LOWKEY SMART

To expand brackets, multiply the outside by EACH term inside: 3(x + 4) = 3x + 12. Two minuses make a plus: 6 - (-3) = 9.

This one is about speed, not method — answer what you know instantly and come back to the rest. It is testing quick multiplication and division facts, percentage change (find 10% first, then build up, and add on or take off) and squares and roots.

Square rooting undoes squaring, so √121 = 11. HCF is the biggest number that divides into both (18 and 24 share 6); LCM is the smallest they both divide into (6 and 8 give 24). Prime factors are found by splitting until every branch is prime: 18 = 2 × 3 × 3.

Every living thing is built from CELLS, and a cell is the smallest thing that counts as alive. Inside one: the NUCLEUS holds the DNA and controls what the cell does, the MITOCHONDRIA release energy by respiration, and a plant cell also has a VACUOLE full of sap that pushes outwards and keeps the plant rigid, plus chloroplasts for photosynthesis. A cell doing a particular job is SPECIALISED, and its shape always follows that job: a red blood cell throws away its nucleus altogether to make room for more haemoglobin, so it carries more oxygen; a sperm cell is packed with mitochondria for the energy to swim; a nerve cell is long and thin to carry signals; a root hair cell is drawn out into a spike to soak up water. Every cell in your body carries the SAME DNA, so they end up different only because different genes are switched on in each. Cells build up in a fixed order: cell, then TISSUE (similar cells together), then ORGAN, then ORGAN SYSTEM. Anything that swaps substances with its surroundings wants a large SURFACE AREA TO VOLUME RATIO, because that means more surface for each unit of inside, so exchange is faster. That is why your lungs are millions of tiny sacs rather than two balloons.

Adding and subtracting: move right for +, left for -, and two minuses together turn into a plus, so 5 - (-4) = 9. Multiplying and dividing: SAME signs give a positive (-12 ÷ -4 = 3), DIFFERENT signs give a negative (-5 × 3 = -15).

Everything is made of ATOMS. An ELEMENT is one type of atom only (oxygen, iron, sodium), and there are about 118 of them, all listed on the periodic table. A COMPOUND is two or more elements CHEMICALLY JOINED, so it can only be split up by a chemical reaction, never by filtering. Water is a compound, not an element, no matter how pure it is. A MIXTURE is substances jumbled together but NOT joined, so it can be separated physically. Air is a mixture. A formula counts the atoms in one molecule: CO2 is one carbon plus two oxygens, so three atoms in total. The big idea is that a compound has COMPLETELY DIFFERENT properties from the elements in it. Sodium is a soft metal that explodes in water and chlorine is a poisonous green gas, yet sodium chloride is the salt you put on chips. To separate a mixture, pick the method that uses the difference between the parts: dissolve the salt and FILTER off the sand, then EVAPORATE the water to get the salt back; CHROMATOGRAPHY to pull apart the coloured dyes in an ink; DISTILLATION to get pure water back out of a solution.

Common denominator to add (2/3 + 1/4 = 8/12 + 3/12 = 11/12); straight across to multiply (3/4 × 2/5 = 3/10); and to DIVIDE, flip the second fraction and multiply (3/4 ÷ 1/8 = 3/4 × 8/1 = 6). Decimal to percentage: × 100, so 0.2 = 20%.

Energy is never created or destroyed, only TRANSFERRED from one store to another. That is the law of conservation of energy. The stores: KINETIC (movement), THERMAL (hot things), CHEMICAL (food, fuel, a battery), GRAVITATIONAL POTENTIAL (height, so it rises as you climb the stairs) and ELASTIC POTENTIAL (a stretched band). Energy transferred by an appliance is worked out from its power and how long it runs: E = P x t, with power in watts, time in SECONDS and energy in joules. A 60 W lamp for 5 minutes is 60 x 300 = 18 000 J. Not all of that does the job you wanted. EFFICIENCY = useful energy out / total energy in, x 100 for a percentage. A lamp that is 20% efficient turns 20% of 60 J into light, which is 12 J; the other 48 J is DISSIPATED, spreading out into the surroundings as thermal energy where it is too spread out to be useful. That is why an old filament lamp is worse than an LED (far more of the energy goes to heat) and why a bouncing ball rises lower each time (energy goes to heating and sound, not that it has vanished).

Find 10% by dividing by 10 and build from there: 35% of 200 is 20 + 20 + 20 + 10 = 70. Working BACKWARDS, scale up to 100%: if 10% = 6 then the whole is 60, and if 20% = 14 then 1% is 0.7 so the whole is 70.

