WMA12 Pure Mathematics 2 (P2): January 2025 Paper 01

The arithmetic series S is given by

S  2  5  8  11   254

Find

(a)the number of terms in the series,[2]

(b)the sum of the series.[2]

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

giving each term in simplest form.[4]

This expansion is to be used to find an approximation for 2.05⁸

(b)State the value of x that should be used.

[1]

Figure 1 shows an open‑topped container used for holding water.

The container is in the shape of a cuboid and is made of sheet metal.

The base of the container is a rectangle 3x metres by x metres.

The height of the container is y metres as shown in Figure 1.

Given that the capacity of the container is 120 m³

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

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

(b)Use calculus to find the value of x for which A has a stationary value, giving your answer to 3 significant figures.[4]

(c)Find d Adx₂ and hence show that the value of x found in part (b) gives the minimum

value of A.[2]

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

Given that, in a particular geometric series,

• the sum of the first three terms is 70.2

• the sum to infinity is 75

find, for this series,

(a)the common ratio,[4]

(b)the first term.[2]

where a and b are constants.

Given that (3x – 4) is a factor of f(x),

(a)show that 16a + 9b = 56[2]

Given further that when f(x) is divided by (x – 2) the remainder is b,

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

(c)Hence, using algebra, fully factorise f(x).[3]

The point A (–1, 2) and the point B (11, 6) both lie on a circle with centre P.

The point M is the midpoint of AB.

Given that the line l passes through M and P, as shown in Figure 2,

(a)find an equation for l, giving your answer in the form y = mx + c, where m and c are constants.[4]

Given that P has coordinates (7, k), where k is a constant,

(b)find the value of k,[1]

(c)find an equation for the circle.[3]

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

(i)The table below shows values of x and y, where y = log₁₀ (x + 5), for x values between –1 and 4

Using the trapezium rule with all the y values in the given table, show that

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

(ii)Find the value of a such that

[5]

(i)A student states

Show, by counter example, that this statement is not always true.[1]

(ii)Prove, using algebra, that for all odd integers n, the value of the expression

is always even but never a multiple of 4[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 with equation

The curve has a turning point at the point M, as shown in Figure 3.

(a)Using calculus, find the coordinates of M.[5]

The curve crosses the x‑axis at the point P, as shown in Figure 3.

(b)Use algebra to find the x coordinate of P.[2]

The finite region R, shown shaded in Figure 3, is bounded by the curve, the line through M parallel to the x‑axis and the line through P parallel to the y‑axis.

(c)Use algebraic integration to find the area of R, giving your answer to one decimal place.[5]

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

(a)Show that

[4]

(b)Hence solve, for 0 < x < 2π, the equation

giving your answers to 3 significant figures.[4]