WMA12 Pure Mathematics 2 (P2): January 2026 Paper 01

(a)Find the first 4 terms, in ascending powers of x, of the binomial expansion of

where k is a non‑zero constant. Give each term in simplest form.[4]

Given that, in this expansion, the coefficient of x² is 3 times the coefficient of x³

(b)find, in simplest form, the value of k.[2]

Figure 1 shows a sketch of part of the curve C with equation y = 2^–0.1x

The table below shows corresponding values of x and y for y = 2^–0.1x

The y values are given to 4 decimal places.

The region R, shown shaded in Figure 1, is bounded by C, the x‑axis and the lines with equations x = –4 and x = 11

(a)Use the trapezium rule with all the values of y in the table to find an estimate for the area of R. Give the answer to 2 decimal places.[3]

(b)State how you would use the trapezium rule to get a more accurate estimate for the true area of R.[1]

Using the answer to part (a) and showing your working,

(c)estimate the value of

[2]

(i)Given that

• f (x) = 4x³+ 6x + k, where k is a constant

• (x + 2) is a factor of f (x)

find the remainder when f (x) is divided by (x – 5)[4]

(ii)Find the remainder, R, and the quotient, Q(x), when

is divided by (2x + 3)[3]

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

Given that k is a positive constant and

find the possible values of k.[5]

The circle C₁ has equation

(a)Find

(i)the coordinates of the centre of C₁

(ii)the radius of C₁[3]

The circle C₂ has

• centre (–k, 0) where k is a positive constant

• radius 5

Given that circles C₁ and C₂ touch

(b)find the exact value of k.[3]

(i)Given that p and q are consecutive odd numbers, where p > q > 0, prove that

is a multiple of 8[4]

(ii)The curve C has equation

Prove that C has no stationary points.[4]

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

(i)Find the exact solution of the equation

giving the answer in the form a + b log₂5 where a and b are rational constants.[4]

(ii)Using the laws of logarithms, solve the equation

[5]

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 < 180

giving the answers to one decimal place.[5]

(ii)Solve, for 0  x < 2π

giving the answers, in radians, in the form kπ where k is a rational constant.[5]

A company mines lithium.

• in year 1, the company mined 12 400 tonnes of lithium

• in year 15, the company is predicted to mine 7 500 tonnes of lithium

Model A assumes that the mass of lithium the company mines will decrease by the same amount each year.

According to model A,

(a)find the mass of lithium the company will mine in year 5[3]

(b)calculate the total mass of lithium the company will mine from year 1 to year 15 inclusive.[2]

Model B assumes that the mass of lithium the company mines will decrease by the same percentage each year.

(c)Find, according to model B, the mass of lithium the company will mine in year 10. Give the answer to the nearest 10 tonnes.[3]

According to model B, there is a limit to the total mass of lithium that can be mined.

(d)Calculate the value of this limit. Give the answer to the nearest 100 tonnes.[2]

Solutions relying entirely on calculator technology are not acceptable.

Figure 2 shows a sketch of the curve C with equation

The point P is a stationary point on C.

(a)Use algebraic differentiation to find the exact x coordinate of P.[4]

The region R₁, shown shaded in Figure 2, is bounded by C and the x‑axis.

The region R₂, also shown shaded in Figure 2, is bounded by C, the x‑axis and the line with equation x = k, where k is a constant.

Given that the area of R₁ is equal to the area of R₂

(b)use algebraic integration to find the exact value of k.[4]