Paper 3: Pure and Statistics

Find the exact value of sin π/3 cos π/6.

Which transformation maps the graph of y = f(x) onto y = f(2x)?

Given y = (x² + 1)/(2x - 1), find dy/dx in its simplest form and its value when x = 1.

(a) Prove that n² + n is even for all integers n. (b) A student claims that n² + n + 41 is prime for every positive integer n. Show that the claim is false.

A sector OAB of a circle has radius 8 cm and angle 1.2 radians. (a) Find the arc length AB and the area of the sector. (b) Find the area of the segment between the chord AB and the arc AB.

(a) Find the first four terms, in ascending powers of x, of the expansion of (1 + 2x)⁸. (b) Use your expansion with a suitable value of x to estimate 1.02⁸, giving your answer to 6 decimal places.

A spherical balloon is inflated so that its volume increases at a constant rate of 10 cm³ s^-1. (a) Find the rate of increase of the radius when the radius is 5 cm. (b) Find the rate of increase of the surface area at this instant. [V = 4/3π r³, S = 4π r²]

(a) Sketch the graph of y = |x² - 4|, stating the coordinates of the points where it meets the axes. (b) Solve |x² - 4| = 3x.

(a) Find the exact value of ∫₀^π/2 sin² x dx. (b) Find ∫ 6x/(3x² + 1) dx. (c) A curve is given parametrically by x = t², y = t³ for t ≥ 0. Find the area of the region bounded by the curve, the x-axis and the line x = 4.

A researcher selects every 20th name from an ordered list after a random start. This is:

For a normal distribution, approximately what percentage of values lie within 2 standard deviations of the mean?

The times, x minutes, taken by 40 people to complete a puzzle are summarised by Σ x = 1520, Σ x² = 59 600. (a) Find the mean and standard deviation. (b) An outlier is defined as a value more than 2 standard deviations from the mean. Determine whether a time of 53 minutes is an outlier.

Events A and B are such that P(A) = 0.4, P(B) = 0.5, P(A ∩ B) = 0.15. (a) Find P(A ∪ B). (b) Find P(A | B). (c) Determine, with a reason, whether A and B are independent. (d) Find the probability that neither A nor B occurs.

A spinner lands on red with probability 0.25. It is spun 15 times. X is the number of reds. (a) State two assumptions needed to model X as B(15, 0.25). (b) Find P(X = 4). (c) Find P(X ≥ 3). (d) Find the mean and variance of X.

The lifetime, L hours, of a type of light bulb is modelled by L ~ N(800, 50²). (a) Find P(L > 900). (b) Find P(750 < L < 880). (c) The manufacturer changes the design. The mean stays at 800 hours but now only 5% of bulbs last less than 720 hours. Find the new standard deviation.

A machine fills bottles with a mean volume of 500 ml and standard deviation 4 ml. After servicing, a random sample of 25 bottles has mean volume 498.2 ml. Assume the standard deviation is unchanged. (a) Test, at the 5% significance level, whether the mean volume has decreased. (b) Find the critical region for the sample mean for this test. (c) State the probability of a Type I error in this test.

A student collects data on daily hours of sunshine and the number of ice creams sold at a kiosk on 12 randomly chosen days. The product moment correlation coefficient is r = -0.62. (a) Test, at the 5% significance level, whether there is evidence of correlation between hours of sunshine and ice cream sales. (Critical value for a two-tailed test at 5% with n = 12: 0.5760.) (b) The student says this proves that sunshine reduces ice cream sales. Comment on this claim. (c) The critical value for a two-tailed test at 1% with n = 12 is 0.7079. State, with a reason, whether your conclusion would change at the 1% level.