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

The point P(6, –2) lies on the continuous curve with equation y = f (x), x ∈ .

Find the point to which P is mapped when the curve with equation y = f (x) is transformed to the curve with equation

(a)y = 2f (3x)[2]

(b)y = f (x – 2) + 8[2]

(c)y = f^–1(x)[1]

The equation f (x) = 0 has a single root α.

(a)Show that α lies in the interval [1, 2][2]

(b)Show that the equation f (x) = 0 can be written as

where k is a constant to be found.[2]

(c)Using the iteration formula

with x₁ = 1 and the value of k found in part (b), find, to 4 decimal places,

(i)the value of x₃

(ii)the value of α.[3]

The curve C has equation

(a)Find dydx in terms of y, giving the answer in simplest form.[4]

A point P lies on C.

Given that

• the gradient of the tangent to C at P is 13

• the point P lies above the x‑axis

(b)find the coordinates of P.

(Solutions relying entirely on calculator technology are not acceptable.)[4]

Figure 1 shows a sketch of part of the curve C with equation y = g (x) where

(a)(i)find the values of the constants A, B and C

(ii)show that D = 0[4]

The finite region R, shown shaded in Figure 1, is bounded by C, the x‑axis and the lines with equations x = 2 and x = 3

(b)Find the exact area of region R using algebraic integration. Give the answer in the form α + ln β where α and β are constants to be found.[4]

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

A curve C has equation

The point A with x coordinate π3 lies on C.

(a)Use algebraic differentiation to find the gradient of the tangent to C at A. Give the answer in simplest form.[3]

(b)Express 5 cos 2x – 12 sin 2x in the form

where R > 0 and 0 < α < π2

Find the exact value of R and find the value of α to 3 decimal places.[3]

A point P lies on C.

Given that the gradient of the tangent to C at P is 6

(c)find the greatest possible value for the x coordinate of P. Give the answer to 2 decimal places.[4]

Figure 2 shows a sketch of the curve with equation y = f (x) where

(a)Show that

where P, Q and n are constants to be found.[4]

(b)Hence find

(i)the x coordinates of the stationary points on the curve with equation y = f (x),

(ii)the range of the function g defined by

(Solutions relying entirely on calculator technology are not acceptable.)[4]

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

(a)Write sin 4θ in the form

where P, Q and n are constants to be found.[3]

(b)Use the result from part (a) to show that, for x ≠ kπ2 where k ∈ , the equation

can be written in the form

[3]

(c)Use the result from part (b) to solve, for 0 < x < π, x ≠ π2 , the equation

giving the answers in radians to 3 significant figures.[4]

The functions f and g are defined by

where a is a positive constant.

(a)Find the exact value of fg (2a)[2]

(b)Find f^–1[3]

Figure 3 shows a sketch of the curve with equation y = g (x)

Indicate clearly on your sketch the equation of the horizontal asymptote.[3]

Given that the larger solution is 8

(d)(i)find the value of a,

(ii)find the value of the smaller solution.[4]

The amount of an antibiotic, x milligrams, in the bloodstream of a horse, t hours after the antibiotic had been administered, is given by the formula

where D milligrams is the dose of the antibiotic given to the horse.

A dose of 30 mg of the antibiotic is given to the horse.

(a)Find the amount of the antibiotic in the bloodstream of the horse 8 hours after the dose is given. Give your answer in mg to 2 decimal places.[2]

A second dose of 20 mg is given to the horse 8 hours after the first dose.

(b)Show that the amount of the antibiotic in the bloodstream of the horse, 2 hours after the second dose is given, is 17.5 mg to one decimal place.[1]

No more doses of the antibiotic are given. At time T hours after the second dose is given, the amount of the antibiotic in the bloodstream is 10 mg.

(c)Find the value of T, giving your answer to 2 decimal places.

(Solutions relying entirely on calculator technology are not acceptable.)[4]