Master the basics - and everything after gets easier

What it is

Light bends (refracts) when its speed changes on moving between media of different densities.

Answer

Light bends because its speed changes when it moves from one medium to another. Glass is optically denser than air, so light slows down on entering glass and bends towards the normal. This bending of light due to a change in speed is called refraction.

What it is

The mirror formula links focal length, object distance and image distance for a curved mirror.

Answer

The mirror formula relates focal length f, image distance v and object distance u as 1/f = 1/v + 1/u. By the New Cartesian sign convention, distances are measured from the pole; distances measured against the incident light (to the left) are negative, so for a concave mirror f and u are negative, and a real image distance v is also negative.

What it is

For a conductor at steady temperature, voltage = current x resistance (V = IR).

Answer

Ohm's law states that, at constant temperature, the current I through a conductor is directly proportional to the potential difference V across it, so V = IR. Resistance R is the property of a conductor that opposes the flow of current; it is the ratio of potential difference to current, R = V/I, measured in ohm.

What it is

Electric power is the rate of using electrical energy: P = VI (also I^2 R or V^2 / R).

Answer

Using Ohm's law, V = IR = 2 x 5 = 10 V. Power dissipated P = VI = 10 x 2 = 20 W (equivalently P = I^2 R = 4 x 5 = 20 W).

What it is

A current in a straight wire creates circular magnetic field lines around it.

Answer

If you hold the straight conductor in your right hand with the thumb pointing in the direction of the current, the curled fingers show the direction of the concentric circular magnetic field lines around the wire.

What it is

A more reactive metal displaces a less reactive one from its salt solution.

Answer

Iron is more reactive than copper, so it displaces copper from copper sulphate: Fe + CuSO4 -> FeSO4 + Cu. The blue colour of CuSO4 fades as it is replaced by the pale green FeSO4, and a brown copper coating forms on the nail. This is a displacement reaction.

What it is

pH measures how acidic or basic a solution is, from 0 (acid) to 14 (base), with 7 neutral.

Answer

The pH scale runs from 0 to 14 and measures the concentration of hydrogen ions (H+) in a solution. A pH of 7 is neutral, below 7 is acidic and above 7 is basic. As a solution becomes more acidic, the H+ concentration increases and the pH value decreases.

What it is

Metals higher in the reactivity series displace those lower down, never the reverse.

Answer

In the reactivity series zinc lies above copper, so zinc is more reactive and loses electrons more readily. Therefore zinc displaces the less reactive copper: Zn + CuSO4 -> ZnSO4 + Cu. Copper, being less reactive than zinc, cannot displace zinc, so Cu + ZnSO4 does not react.

What it is

Carbon forms millions of compounds because it makes four strong covalent bonds and chains (catenation).

Answer

Carbon has four valence electrons, so it shares electrons to form four strong covalent bonds rather than gaining or losing electrons. Two special properties help: catenation (carbon atoms link with other carbon atoms to form long chains, branches and rings) and tetravalency (it can bond with four other atoms such as hydrogen, oxygen, nitrogen). Together these give a huge number of stable compounds.

What it is

In a redox reaction one substance gains oxygen (oxidised) while another loses it (reduced), together.

Answer

Hydrogen gains oxygen to form water, so H2 is oxidised. Copper oxide loses oxygen to form copper, so CuO is reduced. Since oxidation and reduction happen together, this is a redox reaction; here H2 acts as the reducing agent and CuO as the oxidising agent.

What it is

Oxygen rides on haemoglobin in red blood cells from the lungs to every tissue.

Answer

Oxygen diffuses from the air sacs (alveoli) of the lungs into the blood, where it binds to the red pigment haemoglobin in red blood cells to form oxyhaemoglobin. The blood carries it to the tissues, where oxygen is released for respiration. Haemoglobin makes this transport efficient because it has a high affinity for oxygen.

What it is

Arteries carry blood away from the heart under high pressure; veins bring it back and have valves.

Answer

Arteries carry blood away from the heart, usually oxygenated (except the pulmonary artery), have thick elastic walls and no valves because blood flows under high pressure. Veins carry blood towards the heart, usually deoxygenated (except the pulmonary vein), have thinner walls and contain valves to prevent the backflow of low-pressure blood.

