari are the official languages of Afghanistan.

Nawruz (Nowruz), marking the spring equinox around 21 March, is a widely celebrated cultural new year.

Attan is a traditional Afghan circle dance performed at weddings and festivals, often to drum rhythms.

Local governance in Afghanistan has historically included shuras and jirgas, councils that resolve disputes by consensus.