SPEED = distance / time, in metres per second, so 150 m in 12 s is 12.5 m/s. Watch the units first: 2 km in 5 minutes is 2000 m in 300 s, which is 6.7 m/s to 1 dp. Rearranged, distance = speed x time, so 9 m/s for 11 s is 99 m. Force is measured in NEWTONS (N). When several forces act, add the ones pointing the same way and subtract the opposing ones to get the RESULTANT force: 20 N right against 12 N left is 8 N to the right. If the resultant is zero the forces are BALANCED and the motion does not change, which is why a car at a steady speed still has balanced forces on it. An unbalanced force changes speed or direction. MASS is how much matter there is, measured in kilograms, and it is the same everywhere. WEIGHT is the force of gravity pulling on that mass, measured in newtons: weight = mass x g, and on Earth g = 10 N/kg, so a 55 kg person weighs 550 N. A parachute works by hugely increasing air resistance, so the resultant force now acts UPWARDS and the skydiver slows down.

Simplify by dividing both sides by the same number (24:36 both ÷12 = 2:3). To share, add the parts to get the number of shares, divide the total by that, then multiply out: £60 in 2:3 is 5 shares of £12 = £24 and £36. For pricing, find the cost of ONE first.

Everything is made of tiny particles, and the state tells you how they are arranged. In a SOLID they are packed in a fixed pattern, held by strong forces, and can only vibrate on the spot, which is why a solid keeps its shape. In a LIQUID they are still touching but can slide past each other, so a liquid flows but cannot be squashed. In a GAS they are far apart and moving fast in all directions, so a gas can be squashed easily (there is empty space to squash) and it fills its container. Heating gives particles energy: melting, then evaporating or boiling; cooling gives condensing and freezing, and a gas can go straight to a solid without a liquid stage. Changes of state are PHYSICAL, so no mass is lost when a liquid evaporates, the same particles are simply spread out. Water at 0 C has more energy than ice at 0 C, even though the temperature reads the same. Gas particles COLLIDE with the walls of their container, and those collisions are what pressure is, so heating a sealed container raises the pressure. DIFFUSION is particles spreading from where they are crowded to where they are not: perfume across a room, and oxygen from the air in your lungs into your blood. It is faster when warm, because the particles move faster.

Collect like terms (5a + 3a - 2a = 6a). To expand, multiply the outside by EVERY term inside, keeping the signs: 2(3x - 5) = 6x - 10. To solve, undo in reverse and do the same to both sides: 2x + 3 = 11 becomes 2x = 8, so x = 4.

Every question here takes more than one step, so write the steps down. Read it twice, decide what is being asked, then work through. It pulls together percentage change (find 10% first, then build up, and add on or take off), time (count on in whole hours first, then the minutes), quick multiplication and division facts and ratio (add the parts to get the shares, divide, then multiply).

The pH scale runs from 0 to 14. Below 7 is ACIDIC, 7 is NEUTRAL, above 7 is ALKALINE. Universal indicator shows it by colour: red and orange for acid, green for neutral, blue and purple for alkali. An acid is a substance that releases H+ (hydrogen) ions in water, and the more of them it releases the STRONGER the acid and the lower the pH. A weak acid at pH 5 is still an acid, it just releases fewer H+ ions. The scale is not a straight count: each step of 1 is a factor of TEN, so pH 3 is 100 times more acidic than pH 5. The reactions worth knowing: acid + alkali gives SALT + WATER, acid + base gives SALT + WATER, and acid + metal gives SALT + HYDROGEN. All of those first two are NEUTRALISATION, which is what an indigestion tablet does to stomach acid, and what lime does to soil that is too acidic (sulfur would make it MORE acidic). To make a neutral salt solution, add alkali until the indicator reads pH 7, then evaporate off the water. Your body cares about this too: stomach acid sits around pH 2, strongly acidic, while blood is held at about pH 7.4, very slightly alkaline, and it has to stay there.

Standard form is a number between 1 and 10 times a power of ten. A BIG number has a positive power (45000 = 4.5 × 10⁴); a small one has a negative power (0.0032 = 3.2 × 10⁻³). Multiplying powers of the same base adds the indices: x⁴ × x³ = x⁷.

PHOTOSYNTHESIS happens in the CHLOROPLASTS of a plant cell and uses light energy: carbon dioxide + water -> glucose + oxygen. It takes carbon dioxide IN and gives oxygen OUT. RESPIRATION is the opposite trade and happens in the MITOCHONDRIA of EVERY living cell, plant and animal, every moment of the day. Aerobic respiration uses oxygen: glucose + oxygen -> carbon dioxide + water, releasing energy. Plants respire as well as photosynthesise, because they need that energy too; at dawn there is a moment when the two rates match exactly and there is no net gas exchange at all. When muscles cannot get oxygen fast enough during hard exercise they respire ANAEROBICALLY: glucose -> lactic acid. It releases far LESS energy per glucose molecule, and the lactic acid builds up and makes muscles ache and fatigue. So the two differences to hold on to are: photosynthesis stores energy while respiration releases it, and photosynthesis happens only in light in cells with chloroplasts while respiration happens all the time in every cell.

