WMA13 Pure Mathematics 3 (P3): May/June 2023 Paper 01

(a)Show that g(x) = 0 has a root α in the interval [3, 4][2]

Using the iteration formula

(b)(i)find, to 4 decimal places, the value of x₂

(ii)find, by repeated iteration, the value of α. Give your answer to 4 decimal places.[3]

Figure 1 shows the linear relationship between log₆ T and log₆ x

The line passes through the points (0, 4) and (2, 0) as shown.

(a)(i)Find an equation linking log₆ T and log₆ x

(ii)Hence find the exact value of T when x = 216[3]

(b)Find an equation, not involving logs, linking T with x[3]

(i)Find ddx ln sin²3x writing your answer in simplest form.

[2]

(ii)(a)Find ddx 3x²  4

[2]

(b)Hence show that

where R is an integer to be found.

[3]

The function f is defined by

(a)State the range of f[1]

On the following page there is a diagram, labelled Diagram 1, which shows a sketch of the curve with equation y = f(x).

(b)On Diagram 1, sketch the curve with equation y = f^–1(x).[2]

The curve with equation y = f(x) meets the curve with equation y = f^–1(x) at the point P

Using algebra and showing your working,

(c)find the exact x coordinate of P[3]

y = f (x)

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

(i)Solve, for 0 < x < π

[3]

(ii)Solve, for 0 < θ < 360°

[4]

Figure 2 shows a sketch of the graph y = f(x), where

The vertex of the graph is at point P, shown in Figure 2.

(a)Find the coordinates of P[2]

(b)Find ff(0)[2]

(c)Solve the inequality[2]

(d)Solve the equation[3]

A scientist is studying two different populations of bacteria.

The number of bacteria N in the first population is modelled by the equation

where A and k are positive constants and t is the time in hours from the start of the study.

Given that • there were 2500 bacteria in this population at the start of the study • there were 10 000 bacteria 8 hours later

(a)find the exact value of A and the value of k to 4 significant figures.[3]

The number of bacteria N in the second population is modelled by the equation

where t is the time in hours from the start of the study.

(b)Find the rate of decrease of bacteria in this population exactly 5 hours from the start of the study. Give your answer to 3 significant figures.[2]

When t = T, the number of bacteria in the two different populations was the same.

(c)Find the value of T, giving your answer to 3 significant figures.

[3]

Figure 3 shows a sketch of the curve C with equation y = f(x) , where

(a)Show that

where A is a constant to be found.[4]

(b)Hence find the exact coordinates of the two stationary points on C.[3]

The function g is defined by

(c)Find the coordinates of the maximum stationary point on the curve with equation y = g(x) .[2]

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

(a)Show that

[3]

(b)Hence solve, for 0 < θ < π2

giving your answers to 3 significant figures as appropriate.[5]

(c)Using the result from part (a), or otherwise, find the exact value of

[2]

Figure 4 shows a sketch of the curve with equation

(a)Find giving your answer as a fully simplified fraction. dy[4]

The tangents at points P and Q on the curve are parallel to the y-axis, as shown in Figure 4.

(b)Use the answer to part (a) to find the equations of these two tangents.[4]