WMA12 Pure Mathematics 2 (P2): May/June 2023 Paper 01

The continuous curve C has equation y = f(x).

A table of values of x and y for y = f(x) is shown below.

Use the trapezium rule with all the values of y in the table to find an approximation for

giving your answer to 3 decimal places.[3]

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

where a is a constant.

Given that (2x – 3) is a factor of f(x),

(a)use the factor theorem to show that a = –3[2]

(b)Hence show that the equation f(x) = 0 has only one real root.[4]

A circle C has centre (2, 5)

Given that the point P(8, –3) lies on C

(a)(i)find the radius of C

(ii)find an equation for C[3]

(b)Find the equation of the tangent to C at P giving your answer in the form ax + by + c = 0 where a, b and c are integers to be found.[4]

The binomial expansion, in ascending powers of x, of

where p is a constant, can be written in the form

where A, B, C and D are constants.

(a)Find the value of A[1]

Given that

• B = 18D

• p < 0

(b)find

(i)the value of p

(ii)the value of C[6]

Use the laws of logarithms to solve

[5]

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

Solutions relying entirely on calculator technology are not acceptable.

A software developer released an app to download.

The numbers of downloads of the app each month, in thousands, for the first three months after the app was released were

where k is a constant.

Given that the numbers of downloads each month are modelled as a geometric series,

(a)show that k²– 7k – 60 = 0[2]

(b)predict the number of downloads in the 4th month.[4]

The total number of all downloads of the app is predicted to exceed 3 million for the first time in the Nth month.

(c)Calculate the value of N according to the model.[3]

The height of a river above a fixed point on the riverbed was monitored over a 7‑day period.

The height of the river, H metres, t days after monitoring began, was given by

Given that H has a stationary value at t = α

(a)use calculus to show that α satisfies the equation

[5]

(b)Hence find the value of α, giving your answer to 3 decimal places.[1]

(c)Use further calculus to prove that H is a maximum at this value of α.[2]

(i)A student writes the following statement:

“When a and b are consecutive prime numbers, a²+ b² is never a multiple of 10”

Prove by counter example that this statement is not true.[2]

(ii)Given that x and y are even integers greater than 0 and less than 6, prove by exhaustion, that

[3]

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

Solutions relying entirely on calculator technology are not acceptable.

(a)Show that

may be written as

[3]

(b)Hence solve, for 0 < x < 360°

giving your answers to one decimal place.[4]

The curve C has equation

where k is a positive constant.

(a)Show that

where a and b are integers to be found.[5]

Figure 1 shows a sketch of the curve C and the line l.

Given that l intersects C at the point A(1, 9) and at the point B(16, q) where q is a constant,

(b)show that k = 4[2]

The region R, shown shaded in Figure 1, is bounded by C and l

Using the answers to parts (a) and (b),

(c)find the area of region R[3]

A sequence u₁, u₂, u₃, ... is defined by

where a and b are constants.

(a)Find, in terms of a and b,

(i)u₂

(ii)u₃[2]

Given

• u  153 _n

• b = a + 9

(b)show that

[3]

(c)Hence find the larger possible value of u₂[3]