For a linear sequence the common difference is the number in front of n: 7, 10, 13, 16 goes up 3, so start with 3n and adjust by comparing — 3n + 4. If the SECOND differences are constant instead, it is quadratic and built around n².

A wave transfers ENERGY, not matter: the water moves up and down while the wave travels across. In a TRANSVERSE wave the vibration is at right angles to the direction of travel (light, and all electromagnetic waves). In a LONGITUDINAL wave the vibration is along the direction of travel, as a squash and stretch (sound). Light can cross a VACUUM, which is why sunlight reaches us. Sound cannot, because it needs particles to pass the vibration on, which is why astronauts cannot hear each other shout. The wave equation is wave speed = frequency x wavelength, with frequency in HERTZ (waves per second): 10 Hz x 2 m = 20 m/s, and 50 Hz x 4 m = 200 m/s. If the speed is fixed, frequency and wavelength trade off, so raising the frequency SHORTENS the wavelength; if the wavelength doubles and the frequency stays put, the speed doubles. Higher frequency sound is heard as a higher PITCH. Light travels about a million times faster than sound, so you see the lightning long before you hear the thunder. An ECHO is simply sound reflecting off a hard surface and coming back to you.

In a right-angled triangle a² + b² = c², where c is the hypotenuse — the longest side, opposite the right angle. Square the two short sides, add them, then square root: 5 and 12 give 25 + 144 = 169, so c = 13. Learn 3-4-5, 5-12-13 and 8-15-17 by heart.

A current in a wire makes a magnetic field, and around a single straight wire that field is a set of CONCENTRIC CIRCLES. Wind the wire into a coil around an iron core and you have an ELECTROMAGNET. Three ways to make it stronger: more CURRENT, more TURNS on the coil, and an IRON CORE. Iron is used rather than steel because soft iron magnetises strongly and then LOSES its magnetism the instant the current stops, whereas steel would stay magnetic and the crane could never drop the car. A wooden core does nothing at all. The advantage over a permanent magnet is control: it switches on and off, its strength can be turned up, and REVERSING THE CURRENT reverses its poles. It does need a steady direct current, because an alternating one would keep flipping the poles. A permanent magnet needs no power source but can do none of that. A compass is just a tiny magnet, so its NORTH pole swings to point at the SOUTH pole of a nearby magnet, and near a strong electromagnet it will be pulled right round. Uses: scrapyard cranes, loudspeakers, electric motors, relays and circuit breakers.

Probability = wanted ÷ total. MULTIPLY along branches for events one after another: heads then heads is 1/2 × 1/2 = 1/4. For NOT something, subtract from 1 — P(not a six) = 1 - 1/6 = 5/6.

The periodic table is arranged so that a GROUP (a column) holds elements with the same number of electrons in their outer shell, which is why they behave alike. Group 1 has 1 outer electron, Group 2 has 2, Group 7 has 7, Group 0 has a full shell. Going DOWN any group, each element has one more electron shell, so the atom is BIGGER and the outer electron is further from the nucleus and more shielded from its pull. Now the two trends make sense, and they run in OPPOSITE directions. Group 1 metals REACT BY LOSING their outer electron, and the further out it is the easier it goes, so reactivity INCREASES down the group: potassium beats sodium beats lithium, and francium at the bottom would be fiercest of all. Group 7 non-metals react by GAINING an electron, and a bigger atom grabs one less easily, so reactivity DECREASES down the group: fluorine beats chlorine beats iodine. Group 0 already has a full outer shell, so it has no reason to react at all. Group 1 metals are stored under oil to keep air and water away, and dropping potassium into water gives hydrogen, a lilac flame and a lot of fizzing.

Same triangle shape each time: speed = distance ÷ time, density = mass ÷ volume, pressure = force ÷ area. Check the units before dividing, and carry them into the answer — km/h, g/cm³, N/m².

No energy transfer is perfect. EFFICIENCY = useful energy transferred / total energy supplied, x 100 to make it a percentage. A motor given 500 J that usefully transfers 350 J is 350/500 = 70% efficient. A lamp given 50 J that wastes 20 J as heat has 30 J useful, so it is 60%. A power station at 40% efficiency turns 1000 MJ of fuel into 400 MJ of electricity. Whatever is wasted is not destroyed, it is DISSIPATED into the surroundings, almost always as thermal energy, which is why a charger feels warm. Nothing can be 100% efficient. A more efficient device wastes less for the same useful output, so it costs less to run. RENEWABLE resources (wind, solar, hydroelectric, tidal, geothermal, biofuel) are replaced as fast as they are used and have no fuel cost and no carbon dioxide from burning, but the setup cost is high and wind and solar are unreliable. NON-RENEWABLE resources (coal, oil, gas, nuclear fuel) will run out, and burning fossil fuels releases carbon dioxide. Insulation saves energy in a house because it slows the rate at which thermal energy escapes, so the heating has to replace less of it.

