WMA13 Pure Mathematics 3 (P3): January 2024 Paper 01

The point P(– 4, –3) lies on the curve with equation y = f(x), x ∈ 

Find the point to which P is mapped when the curve with equation y = f(x) is transformed to the curve with equation

(a)y = f(2x)[1]

(b)y = 3f(x – 1)[2]

[1]

A curve has equation y = f(x) where

(a)Show that the equation f(x) = 0 has a root, α, in the interval [2, 3][2]

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

[1]

The iterative formula

is used to find α

(c)Starting with x₁= 2 and using the iterative formula,

(i)find, to 4 decimal places, the value of x₂

(ii)find, to 4 decimal places, the value of α[3]

The amount of money raised for a charity is being monitored.

The total amount raised in the t months after monitoring began, £D, is modelled by the equation

(a)Write this equation in the form

where a and b are constants to be found. Give each value to 4 significant figures.[3]

When t = T, the total amount of money raised is £45 000

According to the model,

(b)find the value of T, giving your answer to 3 significant figures.[2]

The charity aims to raise a total of £350 000 within the first 12 months of monitoring.

According to the model,

(c)determine whether or not the charity will achieve its aim.[2]

The function f is defined by

(a)Show that f(x) = 2x3x − 5[3]

(b)Show, using calculus, that f is a decreasing function. You must make your reasoning clear.[3]

The function g is defined by

(c)Find g^–1[3]

(d)Find the exact value of a for which

[4]

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

Solutions relying entirely on calculator technology are not acceptable.

The temperature, T °C, of the air in a room t minutes after a heat source is switched off, is modelled by the equation

where A and B are constants.

Given that the temperature of the air in the room at the instant the heat source was switched off was 18 °C,

(a)find the value of A[1]

Given also that, exactly 45 minutes after the heat source was switched off, the temperature of the air in the room was 16 °C,

(b)find the value of B to 3 significant figures.[3]

Using the values for A and B,

(c)find, according to the model, the rate of change of the temperature of the air in the room exactly two minutes after the heat source was switched off. Give your answer in °C min^–1 to 3 significant figures.[2]

(d)Explain why, according to the model, the temperature of the air in the room cannot fall to 5 °C[1]

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 the curve with equation y = f(x), where

The curve intersects the x‑axis at point R, as shown in Figure 1.

(a)State the coordinates of R[1]

The curve has two turning points, at point P and point Q, also shown in Figure 1.

(b)Show that, at points P and Q,

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

(c)Hence find the x coordinate of point Q, giving your answer to 3 decimal places.[2]

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

Solutions relying entirely on calculator technology are not acceptable.

The curve C has equation

where k is a positive constant not equal to 3

(a)Find dydx giving your answer in simplest form in terms of k.[2]

The point P with x coordinate 1 lies on C.

Given that the gradient of the curve at P is –12

(b)find the two possible values of k[3]

Given also that k < 3

(c)find the equation of the normal to C at P, writing your answer in the form ax  by  c  0 , where a, b and c are integers to be found.[3]

(d)show, using algebraic integration that,

where λ is a constant to be found.[4]

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

The graph shown in Figure 2 has equation

where a and b are positive constants, a > b

(a)Find, giving your answer in terms of a and b,

(i)the coordinates of the maximum point of the graph,

(ii)the coordinates of the point of intersection of the graph with the y‑axis,

(iii)the coordinates of the points of intersection of the graph with the x‑axis.[5]

On page 24 there is a copy of Figure 2 called Diagram 1.

(b)On Diagram 1, sketch the graph with equation

[2]

Given that the graphs y  x  1 and y  a  2x  b intersect at x = –3 and x = 5

(c)find the value of a and the value of b[4]

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

Solutions relying entirely on calculator technology are not acceptable.

(a)Show that the equation

can be written in the form

[3]

(b)Hence solve for π  x  3π2

giving the answer to 3 significant figures.[5]