What it is

A reflex arc gives an instant, automatic response through the spinal cord - no thinking needed.

Answer

A reflex arc is the pathway followed by a nerve impulse during a reflex action, allowing a quick automatic response without conscious thought. When you touch a hot object, receptors in the skin send an impulse along a sensory neuron to the spinal cord; there a relay neuron passes it to a motor neuron, which carries the impulse to the muscles of the hand, causing you to pull your hand away instantly.

What it is

Sexual reproduction mixes two parents' genes to create variety; asexual makes identical copies.

Answer

In sexual reproduction two parents contribute genes, and the mixing of DNA during the formation of gametes and their random combination at fertilisation creates new gene combinations in the offspring. In asexual reproduction a single parent produces genetically identical copies, so there is little variation. Greater variation gives species a better chance of surviving changes in the environment.

What it is

Crossing two hybrids (Tt x Tt) gives a 3:1 tall-to-short ratio because T is dominant over t.

Answer

The F1 generation is all Tt (tall), because T is dominant over t. When F1 plants (Tt x Tt) are crossed, the F2 offspring are TT, Tt, Tt, tt. TT, Tt and Tt appear tall while tt is short, giving a phenotypic ratio of 3 tall : 1 short. The genotypic ratio is 1 TT : 2 Tt : 1 tt.

What it is

Euclid's algorithm finds the HCF by repeated division until the remainder becomes zero.

Answer

Apply repeated division: 404 = 96 x 4 + 20; 96 = 20 x 4 + 16; 20 = 16 x 1 + 4; 16 = 4 x 4 + 0. The last non-zero remainder is 4, so HCF(96, 404) = 4.

What it is

The zeroes of a quadratic are where it equals zero; their sum is -b/a and product is c/a.

Answer

Factorise: x^2 - 7x + 10 = (x - 2)(x - 5), so the zeroes are 2 and 5. Sum of zeroes = 2 + 5 = 7 = -(-7)/1 = -b/a. Product of zeroes = 2 x 5 = 10 = 10/1 = c/a. Both relations hold.

What it is

Two linear equations meet at one point - the (x, y) that satisfies both at once.

Answer

From the second equation, x = y - 1. Substitute into the first: 2(y - 1) + 3y = 13, so 5y - 2 = 13, giving 5y = 15 and y = 3. Then x = y - 1 = 2. Solution: x = 2, y = 3.

What it is

The discriminant D = b^2 - 4ac reveals the nature of a quadratic's roots before you solve it.

Answer

Factorise: (x - 2)(x - 3) = 0, so the roots are x = 2 and x = 3. The discriminant D = b^2 - 4ac = (-5)^2 - 4(1)(6) = 25 - 24 = 1. Since D > 0 and is a perfect square, the roots are real, distinct and rational.

What it is

In an AP each term rises by a fixed step d; the nth term is a + (n - 1)d.

Answer

The nth term of an AP is a_n = a + (n - 1)d. Here a = 3, d = 5, n = 10, so a_10 = 3 + (10 - 1) x 5 = 3 + 45 = 48.

What it is

The sum of an AP pairs the first and last terms: S = n/2 [2a + (n - 1)d].

Answer

Here a = 2 and d = 3. The sum of n terms is S_n = n/2 [2a + (n - 1)d]. So S_20 = 20/2 [2(2) + 19(3)] = 10 [4 + 57] = 10 x 61 = 610.

What it is

A line parallel to one side of a triangle cuts the other two sides in the same ratio.

Answer

The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, then it divides those two sides in the same ratio. In triangle ABC, if DE is parallel to BC and meets AB at D and AC at E, then AD/DB = AE/EC.

What it is

The distance between two points is the hypotenuse of the right triangle of their coordinate gaps.

Answer

The distance formula is d = sqrt[(x2 - x1)^2 + (y2 - y1)^2]. Here d = sqrt[(5 - 2)^2 + (7 - 3)^2] = sqrt[9 + 16] = sqrt[25] = 5 units.

What it is

The identity sin^2 + cos^2 = 1 lets you find one trig ratio from another for the same angle.

