WMA13 Pure Mathematics 3 (P3): January 2020 Paper 01

A population of a rare species of toad is being studied.

The number of toads, N, in the population, t years after the start of the study, is modelled by the equation

According to this model,

(a)calculate the number of toads in the population at the start of the study,[1]

(b)find the value of t when there are 420 toads in the population, giving your answer to 2 decimal places.[4]

(c)Explain why, according to this model, the number of toads in the population can never reach 500[1]

The function f and the function g are defined by

(a)Find, in simplest form, the value of fg(e²)[2]

(b)Find f^–1[3]

(c)Hence, or otherwise, find all real solutions of the equation

[3]

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

The line passes through the points (0, 4) and (6, 0) 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 constants to be found.[3]

(i)f(x) = x ≠ 3

(a)Find f ′(x) in the form P (x)Q (x) where P(x) and Q(x) are fully factorised quadratic

expressions.

(b)Hence find the range of values of x for which f(x) is increasing.[6]

(ii)

The curve with equation y = g(x) has a maximum at the point M.

Show that the x coordinate of M satisfies the equation

where k is a constant to be found.[5]

(a)Use the substitution t = tan x to show that the equation

can be written in the form

[4]

(b)Hence solve, for 0  x < 360°, the equation

Show each stage of your working and give your answers to one decimal place.[4]

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

The vertex of the graph is at point P as shown.

(a)State the coordinates of P.[2]

(b)Solve the equation f(x) = 3x – 2[4]

Given that the equation

where k is a constant, has exactly two roots,

(c)find the range of values of k.[3]

Figure 3 shows a sketch of part of the curve with equation

where x is measured in radians.

The curve crosses the x‑axis at the point P, as shown in Figure 3.

Given that the x coordinate of P is α,

(a)show that α lies between 0.8 and 0.9[2]

The iteration formula

can be used to find an approximate value for α.

(b)Using this iteration formula with x₁= 0.8 find, to 4 decimal places, the value of

(i)x₂

(ii)x₅[3]

The point Q and the point R are local minimum points on the curve, as shown in Figure 3.

Given that the x coordinates of Q and R are β and λ respectively, and that they are the two smallest values of x at which local minima occur,

(c)find, using calculus, the exact value of β and the exact value of λ.[6]

(i)Find, using algebraic integration, the exact value of

giving your answer in simplest form.[4]

(ii)h(x) =₂ x > 1

Given h(x) = Ax + B + C(x − 1 )₂ where A, B and C are constants to be found, find

[6]

(a)Express f(θ) in the form R cos (θ + 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]

The curve with equation y = cos θ is transformed onto the curve with equation y = f(θ) by a sequence of two transformations.

Given that the first transformation is a stretch and the second a translation,

(b)(i)describe fully the transformation that is a stretch,

(ii)describe fully the transformation that is a translation.[2]

Given

(c)find the range of g.[2]