Aniline is less basic than ethylamine (and than ammonia) because in aniline the lone pair of electrons on the nitrogen atom is delocalised (drawn) into the benzene ring by resonance. As a result the lone pair is less available for donation to a proton, so aniline accepts a proton less readily and is a weaker base. In ethylamine, the alkyl group has an electron-releasing (+I) effect that increases the electron density on nitrogen and makes the lone pair more available, so it is more basic.

The coupling reaction is a reaction in which a diazonium salt reacts with an electron-rich aromatic compound such as phenol or an aromatic amine to form a brightly coloured azo compound (containing the -N=N- linkage). For example, benzenediazonium chloride reacts with phenol in a mildly alkaline medium to give p-hydroxyazobenzene (an orange dye). These coupling reactions are used to prepare a large number of azo dyes.

Based on their behaviour on hydrolysis, carbohydrates are classified as: (1) monosaccharides - simple sugars that cannot be hydrolysed further into smaller units (e.g. glucose, fructose); (2) oligosaccharides - which give 2 to 10 monosaccharide units on hydrolysis, the most common being disaccharides (e.g. sucrose, which gives glucose and fructose, and maltose); and (3) polysaccharides - which give a large number of monosaccharide units on hydrolysis (e.g. starch, cellulose and glycogen).

A reducing sugar is a carbohydrate that has a free aldehyde or ketone group (a free hemiacetal), so it can reduce Tollens' reagent and Fehling's solution; all monosaccharides (glucose, fructose) and many disaccharides (maltose, lactose) are reducing sugars. A non-reducing sugar has no free aldehyde or ketone group (the reducing groups of both units are involved in the glycosidic bond), so it does not reduce Tollens' or Fehling's reagents; sucrose is the common example of a non-reducing sugar.

Evidence for an aldehyde (-CHO) group in glucose: (1) it reduces Tollens' reagent to give a silver mirror and Fehling's solution to give a red precipitate; (2) it reacts with hydroxylamine to form an oxime and adds one molecule of HCN to form a cyanohydrin, confirming a carbonyl group. Evidence for alcohol (-OH) groups: glucose reacts with acetic anhydride to form a penta-acetate, showing the presence of five -OH groups. Thus glucose is a polyhydroxy aldehyde.

(1) Starch is a polymer of alpha-glucose units and consists of two components, amylose (a linear chain) and amylopectin (a branched chain), joined by alpha-glycosidic linkages; cellulose is a straight-chain polymer of beta-glucose units joined by beta-glycosidic linkages. (2) Starch is the main storage carbohydrate (food reserve) in plants and can be digested by humans, whereas cellulose is a structural material (the main component of plant cell walls) and cannot be digested by humans (we lack the enzyme cellulase).

(1) Primary structure - the specific sequence in which the amino acids are linked in the polypeptide chain. (2) Secondary structure - the local folding of the chain into shapes such as the alpha-helix or beta-pleated sheet, held together by hydrogen bonds. (3) Tertiary structure - the overall three-dimensional folding of the whole polypeptide chain, giving a globular or fibrous shape. (4) Quaternary structure - the arrangement and association of two or more polypeptide chains (subunits), as in haemoglobin.

Enzymes are biological catalysts that speed up the biochemical reactions occurring in living organisms; chemically almost all enzymes are proteins (globular proteins). Characteristics: (1) they are highly specific, each enzyme usually catalysing only one particular reaction or type of reaction (lock-and-key specificity); and (2) they are highly efficient and work best under mild conditions of an optimum temperature (around body temperature) and an optimum pH; they are denatured and lose activity outside these conditions.

Vitamins are organic compounds required in small amounts in the diet for normal growth and health. They are classified into two groups: (1) fat-soluble vitamins (A, D, E and K), which are soluble in fats and oils and are stored in the liver and fatty tissues; and (2) water-soluble vitamins (the B group and C), which are soluble in water and must be supplied regularly in the diet as they are not stored. Deficiency of vitamin C causes scurvy, and deficiency of vitamin D causes rickets in children.

