Foundation - a strong base in Physics, Chemistry, Maths & Biology

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.