WMA13 Pure Mathematics 3 (P3): Oct/Nov 2023 Paper 01

A curve has equation y = f (x) where

(a)Show that the equation f (x) = 0 has a root, α, in the interval [1, 2][2]

The iterative formula

with x₁= 1 is used to find an approximate value for the root α.

(b)(i)Find the value of x₂ to 4 decimal places.

(ii)Find, by repeated iteration, the value of α, giving your answer to 4 decimal places.[3]

The function f is defined by

(a)Find ff (6)[2]

(b)Find f^–1[3]

The function g is defined by

(c)Find the exact value of a for which

[3]

(a)Using the identity for cos (A + B) , prove that

[2]

(b)Hence, using algebraic integration, find the exact value of

[4]

A new mobile phone is released for sale.

The total sales N of this phone, in thousands, is modelled by the equation

where A is a constant and t is the time in months after the phone was released for sale.

Given that when t = 0 , N = 32

(a)state the value of A.[1]

Given that when t = T the total sales of the phone was 100 000

(b)find, according to the model, the value of T. Give your answer to 2 decimal places.[3]

(c)Find, according to the model, the rate of increase in total sales when t = 7, giving your answer to 3 significant figures.

[2]

The total sales of the mobile phone is expected to reach 150 000

Using this information,

(d)give a reason why the given equation is not suitable for modelling the total sales of the phone.[1]

The curve C has equation

where k is a positive constant.

(a)Show that

where A and B are constants to be found.[3]

Given that C has exactly three turning points,

(b)find the x coordinate of each of these points. Give your answer in terms of k where appropriate.[3]

(c)find the upper limit to the value for k.[1]

An area of sea floor is being monitored.

The area of the sea floor, S km², covered by coral reefs is modelled by the equation

where p and q are constants and t is the number of years after monitoring began.

Given that

(a)find, according to the model, the area of sea floor covered by coral reefs when t = 2[2]

(b)find a complete equation for the model in the form

giving the value of p and the value of q each to 3 significant figures.[3]

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

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

(a)Find the range of f[2]

(b)Show that

where A and B are constants to be found.[4]

Given that the line y = k , where k is a constant, k > 0, intersects the curve at exactly two distinct points,

(c)find the exact range of values of k[4]

(a)Prove that

[4]

(b)Hence solve for 0 < x < 360° , where x ≠ (90n)° , n ∈  , the equation

giving your answers to one decimal place.

[4]

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

Solutions relying on calculator technology are not acceptable.

Figure 2 shows a sketch of the curve with equation

where k is a constant.

Given that the curve

• has an asymptote at x = k

• cuts the y-axis at point A

• meets the x-axis at point B

as shown in Figure 2,

(a)state the value of k[1]

(b)(i)find the y coordinate of A

(ii)find the exact x coordinate of B[3]

(c)Using algebra and showing your working, find the set of values of x such that

[5]

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

Solutions relying on calculator technology are not acceptable.

A curve C has equation

The point P with x coordinate 14 lies on C

(a)Find the exact y coordinate of P[2]

(b)Find dxdy[2]

(c)Hence show that dydx can be written in the form

where q, r and s are constants to be found.[3]

Using the answer to part (c),

(d)(i)state the x coordinate of the point where the value of dydx is a minimum,

(ii)state the value of dydx at this point.[2]