WMA13 Pure Mathematics 3 (P3): Oct/Nov 2024 Paper 01

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

Solutions relying entirely on calculator technology are not acceptable.

Solve, for 0 < θ  360° , the equation

giving your answers to one decimal place.[5]

Figure 1 shows a sketch of the curve with equation

(a)Find dydx in terms of y.[2]

The point P lies on the curve and is shown in Figure 1.

Given that the tangent to the curve at P is parallel to the y‑axis,

(b)find the coordinates of P.[3]

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

Solutions relying entirely on calculator technology are not acceptable.

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

(a)Solve the equation

[3]

(b)Find the set of values of x for which

[4]

The number of bacteria on a surface is being monitored.

The number of bacteria, N, on the surface, t hours after monitoring began is modelled by the equation

Use the equation of the model to answer parts (a) to (c).

(a)Find the initial number of bacteria on the surface.[1]

(b)Show that the equation of the model can be written in the form

where a and b are constants to be found. Give the value of b to 2 decimal places.[3]

(c)Hence find the rate of growth of bacteria on the surface exactly 5 hours after monitoring began.[2]

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

Solutions relying entirely on calculator technology are not acceptable.

(a)Show that sin 3x can be written in the form

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

(b)Hence or otherwise, solve, for 0 < θ  360° , the equation

giving your answers, in degrees, to one decimal place as appropriate.[4]

The functions f and g are defined by

(a)Find gf(2)[2]

(b)Find f^–1[3]

(c)Solve the equation

[3]

The curve C has equation y = f (x) , where

(a)Show that

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

The point P, shown in Figure 3, is the minimum turning point on C.

(b)Find the coordinates of P.[2]

(c)Hence find the range of the function g defined by

[2]

The point Q with coordinates  1 2 , 3 8  lies on C.

(d)Find the coordinates of the point to which Q is mapped when C is transformed to the curve with equation

[2]

The heart rate of a horse is being monitored. The heart rate H, measured in beats per minute (bpm), is modelled by the equation

where t minutes is the time after monitoring began.

Figure 4 is a sketch of H against t.

Use the equation of the model to answer parts (a) to (e).

(a)State the initial heart rate of the horse.[1]

In the long term, the heart rate of the horse approaches L bpm.

(b)State the value of L.[1]

The heart rate of the horse reaches its maximum value after T minutes.

(c)Find the value of T, giving your answer to 3 decimal places.

[5]

The heart rate of the horse is 37 bpm after M minutes.

(d)Show that M is a solution of the equation

[2]

Using the iteration formula

(e)(i)find, to 4 decimal places, the value of t₂

(ii)find, to 4 decimal places, the value of M[3]

Figure 5 shows a sketch of part of the curve C with equation y = f (x) where

(a)Find f′ (x) , giving the answer in simplest form.[3]

The line l is the normal to C at the point P(2, 6)

(b)Show that an equation for l is

[3]

(c)Write f (x) in the form Ax + B + D2x + 1 where A, B and D are constants.[3]

The region R, shown shaded in Figure 5, is bounded by C, l and the x‑axis.

(d)Use algebraic integration to find the exact area of R, giving your answer in the form P + Q ln 3 , where P and Q are rational constants.

[5]