Answer

Using the identity sin^2(theta) + cos^2(theta) = 1, cos^2(theta) = 1 - (3/5)^2 = 1 - 9/25 = 16/25, so cos(theta) = 4/5 (positive since theta is acute). Then tan(theta) = sin/cos = (3/5) / (4/5) = 3/4.

What it is

An angle of elevation turns a triangle into a height problem using tan = opposite / adjacent.

Answer

Let the height be h. tan(30 degrees) = h / 30, and tan(30 degrees) = 1/sqrt(3). So h = 30 x (1/sqrt(3)) = 30/sqrt(3) = 10 sqrt(3) m, which is about 17.3 m.

What it is

A tangent touches a circle at one point and is perpendicular to the radius at that point.

Answer

The tangent is perpendicular to the radius at the point of contact, so the radius, tangent and line OP form a right-angled triangle with OP as the hypotenuse. Length of tangent = sqrt[OP^2 - r^2] = sqrt[13^2 - 5^2] = sqrt[169 - 25] = sqrt[144] = 12 cm.

What it is

A cone's volume is exactly one-third of a cylinder with the same base and height.

Answer

Volume of a cone = (1/3) x pi x r^2 x h = (1/3) x (22/7) x 3^2 x 4 = (1/3) x (22/7) x 36 = (22 x 12) / 7 = 264/7, which is about 37.7 cubic cm.

What it is

For a moderately skewed data set, the three averages link as Mode = 3 Median - 2 Mean.

Answer

The empirical relationship between the three measures of central tendency is Mode = 3 Median - 2 Mean. Substituting, Mode = 3(30) - 2(27) = 90 - 54 = 36.

What it is

Probability = favourable outcomes / total outcomes, always a number between 0 and 1.

Answer

There are 4 kings in a deck of 52 cards. Probability = number of favourable outcomes / total outcomes = 4/52 = 1/13.

What it is

The standard angles 0, 30, 45, 60 and 90 degrees have fixed trig values worth memorising.

Answer

Use standard values: tan(45 deg) = 1, cos(30 deg) = sqrt(3)/2, sin(60 deg) = sqrt(3)/2. So the expression = 2(1)^2 + (sqrt(3)/2)^2 - (sqrt(3)/2)^2 = 2 + 3/4 - 3/4 = 2.

What it is

The power of a lens tells how strongly it bends light; it is the reciprocal of its focal length in metres.

Answer

Power P = 1 / f, where f is the focal length in metres. Here f = 25 cm = 0.25 m, so P = 1 / 0.25 = +4 dioptres (D). The power is positive because it is a convex (converging) lens.

What it is

The sky is blue because tiny air molecules scatter blue light much more than red.

Answer

Sunlight is scattered by the tiny molecules of air in the atmosphere. Blue light has a shorter wavelength and is scattered much more strongly than red light. As this scattered blue light reaches our eyes from all directions, the clear sky appears blue. At sunset the light travels a longer path, most of the blue is scattered away, and the remaining red light makes the sky look reddish.

What it is

A neutralisation reaction is an acid reacting with a base to give a salt and water.

Answer

A neutralisation reaction is the reaction between an acid and a base to form a salt and water. The general equation is: Acid + Base -> Salt + Water (for example, HCl + NaOH -> NaCl + H2O). An everyday use is taking an antacid (a mild base) to neutralise excess acid in the stomach and relieve acidity.

What it is

A homologous series is a family of carbon compounds with the same general formula, differing by CH2.

Answer

A homologous series is a group of organic compounds having the same general formula and the same functional group, in which each successive member differs from the previous one by a -CH2- unit. Two characteristics are: (1) all members have similar chemical properties because of the same functional group, and (2) their physical properties such as boiling point change gradually as the molecule gets bigger. The alkanes (CH4, C2H6, C3H8, ...) are an example.

What it is

Photosynthesis is how green plants make their own food (glucose) from CO2 and water using sunlight.

Answer

Photosynthesis is the process by which green plants make their own food (glucose) from carbon dioxide and water using sunlight, releasing oxygen. The balanced equation is: 6CO2 + 6H2O -> C6H12O6 + 6O2. It requires chlorophyll (the green pigment), sunlight, carbon dioxide and water.

