WMA12 Pure Mathematics 2 (P2): May/June 2026 Paper 01

(a)Use the remainder theorem to find the remainder when f(x) is divided by (x – 2)[2]

(b)Hence, or otherwise, find the quotient when f(x) is divided by (x – 2)[2]

(i)Solve, for 0 < x  2π, the equation

[3]

(ii)Solve, for 0 < θ < 180°, the equation

giving the answer to one decimal place.[3]

The first four terms of a sequence are

(a)Decide, showing detailed reasoning, whether the sequence could be

(i)an arithmetic sequence,

(ii)a geometric sequence.

State, for each part, the value of u₃ if such a sequence is possible.[6]

(b)Given instead that

• the sequence is periodic of period 4

• u  141 ⁱ

find the value of u₃[3]

(i)Show by counterexample that the statement

“ f n  52 n³  52 n² 19n  20 is a perfect cube for all positive integers n”

is not true.[3]

(ii)Prove that the following statement is true

[3]

The table below shows values of x and y for the curve C with equation y = f(x).

(a)Using the trapezium rule with all the y values in the table, find an estimate for the area of the region bounded by C and the x‑axis between the lines with equations x = 0.9 and x = 5.1[3]

0.9 m 0.9 m y

Figure 1 shows a sketch of a solid concrete skateboard ramp. Figure 2 shows a graph of the cross‑section of the same ramp.

The ramp

• has a total length of 6 m

• has a platform at each end at a height of 1.5 m and with a length of 0.9 m

• has a curved section C with equation y = f(x), where x and y are in metres

• has a uniform cross‑section and is 2.5 m wide

as shown in Figure 1 and/or Figure 2.

(b)Using the answer to part (a), find an estimate for the volume of concrete in the whole of the ramp.[3]

(c)State, giving a reason for your answer, whether the answer for part (b) is an overestimate or an underestimate of the true volume of concrete in the whole of the skateboard ramp.[1]

(i)(a)Sketch a graph of the curve with equation

Show on your sketch

• the coordinates of any points at which the curve crosses the axes

• the equation of any asymptotes[3]

(b)Using logarithms and making your method clear, solve the equation

giving the answer to 3 significant figures.[2]

(ii)Given that log₄ x = a, find in simplest form in terms of a

(a)log₄ (64x)  2 [2]

(b)log _{4 2}   x [3]

(a)Find, in ascending powers of x, up to and including the term in x², the binomial expansion of

fully simplifying each coefficient.[4]

(b)Use x = 110 and the answer to part (a) to show that an estimate for  33 is

where k is an integer to be found.[3]

(c)Hence find a rational estimate for¹⁰1389[2]

Figure 3 shows a sketch of a circle C and a parabola S.

The circle C has equation x²+ y²– 8x + 12y + 27 = 0

(a)Find

(i)the coordinates of the centre of C

(ii)the radius of C[3]

The parabola S

• has equation y  ax²  32 x  3

• intersects C on the y‑axis at the point A (0, –3)

• intersects C again at the point B where AB is a diameter of C

Given that B is the minimum point of S,

(b)show that a = 332[3]

The finite region R is bounded by S and C and is shown shaded in Figure 3.

(c)Use algebraic integration to find the exact area of R.[7]

Figure 4 shows a sketch of a design for a logo.

The design consists of a rectangle of base x cm and height y cm with two quarter circles of equal radius r cm joined to the top edge with a gap of 10 cm between the nearest point of each quarter circle, as shown in Figure 4.

(a)Show that the area, A cm², of the design is given by

[2]

Given that

• the total area of the design is one square metre

• the perimeter, P cm, of the design is given by P  2 y   2  π 2  x  5π

(b)show that

[4]

(c)Use algebraic differentiation to find the value of x that gives the minimum value for P, giving the answer to 3 significant figures.[3]

By considering d Pdx₂

(d)justify that the value for x found in part (c) gives the minimum value for P.[2]