WMA13 Pure Mathematics 3 (P3): January 2022 Paper 01

Find, using calculus, the x coordinate of the stationary point on the curve with equation

[4]

(a)Show that the equation

can be written in the form

where k is a constant to be found.[3]

(b)Hence find the smallest positive solution of the equation

giving your answer, in degrees, to one decimal place.[2]

(i)Find, in simplest form,

[2]

(ii)Show, by algebraic integration, that

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

The growth of a weed on the surface of a pond is being studied.

The surface area of the pond covered by the weed, A m², is modelled by the equation

where p is a positive constant and t is the number of days after the start of the study.

Given that

• 30 m² of the surface of the pond was covered by the weed at the start of the study

• 50 m² of the surface of the pond was covered by the weed T days after the start of the study

(a)show that p = 2.4[2]

(b)find the value of T, giving your answer to one decimal place.

[4]

The weed grows until it covers the surface of the pond.

(c)Find, according to the model, the maximum possible surface area of the pond.[1]

Figure 1 shows a sketch of part of the curve with equation

The curve cuts the negative x‑axis at the point P, as shown in Figure 1.

(a)Show that the x coordinate of P lies in the interval [–1.25, –1.2][2]

The curve cuts the positive x‑axis at the point Q, also shown in Figure 1.

Using the iterative formula

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

(ii)find, by continued iteration, the x coordinate of Q. Give your answer to 4 decimal places.[3]

The curve has a maximum turning point at M, as shown in Figure 1.

(c)Using calculus and showing each stage of your working, find the x coordinate of M.[4]

The function f is defined by

(a)Show, by using calculus, that f is a decreasing function.[3]

(b)Find f^–1[3]

(c)(i)Show that ff(x) = where a, b and c are constants to be found.

(ii)Deduce the range of ff.[5]

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

(a)State the coordinates of the vertex, V, of the graph.[2]

(b)Solve, using algebra,

[4]

(c)Sketch the graph with equation

stating the coordinates of the local maximum point and each local minimum point.[4]

A dose of antibiotics is given to a patient.

The amount of the antibiotic, x milligrams, in the patient's bloodstream t hours after the dose was given, is found to satisfy the equation

(a)Show that this equation can be written in the form

where p and q are constants to be found. Give the value of p to the nearest whole number and the value of q to 2 significant figures.[4]

(b)With reference to the equation in part (a), interpret the value of the constant p.[1]

When a different dose of the antibiotic is given to another patient, the values of x and t satisfy the equation

(c)Use calculus to find, to 2 significant figures, the value of dxdt when t = 5[3]

Solutions relying entirely on calculator technology are not acceptable.

(i)Solve, for 0 < x  π, the equation

giving the answers, as appropriate, to 3 significant figures.[4]

(ii)Prove that

[4]

Figure 3 shows a sketch of the curve C with equation

(a)Show that

[4]

Given that the straight line with equation x = k, where k is a constant, cuts C at exactly two points,

(b)find the range of possible values for k.[3]