What it is

A food chain shows who eats whom, and only about 10 percent of energy passes to the next level.

Answer

A food chain is a series of organisms through which energy and nutrients pass as one organism eats another, for example grass -> deer -> lion. The ten percent law states that only about 10 percent of the energy at one trophic level is passed on to the next level; the rest is lost mainly as heat. This is why food chains usually have only three or four links.

What it is

In series, resistances add up; in parallel, the combined resistance is smaller than the smallest one.

Answer

In a series combination the resistors are joined end to end so the same current flows through each, and the total resistance is the sum: R = R1 + R2 + R3. In a parallel combination the resistors are connected between the same two points and share the current, so the reciprocals add: 1/R = 1/R1 + 1/R2 + 1/R3. As a result the total resistance in parallel is always less than the smallest individual resistance. Household circuits are wired in parallel so every appliance gets the full voltage.

What it is

Air molecules scatter shorter (blue) wavelengths much more than red, so the clear sky looks blue.

Answer

As sunlight passes through the atmosphere it is scattered by the tiny molecules of air. Scattering is much stronger for shorter wavelengths, so blue light is scattered far more than red light. This scattered blue light reaches our eyes from every part of the sky, making a clear sky look blue. Near sunrise and sunset the light travels a much longer path through the air, most of the blue is scattered away, and the Sun appears red.

What it is

A changing magnetic field around a conductor induces a current in it - the reverse of the motor effect.

Answer

Electromagnetic induction is the production of an electric current in a conductor whenever the magnetic field around it changes, for example by moving a magnet into or out of a coil. The changing field sets up a potential difference that drives a current. Discovered by Faraday, it is the principle behind the electric generator. The induced current can be increased by moving the magnet faster, using a stronger magnet, or increasing the number of turns in the coil.

What it is

Most reactions are one of four types: combination, decomposition, displacement or double displacement.

Answer

1) Combination: two or more substances combine to form a single product, e.g. CaO + H2O -> Ca(OH)2. 2) Decomposition: a single compound breaks into two or more substances, often on heating, e.g. CaCO3 -> CaO + CO2. 3) Displacement: a more reactive element displaces a less reactive one from its compound, e.g. Fe + CuSO4 -> FeSO4 + Cu. 4) Double displacement: two compounds exchange their ions, e.g. Na2SO4 + BaCl2 -> BaSO4 + 2NaCl.

What it is

The reactivity series lists metals from most reactive (potassium) to least reactive (gold).

Answer

The reactivity series is a list of metals arranged in decreasing order of their reactivity, from potassium and sodium at the top down to silver and gold at the bottom. A metal higher in the series can displace a metal lower down from its salt solution. The series is useful for predicting displacement reactions, choosing the method to extract a metal from its ore, and explaining why very reactive metals corrode easily while gold and silver stay shiny.

What it is

Carbon forms millions of compounds because of catenation and its tetravalency.

Answer

Carbon forms an extremely large number of compounds for two main reasons. First, catenation: carbon atoms can form strong covalent bonds with other carbon atoms, creating long straight chains, branched chains and rings. Second, tetravalency: a carbon atom has four valence electrons, so it can form four strong covalent bonds with other atoms such as hydrogen, oxygen, nitrogen, sulphur and the halogens. Together these give a huge variety of stable molecules.

What it is

A reflex action is a fast, automatic response to a stimulus, controlled by the spinal cord.

Answer

A reflex action is a sudden, automatic response to a stimulus that does not need conscious thinking by the brain. The path taken by the nerve impulse is called the reflex arc: a receptor detects the stimulus, a sensory neuron carries the impulse to the spinal cord, a relay neuron passes it to a motor neuron, and the motor neuron makes the effector (a muscle) act. Because the impulse is processed in the spinal cord and not the brain, the response is very quick.

What it is

Crossing two Tt plants gives a 3:1 ratio of tall to short - the classic monohybrid result.