A triangle is 180° and a quadrilateral is 360°. In general, interior angles sum to (n - 2) × 180°, so a pentagon is 540° and a hexagon is 720°. Exterior angles always sum to 360°, whatever the shape.

DNA carries the genetic information. It is coiled into CHROMOSOMES inside the NUCLEUS of the cell, and a GENE is a short section of DNA that codes for one characteristic. A human body cell has 23 PAIRS of chromosomes, 46 in total, one of each pair from each parent. Because they come in pairs, so do the genes, and the two versions of the same gene are called ALLELES. A DOMINANT allele (capital letter) shows its effect whenever it is present; a RECESSIVE allele (small letter) only shows when BOTH alleles are recessive. So with brown eyes B dominant and blue b recessive, Bb has brown eyes and only bb is blue. GENOTYPE is which alleles you carry (Bb); PHENOTYPE is what you actually see (brown eyes). Two brown-eyed parents can have a blue-eyed child only if both are Bb, and a Bb x Bb cross expects BB, Bb, Bb, bb, so one quarter blue. Identical twins come from one fertilised egg, so their DNA is identical. Sex is decided by one pair: XX is female, XY is male, so it is the sperm, carrying either an X or a Y, that decides it.

This one is about speed, not method — answer what you know instantly and come back to the rest. It is testing quick multiplication and division facts, percentage change (find 10% first, then build up, and add on or take off) and squares and roots.

Atoms bond to end up with a full outer shell of electrons, and there are two ways to get there. IONIC bonding TRANSFERS electrons, and happens between a METAL and a NON-METAL. The metal loses electrons and becomes a POSITIVE ion, the non-metal gains them and becomes a NEGATIVE ion, and the opposite charges attract. Sodium loses one electron to give Na+ and chlorine gains it to give Cl-, making sodium chloride. Those ions stack into a giant LATTICE held together in every direction, which takes a lot of energy to pull apart, so ionic compounds have HIGH melting points. A solid ionic compound does not conduct because the ions are locked in place, but MELT it or DISSOLVE it and the ions are free to move and carry charge, so it conducts. COVALENT bonding SHARES a pair of electrons, and happens between non-metals: water, oxygen, methane. The shared bond inside the molecule is strong, but the forces BETWEEN separate small molecules are weak and easily overcome, which is why simple covalent substances melt and boil at low temperatures and are often gases or liquids at room temperature.

Label the sides from the angle you are using: the HYPOTENUSE is opposite the right angle, the OPPOSITE faces your angle, the ADJACENT is next to it. Then sin = O/H, cos = A/H, tan = O/A — so opposite 6 and hypotenuse 10 gives sin = 0.6.

A MOMENT is the turning effect of a force about a pivot: moment = force x perpendicular distance from the pivot, measured in newton metres (Nm). So 10 N at 2 m gives 20 Nm, and 20 N at 0.5 m gives 10 Nm. Because distance multiplies the force, the same push further out turns harder, which is the whole reason a long spanner undoes a tight bolt more easily and a door handle is fitted as far from the hinges as it will go. Every lever has three parts: the PIVOT (fulcrum), the EFFORT you apply, and the LOAD you are moving. When something balances, the PRINCIPLE OF MOMENTS applies: total clockwise moment = total anticlockwise moment. Use it to find a missing force or distance. Work it in two steps: find the moment you know, then divide. A 250 N child 1.2 m from the pivot gives 250 x 1.2 = 300 Nm, so a 500 N adult balancing that must sit at 300 / 500 = 0.6 m on the other side. Lighter always means further out. The same two steps solve a lever: the load and the effort sit on opposite sides of the pivot, so effort x its distance = load x its distance, and an effort further out than the load can lift more than it weighs. That is why a lever is called a force multiplier.

Find two numbers that MULTIPLY to the last number and ADD to the middle one: for x² + 5x + 6, that is 2 and 3, giving (x + 2)(x + 3). To solve, set each bracket to zero and flip the sign — so x = -2 or x = -3.

The RATE of a reaction is how much product forms, or how much reactant is used up, per unit of time. In the lab you measure it by collecting the gas in a gas syringe and timing it, or by standing the flask on a balance and watching the mass fall as gas escapes. 40 cm3 of gas in 20 s is an average rate of 2 cm3 per second. Plot amount against time and the RATE IS THE GRADIENT: the steeper the line, the faster the reaction. The curve is always steepest at the start, when the reactants are most concentrated, then flattens as they are used up, and goes completely FLAT when a reactant has run out and the reaction has finished. Two experiments can end at the same final volume (same amount of reactant, same product) and still differ in how quickly they got there. Four things speed a reaction up: higher TEMPERATURE, higher CONCENTRATION, larger SURFACE AREA (powder or small chips of marble expose far more of the solid to the acid), and a CATALYST, which speeds the reaction up and is NOT used up, so it can be recovered and used again.

