WMA12 Pure Mathematics 2 (P2): May/June 2025 Paper 01A

Given that a, b and c are positive integers such that

• c = a² – 1

• a + b + c = 20

prove, by exhaustion, that the product abc is always a multiple of 6

You may use the table below to illustrate your answer.[3]

You may not need to use all rows of this table.

A geometric series has first term a and common ratio r.

Given that

• the third term in the series is 64

• the sixth term in the series is –8

(a)show that r = – 12[2]

(b)find the sum to infinity of the series.[4]

where k is a constant.

(a)Deduce the value of the remainder when f (x) is divided by (x – k)[1]

Given that the remainder when f (x) is divided by (x + 2) is 25

(b)show that the value of k is –4[2]

(c)Hence find the quotient and remainder when f (x) is divided by (3x – 1)[4]

Figure 1 shows a sketch of part of the curve with equation

The point P(a, 4) lies on the curve.

(a)Find the exact value of the constant a.

[4]

(b)Use the trapezium rule with 4 strips of equal width to estimate the value of

giving the answer in the form log₃ k , where k is a constant to be found.[4]

(c)Explain how the trapezium rule could be used to obtain a more accurate estimate for

[1]

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

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

(a)Show that this is a periodic sequence of order 3[3]

(b)Find the value of

[4]

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

(i)Given that θ is measured in degrees and

• cos θ = 15

• 180° < θ < 360°

use trigonometric identities to find the exact value of

(a)sin θ

(b)tan θ

giving the answers as fully simplified surds where appropriate.[4]

(ii)

The height of sea water, h metres, on a harbour wall, t hours after midnight on a particular day is given by

A sketch of h against t is shown in Figure 2.

(a)Find the minimum height of sea water on the harbour wall.[1]

(b)Find the exact time of day when this minimum height first occurs.[3]

When t = T, as shown in Figure 2, a boat enters the harbour when the height of sea water on the harbour wall is 3.5 m.

(c)Use Figure 2 and the given equation to find the value of T to 2 decimal places.[4]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The binomial expansion, in ascending powers of x, of (1 + kx)ⁿ is

where k and p are constants and n is an integer greater than 1

(a)Use this information to set up and solve two simultaneous equations to find the value of k and the value of n.[6]

(b)Hence find the value of p.[2]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

A curve C has equation y = f (x) where f (x) is a polynomial in x.

Given

(a)deduce the range of values of x for which y is decreasing.[2]

Given further that C cuts the x‑axis at 72

(b)state a factor of f (x), giving the answer in the form (Ax + B) where A and B are integers.[1]

(c)Hence find an expression for f (x) in a fully factorised form.[6]

The circle C

• has a centre which lies on the x‑axis

• touches the y‑axis

• passes through the point (5, 6)

(a)On Diagram 1, sketch a graph of C.[1]

(b)Find an equation for C.[4]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Figure 3 shows a sketch of the curve C with equation

The point P, shown in Figure 3, is the stationary point on C.

(a)Show, using calculus, that the x coordinate of P is 4[4]

The point Q lies on C and has x coordinate 2

The line segments MQ and NP, shown in Figure 3, are parallel to the x‑axis.

The region R, shown shaded in Figure 3, is bounded by C, the y‑axis and line segments MQ and NP.

(b)Use algebraic integration to find the exact area of R.

You must make your method clear.[5]