Answer

When two Tt plants are crossed (where T for tall is dominant over t for short), the offspring appear in the genotype ratio 1 TT : 2 Tt : 1 tt. Because T is dominant, both TT and Tt plants are tall and only tt is short, giving a phenotype ratio of 3 tall : 1 short (3:1). This shows that a recessive trait, which stays hidden in the F1 generation, can reappear in the F2 generation.

What it is

Arteries carry blood away from the heart; veins carry it back to the heart.

Answer

1) Arteries carry blood away from the heart, while veins carry blood back towards the heart. 2) Arteries carry oxygenated blood (except the pulmonary artery), while veins carry deoxygenated blood (except the pulmonary vein). 3) Arteries have thick, elastic, muscular walls to handle high pressure and have no valves, whereas veins have thinner walls, carry blood at low pressure, and contain valves that stop the blood flowing backwards.

What it is

The power of a lens measures how strongly it converges or diverges light.

Answer

Power = 1 / focal length (in metres) = 1 / 0.5 = +2 dioptre (D). A convex (converging) lens has positive power; the shorter its focal length, the higher its power.

What it is

Population density is the number of individuals living per unit area.

Answer

Population density = number of individuals / area = 500 / 25 = 20 deer per km^2. If the same number of animals had to live in a smaller area, the density - and the competition for food and space - would rise.

What it is

Probability measures how likely an event is - favourable outcomes out of all equally likely outcomes.

Answer

Probability = favourable outcomes / total outcomes = 1 / 6 = 0.167 (about 17%). A fair die has 6 equally likely faces and only one of them is a 4. Every probability lies between 0 (impossible) and 1 (certain).

What it is

The refractive index of a medium is how many times slower light travels in it than in vacuum.

Answer

Refractive index n = speed in vacuum / speed in medium = (3 x 10^8) / (2 x 10^8) = 1.5. A larger refractive index means light travels more slowly in the medium and bends more on entering it.

What it is

Resistors joined end to end (in series) have a combined resistance equal to their sum.

Answer

In series, resistances add: R = R1 + R2 = 10 + 20 = 30 ohm. The total resistance in a series combination is always greater than the largest individual resistor.

What it is

A population changes as births add individuals and deaths remove them.

Answer

New population = initial + births - deaths = 500 + 60 - 40 = 520. When births exceed deaths the population grows; when deaths exceed births it falls (here we ignore migration).

What it is

Alkanes form a homologous series of saturated hydrocarbons with the general formula CnH(2n+2).

Answer

Alkanes are saturated hydrocarbons (only single C-C and C-H bonds) with the general formula CnH(2n+2), where n is the number of carbon atoms. So n = 1 gives CH4 (methane), n = 2 gives C2H6 (ethane), n = 3 gives C3H8 (propane) and n = 4 gives C4H10 (butane). Each successive member differs by one CH2 unit - this family is called a homologous series.

What it is

Electric current is the rate of flow of electric charge through a conductor.

Answer

Current = charge / time = 30 / 60 = 0.5 ampere (A). Current tells us how much charge flows each second, so more charge in less time means a larger current.

What it is

The heat power produced in a resistor depends on the square of the current and the resistance.

Answer

Power = I^2 x R = 2^2 x 5 = 4 x 5 = 20 watt (W). Because the current is squared, doubling the current to 4 A would give 4^2 x 5 = 80 W - four times the heat.

What it is

The number of heartbeats is the heart rate multiplied by the time.

Answer

Heartbeats = heart rate x time = 72 x 10 = 720 beats. A resting heart beating about 72 times a minute beats roughly 100,000 times in a whole day, pumping blood non-stop.

What it is

Resistors connected across the same two points (in parallel) give a combined resistance smaller than any one of them.

Answer

For two resistors in parallel, R = (R1 x R2) / (R1 + R2) = (10 x 20) / (10 + 20) = 200 / 30 = 6.67 ohm. Notice the total is smaller than either resistor, because the current now has more than one path to flow through.

What it is

The charge that flows is the current multiplied by the time it flows for.

Answer

Charge = current x time = 2 x 30 = 60 coulomb (C). A larger current, or a longer time, delivers more charge past a point in the circuit.

What it is

The electrical energy used is the power of a device multiplied by the time it runs, often measured in kilowatt-hours (units).

