OCR A Level Maths A Pure Mathematics June 2019 question paper H240/01

In this question you must show detailed reasoning.

Solve the inequality 10x² + x - 2 > 0.

The point A is such that the magnitude of is 8 and the direction of is 240°.

Show the point A on the axes provided in the Printed Answer Booklet.

Find the position vector of point A.

Give your answer in terms of i and j.

The point B has position vector 6 i.

Find the exact area of triangle AOB.

The point C is such that OABC is a parallelogram.

Find the position vector of C.

Give your answer in terms of i and j.

The function f is defined by f(x) = (x - 3)² - 17 for x ⩾ k, where k is a constant.

Given that f^-1(x) exists, state the least possible value of k.

Evaluate ff(5).

Solve the equation f (x) = x.

Explain why your solution to part (c) is also the solution to the equation f(x) = f^-1(x).

Sam starts a job with an annual salary of £16 000. It is promised that the salary will go up by the same amount every year. In the second year Sam is paid £17 200.

Find Sam’s salary in the tenth year.

Find the number of complete years needed for Sam’s total salary to first exceed £500 000.

Comment on how realistic this model may be in the long term.

A curve has equation x³ - 3x²y + y² + 1 = 0.

Find the equation of the normal to the curve at the point (1, 2).

Let f(x) = 2x³ + 3x. Use differentiation from first principles to show that f ′(x) = 6x² + 3.

A cylindrical tank is initially full of water. There is a small hole at the base of the tank out of which the water leaks.

The height of water in the tank is x m at time t seconds. The rate of change of the height of water may be modelled by the assumption that it is proportional to the square root of the height of water.

When t = 100, x = 0.64 and, at this instant, the height is decreasing at a rate of 0.0032ms^-1.

Find an expression for x in terms of t.

Hence determine at what time, according to this model, the tank will be empty.

Express 3 cos 3x + 7 sin 3x in the form R cos(3x - α), where R > 0 and 0 < α < .

Determine the greatest value of 3 cos 3 x + 7 sin 3 x as x varies and give the smallest positive

value of x for which it occurs.

Determine the least value of 3 cos 3 x + 7 sin 3 x as x varies and give the smallest positive value

of x for which it occurs.

Show that θ = 0.4 + sin θ.

Let F (θ) = 0.4 + sin θ.

By considering the value of F ′ (θ) where θ = 1.2, explain why using an iterative method based on the equation in part (a) will converge to the root, assuming that 1.2 is sufficiently close to

the root.

Use the iterative formula θ_n+1 = 0.4 + sin θ_n with a starting value of 1.2 to find the value of θ correct to 4 significant figures.

You should show the result of each iteration.

Use a change of sign method to show that the value of θ found in part (c) is correct to 4

significant figures.

The diagram shows part of the curve y = ln (x - 4).

State the equation of the vertical asymptote to the curve y = ln (x - 4).

Find the total area enclosed by the curve y = ln (x - 4), the x-axis and the lines x = 4.5 x = 7. Give your answer in the form a ln 3 + b ln 2 + c where a, b and c are constants to be

found.

By considering show that the curve is convex for all values of x.