The angle in a semicircle is always 90°. A tangent always meets the radius at 90°. Two radii and a chord make an isosceles triangle, so the two base angles are equal. The angle at the centre is twice the angle at the circumference on the same arc.

The heart has FOUR chambers: the two ATRIA on top receive blood, the two VENTRICLES below pump it out. Humans have a DOUBLE circulatory system, so blood passes through the heart TWICE for one trip round the body. The PULMONARY circuit sends blood from the right side to the lungs to pick up oxygen; the SYSTEMIC circuit sends it from the left side out to the rest of the body through the AORTA. Keeping the two apart means the body always receives fully oxygenated blood, and the pressure can be topped up between the two loops, which makes it far more efficient than a single circuit. The LEFT ventricle wall is much thicker than the right because it has to push blood all the way round the body, while the right only has to reach the lungs next door. VALVES in the heart and in the veins stop blood flowing backwards. The vessels match their jobs: an ARTERY carries blood away from the heart at high pressure and has thick muscular elastic walls; a VEIN carries it back at low pressure with a thin wall, a wide lumen and valves; a CAPILLARY is only ONE cell thick, so the diffusion distance to the tissues is as short as possible.

Line the equations up and eliminate one letter — subtract when a coefficient matches, add when the signs are opposite. If neither matches, MULTIPLY one whole equation first so they do. Then substitute back to find the second letter, and check in the other equation.

DNA is a DOUBLE HELIX, two strands twisted round each other. It uses FOUR bases, A, T, C and G, and they pair in a fixed way: A always with T, C always with G. That is what lets DNA copy itself, because each strand tells you exactly what the other must read. A strand reading A-T-C-G is matched by T-A-G-C. A GENE is a section of DNA that codes for one PROTEIN, and it is the proteins that build and run the body. The code is read in threes: every THREE bases (a triplet) code for one AMINO ACID, and a chain of amino acids folded up is a protein. The DNA itself stays in the nucleus; the protein is actually assembled at a RIBOSOME in the cytoplasm, from a copy of the gene. Every cell in your body holds a complete copy of your DNA, and it must be copied EXACTLY before a cell divides so that both new cells get the full working set of instructions. A MUTATION is a change in the base sequence. Most do nothing at all, some change the protein, and a few are harmful or, occasionally, useful.

y ∝ x means y = kx; y ∝ 1/x means y = k/x. Always find k FIRST by substituting the pair you are given: x = 4 and y = 20 gives 20 = 4k, so k = 5. Then use y = 5x for any other value — x = 7 gives y = 35.

Every question here takes more than one step, so write the steps down. Read it twice, decide what is being asked, then work through. It pulls together Pythagoras (a² + b² = c², so square, add, then square root), angles (straight line 180°, around a point 360°), percentage change (find 10% first, then build up, and add on or take off) and volume (area of the cross-section × the length).

An unstable NUCLEUS becomes more stable by emitting radiation, and there are three kinds. ALPHA is a helium nucleus (2 protons + 2 neutrons), big and slow, and a sheet of paper stops it. BETA is a fast-moving electron thrown out of the nucleus, and it takes a few millimetres of aluminium to stop it. GAMMA is a high-energy electromagnetic wave and needs thick lead or concrete. So gamma is the most PENETRATING. But penetrating power and danger are not the same thing. Alpha is the most IONISING, dumping all its energy over a very short distance, so outside the body it cannot even get through your skin, while swallowed or inhaled it does far more damage than gamma, which mostly passes straight through. That is also why gamma is the one used for medical imaging: it can get out of the body to reach the detector. HALF-LIFE is the time for half the unstable nuclei in a sample to decay, and you count in halvings: after 3 half-lives you have 1/2 x 1/2 x 1/2 = 1/8 left, so a 5-year half-life leaves 1/8 after 15 years, and a 6-hour half-life leaves 1/16 after 24 hours. Radiation is dangerous to living things because it ionises molecules inside cells and can damage DNA, which kills the cell or leads to cancer.

This one is about speed, not method — answer what you know instantly and come back to the rest. It is testing quick multiplication and division facts, percentage change (find 10% first, then build up, and add on or take off) and squares and roots.