Answer

Energy = power x time = 1000 W x 5 h = 5000 Wh = 5 kilowatt-hours (kWh), that is 5 units. One unit of electricity is one kilowatt-hour - a 1000 W appliance running for one hour.

What it is

Only about 10% of the energy at one trophic level is passed on to the next level.

Answer

Energy passed on = 10% of the energy = 1000 x 10 / 100 = 100 units. The other 90% is used up in life processes and lost as heat, so less and less energy is available at each higher level.

What it is

The potential difference (voltage) between two points is the work done to move each unit of charge between them.

Answer

Potential difference = work done / charge = 60 / 5 = 12 volt (V). Voltage measures the energy given to each coulomb of charge, so 1 V means 1 J of work per coulomb.

What it is

From Ohm's law, the resistance of a conductor is the voltage across it divided by the current through it.

Answer

Resistance = voltage / current = 12 / 2 = 6 ohm. This is Ohm's law V = IR rearranged as R = V / I; for a fixed voltage, a larger current means a smaller resistance.

What it is

The number of breaths taken is the breathing rate multiplied by the time.

Answer

Breaths = breathing rate x time = 16 x 10 = 160 breaths. A resting adult breathes about 12 to 16 times a minute; the rate rises during exercise to bring in more oxygen.

What it is

The volume of a cylinder is the area of its circular base times its height.

Answer

Volume = pi x r^2 x h = 3.14159 x 3^2 x 7 = 3.14159 x 9 x 7 = 197.9 cubic units. The base is a circle of area pi x r^2, and stacking it to a height h gives the volume.

What it is

The surface area of a sphere is four times pi times the square of its radius.

Answer

Surface area = 4 x pi x r^2 = 4 x 3.14159 x 7^2 = 4 x 3.14159 x 49 = 615.75 square units. Remarkably, a sphere's surface is exactly four times the area of its flat circular cross-section (pi x r^2).

What it is

The population growth rate is the net change (births minus deaths) expressed as a percentage of the population.

Answer

Growth rate = (births - deaths) / population x 100 = (60 - 40) / 500 x 100 = 4% per year. If deaths exceed births the rate is negative, meaning the population is shrinking.

What it is

The heat produced in a resistor is the square of the current times the resistance times the time.

Answer

Heat H = I^2 x R x t = 2^2 x 5 x 10 = 4 x 5 x 10 = 200 joule (J). Because the current is squared, doubling it would produce four times as much heat.

What it is

Electric power is the product of the voltage across a device and the current through it.

Answer

Power = voltage x current = 12 x 2 = 24 watt (W). This is the rate at which the device converts electrical energy; combined with Ohm's law it also gives P = I^2 R and P = V^2 / R.

What it is

In an arithmetic progression, each term is found by adding the common difference to the first term repeatedly.

Answer

nth term = a + (n - 1) d = 2 + (5 - 1) x 3 = 2 + 12 = 14. We add the common difference (n - 1) times because the first term already counts as term 1.

What it is

The sum of the first n terms of an arithmetic progression is n/2 times the sum of twice the first term and (n - 1) times the common difference.

Answer

Sum = n/2 x (2a + (n - 1) d) = 5/2 x (2x2 + (5 - 1) x 3) = 5/2 x (4 + 12) = 5/2 x 16 = 40. This equals the number of terms times the average of the first and last term.

What it is

The focal length of a spherical mirror is half its radius of curvature.

Answer

Focal length = radius of curvature / 2 = 40 / 2 = 20 cm. The principal focus of a spherical mirror lies exactly midway between the pole and the centre of curvature.

What it is

The speed of light in a medium is the speed in vacuum divided by the medium's refractive index.

Answer

Speed in medium = c / n = (3 x 10^8) / 1.5 = 2 x 10^8 m/s. The larger the refractive index, the more the light slows down, and the more it bends on entering the medium.

What it is

The slope of a line is how much it rises for each unit it runs across.

Answer

Slope = rise / run = 6 / 3 = 2. A slope of 2 means the line climbs 2 units for every 1 unit across. A negative slope would mean the line falls as you move right.

What it is

The area of a sector is the fraction of the circle its angle covers, times the circle's area.

