The distance between (x1, y1, z1) and (x2, y2, z2) is sqrt[(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2]. Here = sqrt[(4 - 2)^2 + (3 - 3)^2 + (1 - 5)^2] = sqrt[4 + 0 + 16] = sqrt20 = 2 sqrt5.

d = sqrt[(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2]

The distance of a point (x, y, z) from the origin is sqrt(x^2 + y^2 + z^2). Here = sqrt(1^2 + (-2)^2 + 3^2) = sqrt(1 + 4 + 9) = sqrt14 units.

d = sqrt(x^2 + y^2 + z^2)

The midpoint of the segment joining (x1,y1,z1) and (x2,y2,z2) is ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2). = ((2+6)/2, (3+7)/2, (4+8)/2) = (4, 5, 6).

M = ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2)

Direct substitution gives 0/0, so factorise the numerator: x^2 - 4 = (x - 2)(x + 2). Then (x^2 - 4)/(x - 2) = x + 2 for x not equal to 2. Taking the limit as x approaches 2 gives 2 + 2 = 4.

a^2 - b^2 = (a - b)(a + b)

The standard limit is lim as x approaches 0 of (sin x)/x = 1 (with x in radians). This is a fundamental result used throughout limits and derivatives of trigonometric functions.

lim x->0 (sin x)/x = 1

By definition f'(x) = lim as h approaches 0 of [f(x + h) - f(x)]/h = lim [ (x + h)^2 - x^2 ]/h = lim [x^2 + 2xh + h^2 - x^2]/h = lim [2xh + h^2]/h = lim (2x + h) = 2x. So f'(x) = 2x.

f'(x) = lim h->0 [f(x+h) - f(x)]/h

Using the power rule d/dx(x^n) = n x^(n-1) term by term: dy/dx = 3x^2 + 3(2x) - 5 + 0 = 3x^2 + 6x - 5.

d/dx(x^n) = n x^(n-1)

Using the product rule with u = sin x (u' = cos x) and v = cos x (v' = -sin x): dy/dx = cos x cos x + sin x (-sin x) = cos^2 x - sin^2 x = cos 2x. (Alternatively, y = (1/2) sin 2x, whose derivative is cos 2x.)

cos^2 x - sin^2 x = cos 2x

The first 10 natural numbers are 1, 2, ..., 10. Their sum = n(n + 1)/2 = 10 x 11/2 = 55. Mean = sum/number of observations = 55/10 = 5.5.

mean = (sum of observations)/n

Variance is the arithmetic mean of the squares of the deviations of the observations from their mean; for n observations, variance = (1/n) sum of (xi - mean)^2. Standard deviation is the positive square root of the variance; it has the same units as the observations and measures the spread or dispersion of the data about the mean.

SD = sqrt(variance)

The sample space is {1, 2, 3, 4, 5, 6}, so total outcomes = 6. The prime numbers on a die are 2, 3 and 5, which is 3 favourable outcomes. Probability = favourable/total = 3/6 = 1/2.

P(E) = favourable outcomes / total outcomes

Taxonomy is the branch of biology dealing with the identification, nomenclature and classification of organisms based on their characteristics. A taxon is a unit or group of organisms of any rank in classification (for example, mammals, insects, dogs). Taxonomic hierarchy is the arrangement of taxonomic categories in a definite descending order, from kingdom down to species.

Binomial nomenclature is the system of naming organisms with two words, given by Carolus Linnaeus, where the first word is the genus and the second is the species (for example, Mangifera indica for mango). Rules: (1) the generic name begins with a capital letter and the specific name with a small letter; (2) the biological name is printed in italics (or underlined separately when handwritten) and is usually in Latin.

Kingdom Monera includes bacteria, which are prokaryotic (they lack a true membrane-bound nucleus and membrane-bound organelles). They are microscopic, unicellular organisms with a cell wall (except mycoplasma). Their nutrition may be autotrophic (photosynthetic or chemosynthetic) or heterotrophic (saprophytic or parasitic). They are grouped as Archaebacteria (live in extreme habitats) and Eubacteria (true bacteria).

Kingdom Protista includes single-celled eukaryotes. They have a well-defined nucleus and membrane-bound organelles. Some have flagella or cilia for locomotion, and they may be autotrophic or heterotrophic. This kingdom forms a link between plants, animals and fungi. Examples: Chrysophytes (diatoms), dinoflagellates, Euglenoids (Euglena), slime moulds and protozoans (Amoeba, Paramecium).

Viruses are non-cellular infectious agents that are inert outside the host but reproduce inside a living host cell. A virus has a protein coat called a capsid enclosing genetic material of either DNA or RNA (never both). Viroids are the smallest infectious agents, made of free naked RNA without a protein coat (e.g. potato spindle tuber disease). Lichens are symbiotic associations between an alga (phycobiont) and a fungus (mycobiont) and are good indicators of air pollution.

Bryophytes are called the amphibians of the plant kingdom because they can live in soil but need water for sexual reproduction, as the male gametes (antherozoids) require a film of water to swim to the female sex organ. They lack true vascular tissue and true roots (they have rhizoids). Examples: Funaria (moss), Marchantia and Riccia (liverworts).

Pteridophytes are the first terrestrial plants to have true vascular tissues (xylem and phloem). The plant body is a sporophyte differentiated into true root, stem and leaves. They reproduce by spores produced in sporangia, and water is required for fertilisation. Examples: Selaginella, Equisetum and ferns such as Pteris and Dryopteris.

Diploblastic animals have cells arranged in two embryonic germ layers, an outer ectoderm and an inner endoderm (e.g. coelenterates). Triploblastic animals have three germ layers: ectoderm, mesoderm and endoderm (e.g. from flatworms upward). A coelomate animal is one that has a true body cavity (coelom) lined by mesoderm, for example annelids, arthropods and chordates.

Porifera (sponges): (1) they are primitive multicellular animals with a cellular level of organisation and body with pores (ostia) and a canal system; (2) the body is supported by spicules and they show intracellular digestion (e.g. Sycon, Spongilla). Coelenterata (Cnidaria): (1) they are diploblastic with tissue-level organisation and radial symmetry; (2) they possess stinging cells called cnidoblasts and show two body forms, polyp and medusa (e.g. Hydra, jellyfish).