ELECTROLYSIS uses electricity to split an ionic compound into its elements. It only works if the compound is MOLTEN or DISSOLVED, because the ions have to be FREE TO MOVE; in a solid they are locked in the lattice and nothing can flow. The liquid being broken down is the ELECTROLYTE. Positive ions (metals, hydrogen) are attracted to the negative electrode, the CATHODE, where they GAIN electrons. Negative ions are attracted to the positive electrode, the ANODE, where they LOSE electrons. Remember it by charge: opposites attract, so positive ions go to the negative electrode. Electrolysing molten lead bromide gives LEAD at the cathode and BROMINE at the anode. The reason this matters industrially is extraction. A metal LESS reactive than carbon can be pulled out of its ore by heating with carbon, which is cheap. A metal MORE reactive than carbon, like aluminium, cannot, so electrolysis is the only way, and it is expensive because melting the ore and running the current both take enormous amounts of energy. Aluminium oxide is dissolved in molten CRYOLITE first, which lowers the melting point and so saves some of that energy.

A surd is a root that will not come out as a whole number — √4 = 2 is fine, but √5 stays as a surd. To simplify, split out the largest SQUARE factor: √12 = √4 × √3 = 2√3, and √50 = √25 × √2 = 5√2.

MOMENTUM = mass x velocity, measured in kg m/s. A 2 kg ball at 7 m/s has 14 kg m/s. In a closed system momentum is CONSERVED: the total before a collision equals the total after. A 6 kg trolley at 5 m/s has 30 kg m/s; if it hits a stationary 4 kg trolley and they stick together, the pair still has 30 kg m/s but now has 10 kg of mass, so it moves off at 30 / 10 = 3 m/s. Two skaters pushing apart start with zero total momentum, so they must end with zero, which means they move in OPPOSITE directions. A heavy truck at the same speed as a small car has far more momentum, so a crash has far more to get rid of. Force comes into it through the RATE of change of momentum: force = change in momentum / time. A 3 kg ball at 5 m/s has 15 kg m/s, and if it is caught in 0.5 s the average force is 15 / 0.5 = 30 N. Read that formula and every safety feature explains itself: a crumple zone, an airbag and a seatbelt all make the change in momentum take LONGER, and a longer time for the same change in momentum means a SMALLER force on the person.

Similar shapes keep their ANGLES identical and multiply every LENGTH by the scale factor, so 5cm at factor 4 becomes 20cm. AREA scales by the factor squared — factor 2 multiplies area by 4. Going from model to real, multiply; from real to model, divide.

Carbon moves in a loop between the air, living things, the soil and the oceans. PHOTOSYNTHESIS is the only process that takes carbon dioxide OUT of the atmosphere, locking the carbon into plants. Three processes put it back: RESPIRATION by every living thing, COMBUSTION when anything is burned, and DECOMPOSITION, where decomposers (bacteria and fungi) break down dead plants and animals and respire the carbon back out. Eating passes carbon from plants along the food chain. Left alone the loop is balanced. FOSSIL FUELS are the exception, and this is the key point: the carbon in coal, oil and gas came from living things that died hundreds of millions of years ago and was buried out of the cycle ever since, so burning them is not just moving today's carbon around, it ADDS carbon that had been locked away, and atmospheric CO2 rises. DEFORESTATION raises it in TWO separate ways at once: fewer trees are left to photosynthesise and remove CO2, and the felled wood is burned or rotted, releasing what it stored. Farming adds greenhouse gases too, mostly methane from cattle and from rice paddies.

Without replacement, both the wanted count AND the total go down by one after the first pick — 5 red from 8 leaves 4 red from 7. Multiply along the branch: 5/8 × 4/7 = 5/14. So yes, the first pick changes the second.

DIFFUSION is the movement of particles from HIGH concentration to LOW concentration, down the concentration gradient. It is PASSIVE, so it needs no energy from the cell. OSMOSIS is a special case of it: the movement of WATER, across a PARTIALLY PERMEABLE membrane, from a dilute solution (lots of water) to a more concentrated one (less water). Put a cell in very salty water and the water leaves the cell; put it in pure water and water moves in. In medicine that is not a detail, it is a rule. A drip has to be at roughly the same concentration as blood, because plain water would flood into the red blood cells until they BURST, and a strong salt solution would draw water out and shrivel them. A plant cell in pure water does not burst, because the rigid CELL WALL holds it, and it becomes firm, or TURGID, which is what holds a plant up. In a strong sugar solution a plant cell loses water and goes limp, or FLACCID, and eventually PLASMOLYSED as the membrane pulls away from the wall. That is what wilting is, and why watering reverses it.

Factorise the top and the bottom first, then cancel what they share. (x² + 5x) ÷ x = x(x + 5) ÷ x = x + 5, and (x² - 16) ÷ (x - 4) = (x - 4)(x + 4) ÷ (x - 4) = x + 4. Only cancel whole FACTORS, never single terms.

