WMA13 Pure Mathematics 3 (P3): Oct/Nov 2020 Paper 01

Solve, for 0  x < 360°, the equation

giving your solutions to one decimal place.

[5]

A scientist monitored the growth of bacteria on a dish over a 30‑day period.

The area, N mm², of the dish covered by bacteria, t days after monitoring began, is modelled by the equation

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

where a and b are constants to be found. Give the value of a to the nearest integer and give the value of b to 3 significant figures.[4]

(b)Use the model to find the area of the dish covered by bacteria 30 days after monitoring began. Give your answer, in mm², to 2 significant figures.[2]

Figure 1 shows a sketch of a curve with equation y = f(x) where

(a)Find, in simplest form, f ʹ(x).[4]

(b)Hence find the range of f.[3]

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

(a)Find f f (6)[2]

(b)Solve the equation f(x) = 5x[2]

Given that the equation f(x) = k , where k is a constant, has exactly two roots,

(c)state the set of possible values of k.[2]

The graph with equation y = f(x) is transformed onto the graph with equation y = af(x − b)

The vertex of the graph with equation y = af(x − b) is (6, 3).

Given that a and b are constants,

(d)find the value of a and the value of b.[2]

(a)Show that

[4]

(b)Hence find, using algebraic integration,

[4]

Figure 3 shows a sketch of curve C₁ with equation y = 5e^x−1 + 3

The point P lies on C₁ and has y coordinate 18

(a)Find the x coordinate of P, writing your answer in the form ln k, where k is a constant to be found.[3]

The curve C₁ meets the curve C₂ at x = α and at x = β , as shown in Figure 3.

(b)Using a suitable interval and a suitable function that should be stated, show that to 3 decimal places α = 1.134[3]

The iterative equation

is used to find an approximation to β .

Using this iterative formula with x₁ = −3

(c)find the value of x₂ and the value of β , giving each answer to 6 decimal places.[3]

(a)Express cos x + 4 sin x in the form R cos (x − α) where 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]

A scientist is studying the behaviour of seabirds in a colony.

She models the height above sea level, H metres, of one of the birds in the colony by the equation

where t seconds is the time after it leaves the nest.

Find, according to the model,

(b)the minimum height of the seabird above sea level, giving your answer to the nearest cm,[2]

(c)the value of t, to 2 decimal places, when H = 10[4]

(i)The curve C has equation y = g(x) where

(a)Find gʹ(x)[2]

(b)Hence find the x coordinate of the stationary point of C.[3]

(ii)A different curve has equation

where f (x) is a function of e^x that should be found.[4]

(a)Given that

find the value of the constant P and show that Q = 5[4]

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

g(x) = − 3 < x < 3.5 x ∈ 

(b)Find the equation of the tangent to C at the point where x = 2 Give your answer in the form y = mx + c , where m and c are constants to be found.[5]

Figure 4 shows a sketch of the curve C. The region R, shown shaded in Figure 4, is bounded by C, the y‑axis, the x‑axis and the line with equation x = 2

(c)Find the exact area of R, writing your answer in the form a + b ln2 , where a and b are constants to be found.[5]