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

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

giving each term in simplest form.[4]

In the series expansion of

the coefficient of the term in x² is 1316

(b)Use the answer to part (a) to find the value of k.[2]

The line joining the points (–2, 5) and (4, 15) is the diameter of a circle C.

(a)Find an equation for C.[5]

(b)Hence find the exact coordinates of the point on C that is nearest the x-axis.[2]

y

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 C with equation

The points P (1, 11) and Q (4, 5) lie on C and are shown in Figure 1.

The region R, shown shaded in Figure 1, is bounded by C and line segment PQ.

Use algebraic integration to find the area of R.[5]

The function f(x) is defined by

where a and b are constants.

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

(a)show that

[2]

Given further that when f(x) is divided by (2x – 1) the remainder is 74

(b)find the value of a and the value of b.[4]

(c)Using algebra, find the quotient and the remainder when f(x) is divided by (x – 2)[3]

The table above shows corresponding values of x and y for

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

(a)Using the trapezium rule with all the values of y in the given table, obtain an estimate for

giving the answer to one decimal place.[3]

Using the answer to part (a) and making your method clear, estimate

 1

(b)(i)3 dx    2

(ii)2^x  2x dx

[3]

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

(i)Solve

[5]

(ii)The curves C₁ and C₂ with equations

meet at the point P.

Find the exact coordinates of P, writing your answer in the form (log₃ a, b) where a and b are integers.[5]

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

(i)A geometric series begins

(a)Find the sum to infinity of this series.[2]

Given that the k th term of this series is less than 0.0005

(b)use algebra to find the smallest possible value of k.[3]

(ii)An arithmetic series begins

Given that the sum of the first n terms of this series is S_n find the greatest possible value of S_n[4]

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

(i)Solve, for 0  x < π, the equation

giving the answers in radians in the form kπ, where k is a rational constant to be found.[3]

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

giving your answers, in degrees, to one decimal place.[5]

Figure 2 shows a sketch of an open container.

The sides CDEF and ABFE are rectangles.

The ends ADE and BCF are congruent (identical) right-angled triangles.

The container is made from metal of negligible thickness.

Given that

• AE = BF = 3x metres

• DE = CF = 2x metres

• AB = DC = EF = L metres

and the capacity of the container is 12 m³

(a)show that the area of metal used to make the container, S m², is given by

where P and Q are positive integers to be found.[4]

Given that x can vary,

(b)use algebraic calculus to find the minimum value of S, giving your answer to one decimal place.[5]

(c)Justify that the value of S found in part (b) is a minimum.[2]

Use algebra to prove by exhaustion that,

is always a multiple of 3[4]