WMA13 Pure Mathematics 3 (P3): Oct/Nov 2022 Paper 01

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

Solutions relying entirely on calculator technology are not acceptable.

(a)Show that

where A, B and C are integers to be found.[4]

(b)Hence, find

giving your answer in the form p + ln q , where p and q are integers.[5]

The functions f and g are defined by

(a)Find the range of f[2]

(b)(i)Find f⁻¹(x)

(ii)Write down the domain of f⁻¹[3]

(c)Find fg(−π)[2]

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

Solutions relying entirely on calculator technology are not acceptable.

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

The curve has a maximum turning point at A and a minimum turning point at (2, 0)

(a)Use calculus to find the exact coordinates of A.[5]

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

(b)state the range of possible values for k.[2]

(a)Express x in terms of y.[2]

(b)Hence, giving your answer in terms of x, find dydx[3]

The profit made by a company, £ P million, t years after the company started trading, is modelled by the equation

The graph of P against t is shown in Figure 2.

According to the model,

(a)show that exactly one year after it started trading, the company had made a loss of approximately £ 830 000[2]

A manager of the company wants to know the value of t for which P = 0

(b)Show that this value of t occurs in the interval [6, 7][2]

(c)Show that the equation P = 0 can be expressed in the form

[2]

(d)Using the iteration formula

find the value of t₂ and the value of t₆ , giving your answers to 3 decimal places.[3]

(e)Hence find, according to the model, how many months it takes in total, from when the company started trading, for it to make a profit.[2]

where a, b and c are integers to be found.[6]

Figure 3 shows a sketch of the graph of C₁ with equation

(a)Write down the coordinates of

(i)the vertex of C₁

(ii)the intersection of C₁ with the y-axis.[2]

(b)Find the x coordinates of the intersections of C₁ with the x-axis.[2]

Diagram 1, shown on page 21, is a copy of Figure 3.

(c)On Diagram 1, sketch the curve C₂ with equation

Identify clearly the coordinates of any points of intersection of C₂ with the coordinate axes.[3]

(d)Find the coordinates of the points of intersection of C₁ and C₂ (Solutions relying entirely on calculator technology are not acceptable.)[5]

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

Solutions relying entirely on calculator technology are not acceptable.

(a)Express 8 sin x − 15 cos x in the form R sin (x − α) , where R > 0 and 0 < α < π2

Give the exact value of R, and give the value of α, in radians, to 4 significant figures.[3]

(b)Find

(i)the minimum value of f(x)

(ii)the smallest value of x at which this minimum value occurs.[4]

(c)State the y coordinate of the minimum points on the curve with equation

[1]

(d)State the smallest value of x at which a maximum point occurs for the curve with equation

[1]

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

Solutions relying entirely on calculator technology are not acceptable.

Given that cos 2θ − sin 3θ ≠ 0

(a)prove that

[4]

(b)Hence solve, for 0 < θ  360°

Give your answers to one decimal place.[5]