WMA13 Pure Mathematics 3 (P3): January 2021 Paper 01

Find

giving your answer in simplest form.[3]

Figure 1 shows a sketch of the curve with equation y = f(x), where x ∈  and f(x) is a polynomial.

The curve passes through the origin and touches the x-axis at the point (3, 0)

There is a maximum turning point at (1, 2) and a minimum turning point at (3, 0)

On separate diagrams, sketch the curve with equation

(i)y = 3f(2x)[3]

(ii)y = f(−x) − 1[3]

On each sketch, show clearly the coordinates of

● the point where the curve crosses the y-axis ● any maximum or minimum turning points

(a)Show that

where a, b, c and d are integers to be found.[4]

(b)Hence find f⁻¹(x)[2]

(c)Find the domain of f⁻¹[2]

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

and where a is a positive constant.

The graph has a vertex at the point P, as shown in Figure 2.

(a)Find, in terms of a, the coordinates of P.[2]

(b)Sketch the graph with equation y = g(x), where

On your sketch, show the coordinates, in terms of a, of each point where the graph cuts or meets the coordinate axes.[2]

The graph with equation y = g(x) intersects the graph with equation y = f(x) at two points.

(c)Find, in terms of a, the coordinates of the two points.[5]

The temperature, θ °C, inside an oven, t minutes after the oven is switched on, is given by

where A and k are positive constants.

Given that the temperature inside the oven is initially 18 °C,

(a)find the value of A.[2]

The temperature inside the oven, 5 minutes after the oven is switched on, is 90 °C.

(b)Show that k = p ln q where p and q are rational numbers to be found.[4]

Hence find

(c)the temperature inside the oven 9 minutes after the oven is switched on, giving your answer to 3 significant figures,[2]

(d)the rate of increase of the temperature inside the oven 9 minutes after the oven is switched on. Give your answer in °C min^–1 to 3 significant figures.[3]

(a)Find f ʹ(x)[2]

(b)Show that the equation f ʹ(x) = 0 can be written as

where k is an integer to be found.[2]

(c)Starting with x₁= 2.5 use the iteration formula

with the value of k found in part (b), to calculate the values of x₂ and x₆ giving your answers to 3 decimal places.[2]

(d)Using a suitable interval and a suitable function that should be stated, show that a root of fʹ (x) = 0 is 2.581 correct to 3 decimal places.[2]

(a)Prove that

[3]

(b)Hence solve, for − < θ <

giving your answers in radians to 3 significant figures where appropriate.[6]

The percentage, P, of the population of a small country who have access to the internet, is modelled by the equation

where a and b are constants and t is the number of years after the start of 2005

Using the data for the years between the start of 2005 and the start of 2010, a graph is plotted of log₁₀ P against t.

The points are found to lie approximately on a straight line with gradient 0.09 and intercept 0.68 on the log₁₀ P axis.

(a)Find, according to the model, the value of a and the value of b, giving your answers to 2 decimal places.[4]

(b)In the context of the model, give a practical interpretation of the constant a.[1]

(c)Use the model to estimate the percentage of the population who had access to the internet at the start of 2015[2]

Find

3x − 2

(i)₂ ∫ 3x − 4x + 5[2]

(ii)_2xe− 1)₃ ∫ (e[2]

The curve C has equation

(a)Find dxdy in terms of y.[2]

(b)Hence show that

where p is irrational and q is an integer, stating the values of p and q.[3]

(c)Find the equation of the normal to C at the point where y = π12, giving your answer

in the form y = mx + c, giving m and c as exact irrational numbers.[5]