WMA13 Pure Mathematics 3 (P3): May/June 2025 Paper 01

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The functions f and g are defined by

(a)Find the value of gf (1)[2]

(b)Find the range of f[2]

(c)Find f^–1(x)[2]

(d)Solve f^–1(x) = f(x)[2]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

(a)Express f (x) in the form R cos(x + α) where R and α are constants, R > 0

and 0 < α < π2

Give the exact value of R and give the value of α, in radians, to 3 decimal places.[3]

(b)Using the answer to part (a), find

(i)the minimum value of g (x), giving your answer as a fully simplified fraction,

(ii)the smallest positive value of x for which this minimum value occurs, giving your answer to 3 decimal places.[4]

Figure 1 shows a linear relationship between log₁₀ y and log₁₀ x

The line passes through the points (–3.5, 0) and (0, –2) as shown.

(a)Find an equation linking log₁₀ y with log₁₀ x[2]

(b)Hence, or otherwise, express y in the form px^q where p and q are rational constants.[3]

The function f is defined by

(a)Show that

[3]

(b)Hence find fʹ(x) giving your answer in simplest form.[2]

Find

(i) ∫[2]

(ii)²_3/2

[2]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The temperature, θ°C, of a computer processor, t minutes after the computer is switched off, is modelled by the equation

where A and k are positive constants.

Given that the temperature of the processor was 75°C when the computer was switched off,

(a)find the value of A.[2]

Given also that it takes 5 minutes for the temperature of the processor to decrease from 75°C to 25°C,

(b)find the value of k, giving your answer to 3 significant figures.[3]

At time T minutes, the temperature of the processor is decreasing at a rate of 9°C per minute.

(c)Find the value of T according to the model, giving your answer to 2 decimal places.[3]

A continuous curve has equation

The curve has a stationary point at the point P.

(a)Show, using calculus, that the x coordinate of P is a solution of the equation

[4]

Using the iteration formula

(b)find the value of

(i)x₂

(ii)x₄

giving your answers to 4 decimal places.[3]

(c)Using a suitable interval and a suitable function which should be stated, show that the x coordinate of P is 0.430 correct to 3 decimal places.[2]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

(a)Prove that

[3]

(b)Hence solve, for 0 < θ < π2

giving your answers to 2 decimal places.[5]

Figure 2 shows a sketch of the curve C with equation

The curve intersects the y-axis at the points A and B as shown.

(a)Find the exact value of the y coordinate of

• point A

• point B[3]

(b)Show that

where p and q are integers to be found.[4]

The normal to C at A and the tangent to C at B intersect at the point D.

Using

• the answer to part (b)

• the sketch of curve C in Figure 2

(c)find, in simplest form, the exact x coordinate of D.[4]

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

and k is a positive constant.

The graph

• cuts the y-axis at the point A

• has a vertex at the point P

(a)Find, in simplest form in terms of k,

(i)the y coordinate of A

(ii)the coordinates of P[3]

(b)Find, in terms of k, the range of values of x which satisfy[3]

Given that the line with equation y = 3x + 1 intersects the graph of y = f (x) at 2 distinct points,

(c)find the range of values of k.[4]