OCR A Level Maths A Pure Mathematics and Statistics June 2018 question paper H240/02

Express 2x² - 12x + 23 in the form a(x + b)² + c.

Use your result to show that the equation 2x² - 12x + 23 = 0 has no real roots.

Find the exact length of AB.

Show that ABPQ is a parallelogram.

Assuming Ayesha’s model is correct, how long would you recommend that she spends revising?

State one feature of Dave’s model that is likely to be unrealistic.

Suggest a reason for the shape of Bob’s graph as compared with Ayesha’s graph.

What does Chloe’s model suggest about her attitude to revision?

Charlie claims to have proved the following statement.

“The sum of a square number and a prime number cannot be a square number.”

Give an example to show that Charlie’s statement is not true.

Charlie’s attempt at a proof is below.

Assume that the statement is not true. ⇒ There exist integers n and m and a prime p such that n² + p = m². ⇒ p = m² - n² ⇒ p = (m - n)(m + n) ⇒ p is the product of two integers. ⇒ p is not prime, which is a contradiction. ⇒ Charlie’s statement is true.

Explain the error that Charlie has made.

Given that 853 is a prime number, find the square number S such that S + 853 is also a square number.

In this question you must show detailed reasoning.

Find the x-coordinate of the point where the curve crosses the x axis.

The points A and B lie on the curve and have x coordinates 2 and 4. Show that the line AB is parallel to

the x-axis.

Find the coordinates of the turning point on the curve.

Determine whether this turning point is a maximum or a minimum.

The diagram shows a part ABC of the curve y = 3 - 2x², together with its reflections in the lines y = x, y = - x and y = 0.

Find the area of the shaded region.

The variable X has the distribution N(20, 9).

Find P(X > 25).

Given that P(X > a) = 0.2, find a.

Find b such that P(20 - b < X < 20 + b) = 0.5.

Briony suspects that a particular 6-sided dice is biased in favour of 2. She plans to throw the dice 35 times and note the number of times that it shows a 2. She will then carry out a test at the 4% significance level.

Find the rejection region for the test.

A certain forest contains only trees of a particular species. Dipak wished to take a random sample of 5 trees from the forest. He numbered the trees from 1 to 784. Then, using his calculator, he generated the random digits 14 781 049. Using these digits, Dipak formed 5 three-digit numbers. He took the first, second and third digits, followed by the second, third and fourth digits and so on. In this way he obtained the following list of numbers for his sample.

147 478 781 104 49

Explain why Dipak omitted the number 810 from his list.

Explain why Dipak’s sample is not random.

The mean height of all trees of this species is known to be 4.2 m. Dipak wishes to test whether the mean height of trees in the forest is less than 4.2 m. He now uses a correct method to choose a random sample of 50 trees and finds that their mean height is 4.0 m. It is given that the standard deviation of trees in the forest is 0.8 m.

Carry out the test at the 2% significance level.

Christa used Pearson’s product-moment correlation coefficient, r, to compare the use of public transport with the use of private vehicles for travel to work in the UK.

Using the pre-release data set for all 348 UK Local Authorities, she considered the following four variables.

Explain, in context, why you would expect strong, positive correlation between x and y.

Explain, in context, what kind of correlation you would expect between a and b.

Suggest what effect this outlier is likely to have on the value of r for the 32 London Boroughs.

Suggest what effect this outlier is likely to have on the value of r for the whole country.

What can you deduce about the area of the London Borough represented by the outlier?

Explain your answer.

Find the probability that S is odd.

Find the probability that S is greater than 8, given that S is odd.

Sheila sometimes needs several attempts to start her car in the morning. She models the number of attempts she needs by the discrete random variable Y defined as follows. P(Y = y+1) = (Y = y) for all positive integers y.

Find P(Y = 1).

Give a reason why one of the variables, X or Y, might be more appropriate as a model for the number

of attempts that Sheila needs to start her car.

In this question you must show detailed reasoning.

The probability that Paul’s train to work is late on any day is 0.15, independently of other days.

The number of days on which Paul’s train to work is late during a 450-day period is denoted by the random variable Y. Find a value of a such that P(Y > a) ≈

The number of days on which Paul’s train to work is late during a 50-day period is modelled by the random variable X.

Find the values of r for which P (X = r) ⩽ P (X = r + 1).

Hence find the most likely number of days on which the train will be late during a 50-day period.