FISSION SPLITS a large unstable nucleus, usually uranium-235, into two smaller nuclei. It is started by firing a NEUTRON at the nucleus, and the split releases energy plus two or three more neutrons, which can go on to split further nuclei. That is a CHAIN REACTION, and it is what a nuclear power station runs on. CONTROL RODS, usually boron, are lowered in to ABSORB spare neutrons and keep the chain reaction steady rather than runaway. FUSION JOINS two small nuclei, hydrogen isotopes, into a larger one, and releases even MORE energy per reaction than fission. Fusion is what powers the Sun and every other star. It is not yet used to generate electricity commercially on Earth, and the reason is the conditions: both nuclei are positively charged, so they REPEL each other fiercely, and only enormous temperature and pressure can force them close enough to fuse. Building something that holds those conditions long enough to get more energy out than you put in is the hard part.

A rounded value could have been anything within half a unit either side. To the nearest cm, 20cm runs from 19.5cm up to 20.5cm. To 1 decimal place, half a unit is 0.05 — so 6.2kg runs from 6.15kg to 6.25kg.

Every question here takes more than one step, so write the steps down. Read it twice, decide what is being asked, then work through. It pulls together squares and roots, area and perimeter (triangle area is base × height ÷ 2, and area is in square units), ratio (add the parts to get the shares, divide, then multiply) and percentage change (find 10% first, then build up, and add on or take off).

SPECIFIC HEAT CAPACITY, c, is the energy needed to raise the temperature of 1 kg of a substance by 1 degree Celsius, measured in J/kg C. The formula is energy = mass x specific heat capacity x temperature change, E = m x c x change in temp. Heating 2 kg of water (c = 4200) by 10 degrees needs 2 x 4200 x 10 = 84 000 J. Heating 5 kg of a metal with c = 900 by 20 degrees needs 5 x 900 x 20 = 90 000 J. Water has an unusually HIGH specific heat capacity, so it takes a lot of energy to warm up and gives out a lot as it cools. That makes it an excellent coolant and the right thing to pump round a heating system, and it is why the sea warms and cools so slowly that coastal areas have milder winters and cooler summers than places inland. A substance with a LOW specific heat capacity, like cooking oil, heats up quickly for the same energy, so given equal energy the oil ends up hotter than the water. Two blocks given the same energy: the one with the higher c rises LESS in temperature. A real kettle always needs more energy than the calculation says, because some energy heats the kettle itself and some escapes to the room.

This one is about speed, not method — answer what you know instantly and come back to the rest. It is testing quick multiplication and division facts, percentage change (find 10% first, then build up, and add on or take off) and squares and roots.

An atom has a tiny central NUCLEUS containing PROTONS (charge +1) and NEUTRONS (charge 0), with ELECTRONS (charge -1) in shells around it. Nearly all of the mass is in the nucleus, because an electron weighs almost nothing. The ATOMIC NUMBER is the number of protons, and it is what makes an element that element. The MASS NUMBER is protons + neutrons added together, so 5 protons and 6 neutrons gives a mass number of 11, and 19 protons with 21 neutrons gives 40. In a neutral atom the number of electrons EQUALS the number of protons, so the + and - charges cancel exactly and the atom has no overall charge. Lose or gain an electron and you have an ION: 13 protons with only 10 electrons is three positive charges short of balance, so it is 3+. ISOTOPES are atoms of the SAME element, so the same number of protons, but a DIFFERENT number of neutrons, and therefore different mass numbers, like carbon-12 and carbon-14. Because chemistry is decided by the ELECTRONS and isotopes have identical electron arrangements, isotopes of an element behave the same chemically. Only their mass, and sometimes their stability, differs.

x = (-b ± √(b² - 4ac)) ÷ 2a. Write down a, b and c from ax² + bx + c = 0 first — for x² + 6x + 2 they are 1, 6 and 2. The part under the root, b² - 4ac, is the DISCRIMINANT: here 36 - 8 = 28. The ± is what gives two solutions.

COLLISION THEORY explains every rate rule with one idea: a reaction only happens when particles COLLIDE, and collide with at least a minimum amount of energy called the ACTIVATION ENERGY. Anything that makes collisions more FREQUENT or more ENERGETIC speeds the reaction up. Raising the TEMPERATURE does both: the particles move faster, so they collide more often AND a bigger share of those collisions carries enough energy to get over the activation energy. Raising the CONCENTRATION (or the pressure of a gas) packs more particles into the same space, so collisions are more frequent. Increasing the SURFACE AREA by powdering a solid exposes far more of it, so more particles are available to be hit at any moment. A CATALYST works differently: it does not push the particles harder, it offers an alternative route with a LOWER activation energy, so a greater proportion of ordinary collisions is now energetic enough. A catalyst is NOT used up, and it does not change how much product you finish with, only how fast you get there. Enzymes are the body's catalysts and work the same way.