Answer

Sector area = (angle / 360) x pi x r^2 = (90 / 360) x 3.14159 x 7^2 = 0.25 x 3.14159 x 49 = 38.48 square units. A 90 degree sector is one quarter of the circle, so it has a quarter of the circle's area.

What it is

The length of an arc is the fraction of the circumference that its angle subtends.

Answer

Arc length = (angle / 360) x 2 x pi x r = (90 / 360) x 2 x 3.14159 x 7 = 0.25 x 43.98 = 11.0 units. A 90 degree arc is one quarter of the full circumference of the circle.

What it is

The cost of electricity is the number of units (kilowatt-hours) used multiplied by the rate per unit.

Answer

Cost = units x rate = 100 x 6 = 600 rupees. One unit is one kilowatt-hour, so using fewer units - by switching off idle appliances - directly reduces the bill.

What it is

The discriminant of a quadratic equation tells how many real roots it has.

Answer

Discriminant D = b^2 - 4ac = 5^2 - 4 x 1 x 6 = 25 - 24 = 1. Since D is positive, the equation has two distinct real roots. If D were 0 there would be one repeated root, and if D were negative there would be no real roots.

What it is

The sum of the roots of a quadratic equation is minus b divided by a.

Answer

Sum of roots = -b / a = -(-5) / 1 = 5. Indeed the roots of x^2 - 5x + 6 = 0 are 2 and 3, which add to 5. The product of the roots is c / a = 6, matching 2 x 3.

What it is

A Punnett square predicts the genotypes and phenotypes of offspring from the alleles of the two parents.

Answer

Each Tt parent gives half T and half t gametes. Filling the Punnett square gives TT, Tt, Tt, tt - a genotype ratio of 1 : 2 : 1. Since T (tall) is dominant, three of the four are tall and one is short, a phenotype ratio of 3 : 1.

What it is

A balanced chemical equation has the same number of atoms of each element on both sides, as required by the law of conservation of mass.

Answer

Balanced, it is 2H2 + O2 -> 2H2O. That gives 4 hydrogen atoms and 2 oxygen atoms on each side. We only adjust the coefficients in front of each formula, because changing a subscript would change the substance itself.

What it is

The current a device draws is its power divided by the supply voltage.

Answer

Current = power / voltage = 1100 / 220 = 5 A. This comes from P = V x I rearranged as I = P / V, and it decides the fuse rating the appliance needs.

What it is

The power dissipated in a resistor is the square of the voltage divided by the resistance.

Answer

Power = V^2 / R = 220^2 / 484 = 48400 / 484 = 100 W. This form comes from combining P = VI with Ohm's law V = IR, and shows that a smaller resistance draws more power at a fixed voltage.

What it is

In a right triangle, the sine of an angle is the opposite side divided by the hypotenuse.

Answer

sin(theta) = opposite / hypotenuse = 3 / 5 = 0.6. The sine of an acute angle is always between 0 and 1, because the opposite side is never longer than the hypotenuse.

What it is

In a right triangle, the cosine of an angle is the adjacent side divided by the hypotenuse.

Answer

cos(theta) = adjacent / hypotenuse = 4 / 5 = 0.8. Like sine, it lies between 0 and 1, and the two obey the identity sin^2(theta) + cos^2(theta) = 1.

What it is

In a right triangle, the tangent of an angle is the opposite side divided by the adjacent side.

Answer

tan(theta) = opposite / adjacent = 3 / 4 = 0.75. It also equals sin(theta) / cos(theta), and it grows without limit as the angle approaches 90 degrees.

What it is

The distance between two points is the square root of the sum of the squares of the differences in their coordinates.

Answer

Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((3 - 0)^2 + (4 - 0)^2) = sqrt(9 + 16) = sqrt(25) = 5 units. It is simply the Pythagoras theorem applied to the horizontal and vertical gaps between the points.

What it is

The sum of the first n natural numbers is n times n plus one, divided by two.

Answer

Sum = n(n + 1) / 2 = 10 x 11 / 2 = 55. This is an arithmetic progression with first term 1 and common difference 1; pairing the first and last terms gives the neat n(n + 1) / 2 formula.

What it is

The perimeter of a sector is its two straight radii plus the curved arc.

