TG POLYCET cheat sheet

Quadratic formula: roots of ax² + bx + c = 0 are x = [−b ± √(b² − 4ac)] / 2a; the discriminant b² − 4ac decides whether roots are real and distinct (>0), equal (=0), or non-real (<0).

Sum and product of roots: α + β = −b/a and αβ = c/a; a quadratic with given roots is x² − (α+β)x + αβ = 0.

Arithmetic progression: nth term a + (n−1)d; sum of n terms = (n/2)[2a + (n−1)d] = (n/2)(first + last).

Geometric progression: nth term ar^(n−1); sum of n terms = a(rⁿ − 1)/(r − 1) for r ≠ 1; infinite sum = a/(1 − r) only when |r| < 1.

Distance between points = √[(x₂−x₁)² + (y₂−y₁)²]; section formula for point dividing AB in ratio m:n is ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).

sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ; sin(90° − θ) = cos θ, tan(90° − θ) = cot θ.

Standard values: sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, cos 30° = √3/2, tan 45° = 1.

Volume formulas: cylinder πr²h, cone (1/3)πr²h, sphere (4/3)πr³, hemisphere (2/3)πr³; curved surface of cone = πrl with l = √(r² + h²).

Probability of an event = favourable outcomes / total equally likely outcomes; P(E) + P(not E) = 1, so 0 ≤ P(E) ≤ 1.

Equations of motion (uniform acceleration): v = u + at, s = ut + ½at², v² = u² + 2as; for free fall take a = g ≈ 9.8 m/s².

Newton's second law: F = ma; momentum p = mv is conserved in collisions with no external force, and impulse = change in momentum = F·t.

Work W = Fs cos θ; power P = W/t; kinetic energy = ½mv²; gravitational potential energy near Earth = mgh; 1 kWh = 3.6 × 10⁶ J.

Ohm's law V = IR; series resistance R = R₁ + R₂ + …, parallel 1/R = 1/R₁ + 1/R₂ + …; electrical power P = VI = I²R = V²/R.

Reflection: angle of incidence equals angle of reflection, both measured from the normal; a plane mirror forms a virtual, erect, laterally inverted image of equal size at equal distance.

Snell's law n₁ sin i = n₂ sin r; refractive index n = c/v = real depth / apparent depth; light bends toward the normal when entering a denser medium.

Lens and mirror formula 1/v − 1/u = 1/f (lens) and 1/v + 1/u = 1/f (mirror); power of a lens P = 1/f (in metres), unit dioptre.

Wave relation v = fλ; sound needs a material medium and travels fastest in solids; audible range is about 20 Hz to 20 kHz.

Mole concept: 1 mole = 6.022 × 10²³ particles = molar mass in grams; at STP one mole of an ideal gas occupies 22.4 L.

Molarity = moles of solute / volume of solution in litres; mass percent = (mass of solute / mass of solution) × 100.

pH = −log[H⁺]; pH < 7 acidic, pH = 7 neutral, pH > 7 basic; each unit change means a tenfold change in [H⁺].

Oxidation is loss of electrons (increase in oxidation number); reduction is gain of electrons; in a redox reaction the oxidising agent itself gets reduced.

Periodic trends across a period: atomic radius decreases, ionisation energy and electronegativity increase; down a group the reverse occurs.

Ionic bonds form by electron transfer between a metal and a non-metal; covalent bonds form by sharing between non-metals; metals bond via a delocalised electron sea.

Balancing rule: atoms of each element and total charge must be equal on both sides; in ionic equations spectator ions are cancelled.

Common gas tests: CO₂ turns lime water milky; O₂ relights a glowing splint; H₂ burns with a pop; moist blue litmus turning red indicates an acidic gas such as HCl.

Photosynthesis equation: 6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂ in the presence of chlorophyll and light; light reaction occurs in the grana, the Calvin cycle in the stroma.

Cellular respiration: C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O + energy; glycolysis occurs in the cytoplasm, the Krebs cycle and electron transport in mitochondria.

Human blood flow: right atrium → right ventricle → lungs → left atrium → left ventricle → aorta; arteries carry blood away from the heart, veins toward it.

Mendelian ratios: monohybrid F₂ phenotype ratio 3:1 and genotype ratio 1:2:1; dihybrid F₂ phenotype ratio 9:3:3:1.

DNA structure: double helix of two antiparallel strands; adenine pairs with thymine (two hydrogen bonds), guanine with cytosine (three).

Human digestive order: mouth → oesophagus → stomach → small intestine → large intestine; most absorption occurs in the small intestine via villi.

Nephron function: filtration in the glomerulus, selective reabsorption of glucose, amino acids and most water in the tubules; urine leaves via the ureter to the bladder.

Plant tissue roles: xylem transports water and minerals upward, phloem translocates food; stomata, guarded by guard cells, control gas exchange and transpiration.

Cell organelles: mitochondria release energy, ribosomes synthesise protein, chloroplasts carry out photosynthesis, lysosomes digest waste; the nucleus contains DNA.

Prokaryotic cells lack a membrane-bound nucleus and organelles; eukaryotic cells have a true nucleus and membrane-bound organelles.

Mitosis produces two genetically identical diploid cells for growth and repair; meiosis produces four haploid gametes with genetic variation.

Blood groups follow ABO alleles: I^A and I^B are codominant, i is recessive; parents A (I^A i) and B (I^B i) can have an O (ii) child.

Human karyotype has 46 chromosomes (23 pairs); sex determination: XX female, XY male, with the father's sperm determining the child's sex.

Enzymes are protein catalysts with an active site; activity is highest at optimum temperature and pH, and denatures (loses shape) at high temperature.

Hormones: insulin from the pancreas lowers blood glucose, glucagon raises it; thyroxine from the thyroid controls metabolic rate.

Ecological terms: producers make food, consumers eat others, decomposers break down dead matter; energy flow between trophic levels is roughly 10% per level.