MEAN: total ÷ how many. MEDIAN: order the list, take the middle. MODE: most frequent. RANGE: biggest minus smallest, and it measures SPREAD rather than average. Ordering the list first is what most marks hang on.

HOMEOSTASIS is the body keeping its internal conditions steady whatever is happening outside it. The three to know are body TEMPERATURE (about 37 C), BLOOD GLUCOSE and WATER level. It runs on NEGATIVE FEEDBACK: a receptor detects that a level has drifted, and the body's response pushes it back the OTHER WAY, towards normal. That is why it is called negative, the response OPPOSES the change rather than adding to it. Temperature is monitored by the HYPOTHALAMUS in the brain. Too hot and you SWEAT (evaporating water takes energy away from the skin) and the blood vessels near the skin WIDEN, VASODILATION, so more warm blood flows near the surface and more energy is lost. Too cold and you SHIVER, because muscles contracting rapidly respire faster and respiration releases energy as heat, and the surface vessels NARROW to keep blood away from the skin. Blood glucose works the same way in the other system: too high and INSULIN brings it down, too low and GLUCAGON brings it up.

TRANSLATION slides (write it as a column vector), REFLECTION flips over a mirror line, ROTATION turns about a centre, ENLARGEMENT resizes. A translation of 5 right and 1 down is the vector (5, -1).

HORMONES are chemical messengers. They are made by GLANDS, released straight into the BLOOD, and carried all round the body to act on their target organs. That makes them SLOWER to act than nerve impulses, which travel as electrical signals along neurones, but their effects last LONGER and are more general. Nerves are fast and short-lived and precise; hormones are slow and long-lasting and widespread. The ones to know: INSULIN, made by the PANCREAS, LOWERS blood glucose by telling cells to take glucose in and the liver to store it as glycogen. GLUCAGON, also from the pancreas, RAISES it again by making the liver release glucose. ADRENALINE, from the adrenal glands, prepares the body for fight or flight by raising heart rate and breathing rate. That is why the two kinds of diabetes are different problems. In TYPE 1 the pancreas cannot make enough insulin at all, so it has to be injected. In TYPE 2 the pancreas still makes insulin but the body's cells have stopped RESPONDING to it properly, and it is usually managed with diet, exercise and weight loss first.

Use compasses and a straight edge, and leave your arcs showing — they are the working. A perpendicular bisector cuts a line at 90° through its midpoint; an angle bisector halves an angle. A locus is every point obeying one rule, so a fixed distance from a point is a circle.

Read the axes first, because the two graphs mean different things. On a DISTANCE-TIME graph the GRADIENT is the SPEED: a horizontal line means stationary, a steeper line means faster, and a line curving upwards means the speed is increasing, so the object is accelerating. On a VELOCITY-TIME graph the GRADIENT is the ACCELERATION and the AREA under the line is the DISTANCE travelled. So a horizontal line above zero on a velocity-time graph is not stationary, it is a steady speed, and a straight sloping line is constant acceleration. ACCELERATION = change in velocity / time taken, in m/s2: 0 to 18 m/s in 3 s is 18 / 3 = 6 m/s2, and 0 to 21 m/s in 6 s is 21 / 6 = 3.5 m/s2. For the distance, the area under a rectangle is just height x width, so a constant 8 m/s for 15 s covers 8 x 15 = 120 m. A line dropping to zero and staying there means the object slowed down and stopped.

m is the GRADIENT — how much y goes up for each 1 across — and c is the Y-INTERCEPT, where the line crosses the y-axis. So y = 6x + 2 climbs 6 for every 1 across and passes through (0, 2).

Every question here takes more than one step, so write the steps down. Read it twice, decide what is being asked, then work through. It pulls together angles (straight line 180°, around a point 360°), percentage change (find 10% first, then build up, and add on or take off) and area and perimeter (triangle area is base × height ÷ 2, and area is in square units).

OHM'S LAW is V = I x R: potential difference (volts) = current (amps) x resistance (OHMS). Rearrange it as needed: R = V / I, so 24 V driving 4 A means R = 6 ohms; I = V / R, so 20 V across a 5 ohm resistor gives 4 A. Higher resistance for the same voltage means LESS current, not more. Plot current against voltage for a fixed resistor at constant temperature and you get a STRAIGHT LINE THROUGH THE ORIGIN, showing current is directly PROPORTIONAL to voltage. Compare two such lines and the STEEPER one is the LOWER resistance, because more current is flowing for the same voltage. A FILAMENT LAMP is the exception: as more current flows the filament gets hot, the resistance RISES, and the graph curves over into an S shape rather than staying straight. Resistance depends on the wire too: a longer wire has MORE resistance, because the electrons collide with more ions on the way through. An AMMETER goes in series and must have as close to ZERO resistance as possible, so that adding it does not change the current it is supposed to be measuring.