Paper 2: Pure Mathematics and Statistics

In the expansion of (2 + kx)⁶, the coefficient of x² is 240. (a) Find the possible values of k. (b) Given k > 0, find the coefficient of x³.

A sector of a circle has area 24 cm² and perimeter 20 cm. Find the two possible pairs of values of the radius r cm and the angle θ radians.

A curve has equation y³ + 3xy - x² = 3. (a) Find dy/dx in terms of x and y. (b) Verify that (1, 1) lies on the curve and find the equation of the tangent there.

(a) Prove by contradiction that if n³ is even then n is even. (b) Prove that there are no integers a and b such that 6a + 9b = 1.

(a) The curves y = sin x and y = cos x intersect at x = π/4 and x = 5π/4. Find the exact area enclosed between the curves for π/4 ≤ x ≤ 5π/4. (b) Use integration by parts to find the exact value of ∫₀^π/4 xcos 2x dx.

The functions f and g are defined by f(x) = e^2x - 3 for x ∈ mathbb R, and g(x) = ln(x + 4) for x > -4. (a) State the range of f. (b) Find f^-1(x) and state its domain. (c) Find gf(x) in its simplest form. (d) Solve gf(x) = 2, giving your answer in exact form and to 3 decimal places. (e) Describe a sequence of two transformations that maps y = e^x onto y = f(x).

A council surveyed residents' ages in two towns using a sample of 80 people from each town, chosen from people walking through the town centre on a Saturday morning.

	Town A	Town B
Median age	32	28
Interquartile range	14	22

(a) Compare the ages in the two towns. (b) Name the sampling method and give one reason why it may not give a representative sample.

The random variable X has probability distribution P(X = x) = kx for x = 1, 2, 3, 4. (a) Find k. (b) Find E(X). (c) Find P(X ≥ 3). (d) Two independent values of X are taken. Find the probability that their sum is 5.

A machine produces bolts, of which 8% are defective. After an adjustment, a random sample of 50 bolts contains 1 defective bolt. (a) Test, at the 5% significance level, whether the adjustment has reduced the proportion of defective bolts. (b) Find the critical region for this test. (c) Explain why the significance level of the test is not exactly 5%.

The heights, H cm, of a species of plant are modelled by H ~ N(25, 4²). (a) Find P(H < 20). (b) Find P(|H - 25| < 3). (c) Six plants are chosen at random. Find the probability that exactly two of them are shorter than 20 cm. (d) Find the height h such that 80% of plants are taller than h. (e) A new fertiliser is used. A random sample of 20 plants has mean height 26.8 cm. Assuming the standard deviation is unchanged, test at the 5% level whether the mean height has increased.

(a) Events A and B are such that P(A) = 0.5, P(B | A) = 0.3 and P(B | A') = 0.6. (i) Draw a tree diagram to show this information. (ii) Find P(B). (iii) Find P(A | B). (iv) Determine whether A and B are independent. (b) For a random sample of 20 students, the product moment correlation coefficient between hours of revision and test score is 0.42. Test, at the 5% level, whether there is positive correlation in the population. (One-tailed critical value: 0.3783.)