OCR A Level Maths B (MEI) Pure Mathematics and Comprehension November 2020 question paper H640/03

The graph of y = | 1 - x | - 2 is shown in Fig. 2.

Determine the set of values of x for which | 1 - x | > 2.

A particular phone battery will last 10 hours when it is first used. Every time it is recharged, it will only last 98% of its previous time.

Find the maximum total length of use for the battery.

Fig. 4 shows the regular octagon ABCDEFGH.

= i = j, where i is a unit vector parallel to the x-axis and j is a unit vector parallel to the y-axis.

Find an exact expression for in terms of i and j.

Fig. 5 shows part of the curve y = cosec x together with the x - and y-axes.

For the section of the curve which is shown in Fig. 5, write down

the equations of the two vertical asymptotes,

the coordinates of the minimum point.

Show that the equation x = cosec x has a root which lies between x = 1 and x = 2.

Use the iteration x_n+1 = cosec(x_n), with x₀ = 1, to find

the values of x₁ and x₂, correct to 5 decimal places,

this root of the equation, correct to 3 decimal places.

There is another root of x = cosec x which lies between x = 2 and x = 3.

Determine whether the iteration x_n+1 = cosec(x_n) with x₀ = 2.5 converges to this root.

Sketch the staircase or cobweb diagram for the iteration, starting with x₀ = 2.5, on the diagram

in the Printed Answer Booklet.

Write down the derivative of e^kx, where k is a constant.

A business has been running since 2009. They sell maths revision resources online.

Give a reason why an exponential growth model might be suitable for the annual profits

for the business.

Estimate the values of A and k.

Use the model to predict the profit in the year 2020.

How reliable do you expect the prediction in part (d) to be? Justify your answer.

The curve y = is shown in Fig. 8.

In this question you must show detailed reasoning.

Find the set of values of x for which the curve is concave downwards.

Show that if a = 1 and b > 1 then a^b < b^a.

Find integer values of a and b with b > a > 1 and a^b not greater than b^a (a counter example

to the conjecture given in lines 7–8).

In this question you must show detailed reasoning.

Show that e^x is an increasing function for all values of x, as stated in line 39.

Show that the only stationary point on the curve y = occurs where x = e, as given in

Show that the stationary point is a maximum.