WMA12 Pure Mathematics 2 (P2): Oct/Nov 2025 Paper 01

An arithmetic series starts

(a)the 18^th term in the series,[2]

(b)the sum of the first 100 terms of the series.[2]

The first 4 terms, in ascending powers of x, in the binomial expansion of

where p, q and r are constants.

Find the value of p, the value of q and the value of r.[6]

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 = 3 × 2 ^–x

The point P(k, 300 000) lies on the curve.

(a)Use logarithms to find the value of k to 2 decimal places.[2]

The table shows corresponding values of x and y for y = 3 × 2^–x

The values of y are given to 3 decimal places where appropriate.

(b)(i)Use the trapezium rule, with all the values of y from the table, to find an approximate value, to 2 decimal places, for

[3]

(ii)Use your answer to part (b)(i) to estimate

[2]

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

(a)When f(x) is divided by (x – 2) the remainder is R and the quotient is Q(x).

(i)Find Q(x).

(ii)Find R.[4]

(b)(i)Use the factor theorem to show that (x + 4) is a factor of f(x).

(ii)Hence prove, using algebra, that the equation f(x) = 0 has only one real solution.[5]

(c)Find the range of values of x for which f(x) is decreasing.[3]

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

(i)Solve, for 0 < θ  360°, the equation

giving any non‑exact answers to one decimal place.[5]

(ii)Solve, for 0 < x < π, the equation

giving the answers, in radians, to 3 significant figures.[4]

The points A(–2, 18) and B(14, 6) lie on a circle C₁ as shown in Figure 2.

Given that AB is a diameter of the circle C₁

(a)find an equation for C₁ making your method clear.[5]

A circle C₂ has its centre at the origin.

Given that circles C₁ and C₂ touch,

(b)find possible equations for C₂[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 part of the curve C with equation

where k is a positive constant.

(a)Find in simplest form.

[2]

The point T, shown in Figure 3, is a minimum stationary point on C.

Given that the x coordinate of T is 9

(b)show that k = 6[2]

The line through T parallel to the x‑axis meets the y‑axis at the point N.

The finite region R, shown shaded in Figure 3, is bounded by C, the y‑axis and the line segment NT.

(c)Use algebraic integration to find the area of R, giving the answer to 3 significant figures.[6]

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

Prove that the sum of the first n terms of this series S_n is given by

[3]

(ii)A liquid is to be stored in a barrel.

Due to evaporation, the volume of the liquid in the barrel at the end of each year is 8% less than the volume of the liquid in the barrel at the start of the year.

At the start of the first year, the barrel is filled with 150 litres of the liquid.

(a)Show that the amount of the liquid in the barrel at the end of 6 years is approximately 91 litres.[2]

At the start of each year a new barrel is filled with 150 litres of the liquid so that, at the end of 40 years, there are 40 barrels containing the liquid.

(b)Calculate the total amount of the liquid, to the nearest litre, in the 40 barrels at the end of 40 years.[3]

Solutions relying on calculator technology are not acceptable.

(i)Solve

[5]

(ii)Given that a > 0, b > 0 and

(a)prove that a  bb 1[3]

(b)Hence write down the full restriction on the value of b, giving a reason for your answer.[2]