WMA13 Pure Mathematics 3 (P3): Oct/Nov 2021 Paper 01

The function f is defined by

(a)Show that f (x) = 5xx + 3[3]

(b)Find f ^–1[3]

(c)(i)Find, in simplest form, f ′(x).

(ii)Hence, state whether f is an increasing or a decreasing function, giving a reason for your answer.[3]

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

The vertex of the graph is at point P, as shown in Figure 1.

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

(b)(i)State the range of f.

(ii)Find the value of ff(4)[2]

(c)Solve, using algebra and showing your working,[4]

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 (4, 20)

Given that a and b are constants,

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

The total mass of gold, G tonnes, extracted from a mine is modelled by the equation

where t is the number of years after 1st January 1800.

Figure 2 shows a sketch of G against t.

(a)(i)Find the value of k.

(ii)Hence find the year and month in which gold started being extracted from the mine.[3]

(b)Find the total mass of gold extracted from the mine up to 1st January 1870.[2]

There is a limit to the mass of gold that can be extracted from the mine.

(c)State the value of this limit.[1]

(a)Show that the equation

can be written in the form

[4]

(b)Hence, or otherwise, solve for 0  x  360°

giving your answers to one decimal place.[3]

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

[4]

(ii)Find, in simplest form,

[2]

(i)The curve C₁ has equation

Show that C₁ has a stationary point at x = p2 where p is a constant to be found.[4]

(ii)A different curve C₂ has equation

(a)Show that, for this curve,

where A and B are constants to be found.

(b)Hence, state the maximum gradient of this curve.[4]

where r cm is the base radius of the tree.

The base radius of a particular tree of this species is 45 cm.

According to the model,

(a)find the mass of this tree, giving your answer to 2 significant figures.[2]

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

giving the values of the constants p and q to 3 significant figures.[3]

(c)With reference to the model, interpret the value of the constant p.[1]

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

(a)Sketch C.[1]

(b)Given x = 2 sin y, show that

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

The point P lies on C and has y coordinate π4

(c)Find the equation of the tangent to C at P. Write your answer in the form y = mx + c, where m and c are constants to be found.[3]

Figure 3 shows a sketch of part of the curve with equation y = f(x), where

(a)Find f ′(x).[2]

The line l is the normal to the curve at O and meets the curve again at the point P.

The point P lies in the 3rd quadrant, as shown in Figure 3.

(b)Show that the x coordinate of P is a solution of the equation

[4]

(c)Using the iterative formula

find, to 4 decimal places,

(i)the value of x₂

(ii)the x coordinate of P.[3]

Figure 4 shows a sketch of part of the curve with equation

(a)Show that

where p, q and r are constants to be found.[2]

The curve touches the positive x‑axis for the second time when x = a, as shown in Figure 4.

The regions bounded by the curve, the y‑axis and the x‑axis up to x = a are shown shaded in Figure 4.

(b)Find, using algebraic integration and making your method clear, the exact total area of the shaded regions. Write your answer in simplest form.[5]