Answer

Perimeter = 2r + arc = 2 x 7 + (90 / 360) x 2 x 3.14159 x 7 = 14 + 11.0 = 25.0 units. It is the two radii bounding the sector plus the curved arc along the circle.

What it is

In a series circuit the components share one path, while in a parallel circuit each has its own path.

Answer

In a series circuit there is only one path for the current, so if one bulb breaks the circuit is open and both bulbs go out. In a parallel circuit each bulb sits on its own branch and gets the full voltage, so if one breaks the other keeps glowing. That is why household wiring is done in parallel and each appliance also gets the same voltage.

What it is

The probability that an event does not happen is one minus the probability that it does.

Answer

P(not E) = 1 - P(E) = 1 - 1/6 = 5/6 (about 0.833). The event and its complement together cover every outcome, so their probabilities always add up to 1.

What it is

The product of the roots of a quadratic equation is c divided by a.

Answer

Product of roots = c / a = 6 / 1 = 6. The roots of x^2 - 5x + 6 = 0 are 2 and 3, and indeed 2 x 3 = 6. Their sum is -b / a = 5, matching 2 + 3.

What it is

Two angles are complementary when they add up to 90 degrees.

Answer

Complement = 90 - angle = 90 - 30 = 60 degrees. Complementary angles add to a right angle, which is why the sine of an angle equals the cosine of its complement: sin(30) = cos(60).

What it is

The perimeter of a semicircle is the curved half of the circumference plus the straight diameter.

Answer

Perimeter = pi x r + 2r = 3.14159 x 7 + 2 x 7 = 21.99 + 14 = 35.99 units. The curved part is half the circumference (pi x r), and we must add the straight diameter (2r) that closes the shape.

What it is

The area of a semicircle is half the area of the full circle.

Answer

Area = 1/2 x pi x r^2 = 1/2 x 3.14159 x 7^2 = 1/2 x 3.14159 x 49 = 76.97 square units. A semicircle is exactly half a circle, so we take half of pi x r^2.

What it is

A reflex arc is the short nerve pathway that produces an automatic, rapid response to a stimulus.

Answer

The stimulus (heat) is detected by receptors in the skin. A sensory neuron carries the message to the spinal cord, which acts as the relay and decides the response without waiting for the brain. A motor neuron then carries the command to the effector - a muscle - which contracts and pulls your hand away. This whole pathway is the reflex arc, and it makes the response fast and automatic.

What it is

The cosecant of an angle is the hypotenuse divided by the opposite side, the reciprocal of sine.

Answer

cosec(theta) = hypotenuse / opposite = 5 / 3 = 1.667. It is the reciprocal of the sine ratio, so cosec(theta) = 1 / sin(theta).

What it is

The secant of an angle is the hypotenuse divided by the adjacent side, the reciprocal of cosine.

Answer

sec(theta) = hypotenuse / adjacent = 5 / 4 = 1.25. It is the reciprocal of the cosine ratio, so sec(theta) = 1 / cos(theta).

What it is

The cotangent of an angle is the adjacent side divided by the opposite side, the reciprocal of tangent.

Answer

cot(theta) = adjacent / opposite = 4 / 3 = 1.333. It is the reciprocal of the tangent ratio, so cot(theta) = 1 / tan(theta).

What it is

The height of an object is the horizontal distance to it times the tangent of the angle of elevation.

Answer

Height = distance x tan(theta) = 50 x tan(30) = 50 x 0.577 = 28.87 m. The horizontal distance is the adjacent side and the height is the opposite side, so tan links them.

What it is

A quadrant is a quarter of a circle, so its area is a quarter of the circle's area.

Answer

Area = 1/4 x pi x r^2 = 1/4 x 3.14159 x 7^2 = 1/4 x 3.14159 x 49 = 38.48 square units. A quadrant is a 90 degree sector, which is one quarter of the whole circle.

What it is

The sum of the first n odd numbers is always n squared.

Answer

The sum of the first n odd numbers = n^2. For n = 5 that is 1 + 3 + 5 + 7 + 9 = 25 = 5^2. It is an arithmetic progression with first term 1 and common difference 2.