(1) DNA (deoxyribonucleic acid) contains the sugar deoxyribose, whereas RNA (ribonucleic acid) contains the sugar ribose. (2) DNA contains the nitrogenous bases adenine, guanine, cytosine and thymine, whereas RNA contains adenine, guanine, cytosine and uracil (uracil replaces thymine). (3) DNA is usually a double-stranded helix and stores the genetic (hereditary) information, whereas RNA is usually single-stranded and mainly takes part in protein synthesis.

A relation R on a set A is reflexive if (a, a) belongs to R for every a in A. It is symmetric if (a, b) in R implies (b, a) in R. It is transitive if (a, b) in R and (b, c) in R together imply (a, c) in R. A relation that is reflexive, symmetric and transitive at the same time is called an equivalence relation.

A function f from A to B is one-one (injective) if distinct elements of A have distinct images, that is, f(x1) = f(x2) implies x1 = x2. It is onto (surjective) if every element of B has at least one pre-image in A, that is, the range of f equals the codomain B. A function that is both one-one and onto is called bijective (a one-to-one correspondence).

(g o f)(x) means g(f(x)). So (g o f)(x) = g(f(x)) = g(x + 1) = 2(x + 1) = 2x + 2. (f o g)(x) means f(g(x)). So (f o g)(x) = f(g(x)) = f(2x) = 2x + 1. Note that (g o f)(x) is not equal to (f o g)(x), so composition of functions is not commutative in general.

(g o f)(x) = g(f(x))

The principal value branch of sin inverse is [-pi/2, pi/2]. Since sin(pi/6) = 1/2, the principal value of sin inverse (1/2) = pi/6. The principal value branch of cos inverse is [0, pi]. Since cos(2pi/3) = -1/2, the principal value of cos inverse (-1/2) = 2pi/3.

The principal values are: tan inverse (1) = pi/4, cos inverse (1/2) = pi/3, and sin inverse (1/2) = pi/6. Adding them: pi/4 + pi/3 + pi/6. Taking the LCD 12: 3pi/12 + 4pi/12 + 2pi/12 = 9pi/12 = 3pi/4.

Let sin inverse (x) = theta, so sin theta = x, where theta lies in [-pi/2, pi/2]. Then x = sin theta = cos(pi/2 - theta). Since (pi/2 - theta) lies in [0, pi], we have cos inverse (x) = pi/2 - theta = pi/2 - sin inverse (x). Rearranging gives sin inverse (x) + cos inverse (x) = pi/2.

sin inverse (x) + cos inverse (x) = pi/2

The principal value branch of cos inverse is [0, pi], but 7pi/6 does not lie in this interval. First find cos(7pi/6) = cos(pi + pi/6) = -cos(pi/6) = -sqrt(3)/2. Now cos inverse (-sqrt(3)/2) = 5pi/6, since 5pi/6 lies in [0, pi] and cos(5pi/6) = -sqrt(3)/2. Therefore cos inverse (cos(7pi/6)) = 5pi/6.

For sin inverse (x): the domain is [-1, 1] and the principal value range is [-pi/2, pi/2]. For tan inverse (x): the domain is all real numbers R and the principal value range is the open interval (-pi/2, pi/2). Inverse trigonometric functions are defined by restricting the domain of the corresponding trigonometric functions so that they become one-one.

A diagonal matrix is a square matrix in which all the non-diagonal (off-diagonal) elements are zero. A scalar matrix is a diagonal matrix in which all the diagonal elements are equal (to the same non-zero scalar). An identity (unit) matrix is a scalar matrix in which all the diagonal elements are equal to 1; it is denoted by I and acts as the multiplicative identity, since AI = IA = A.

Two matrices are equal when their corresponding elements are equal. Comparing: x + y = 6 and x - y = 2. Adding these two equations: 2x = 8, so x = 4. Subtracting the second from the first: 2y = 4, so y = 2. Hence x = 4 and y = 2.