OCR A Level Maths B (MEI) Pure Mathematics and Comprehension June 2019 question paper H640/03

The function f (x) is defined for all real x by

f (x) = 3 x - 2.

Find an expression for f^-1(x).

Sketch the graphs of y = f(x) and y = f^-1(x) on the same diagram.

Find the set of values of x for which f(x) > f^-1(x).

Find the transformation which maps the curve y = x² to the curve y = x² + 8x - 7.

Write down the coordinates of the turning point of y = x² + 8x - 7.

In this question you must show detailed reasoning.

A student’s attempt to prove by contradiction that there is no largest prime number is shown below.

If there is a largest prime, list all the primes. Multiply all the primes and add 1. The new number is not divisible by any of the primes in the list and so it must be a new prime.

The proof is incorrect and incomplete.

Write a correct version of the proof.

A circle has centre C (10, 4). The x-axis is a tangent to the circle, as shown in Fig. 6.

Find the equation of the circle.

Show that the line y = x is not a tangent to the circle.

Write down the position vector of the midpoint of OC.

In this question you must show detailed reasoning.

Express ln 3 × ln 9 × ln 27 in terms of ln3.

Hence show that ln 3 × ln 9 × ln 27 > 6.

Find the area of triangle ODE.

Find the area of the region bounded by the curve y = x³, the tangent at B and the y-axis.

In this question you must show detailed reasoning.

The curve xy + y² = 8 is shown in Fig. 9.

Find the coordinates of the points on the curve at which the normal has gradient 2.

Show that f(x) = is an increasing function for all values of x.

Show that the equation of the line in Fig. C2 is ry + hx = hr, as given in line 24.

Show that the cross-sectional area in Fig. C3.2 is π x (2 r - x).

Fig. C3.1,

Fig. C3.3.

Hence show that V = r²h, as given in line 41.

A typical tube of toothpaste measures 5.4 cm across the straight edge at the top and is 12 cm high. It contains 75 ml of toothpaste so it needs to have an internal volume of 75cm³.

Comment on the accuracy of the formula V = r²h, as given in line 41, for the volume in this

case.