WMA12 Pure Mathematics 2 (P2): May/June 2024 Paper 01R

The sequence u₁ , u₂ , u₃ , … satisfies

• u₁ = 7

• u₃ = 4

(a)find the value of u₂[2]

(b)find u  2r 

[3]

The table shows corresponding values of x and y for a continuous curve with equation y = f ( x) between x = –4 and x = 5, where a is a constant.

The trapezium rule is used with all the y values in the table to find an approximation for

Given that the value of this approximation is 19.3

(a)find the value of the constant a to 3 significant figures.[3]

(b)Use the given answer of 19.3 to find an approximate value for

[2]

A circle has equation

(a)Find

(i)the coordinates of the centre of the circle,

(ii)the radius of the circle.[3]

Given that P is the point on the circle that is nearest the origin O,

(b)find the exact length of OP[2]

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

giving each coefficient in simplest form.[4]

(b)Hence find the coefficient of x² in the expansion of

[3]

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

The depth of water, D metres, in a harbour on a particular day is given by the equation

where t is the number of hours after midnight.

(a)Show that the depth of water in the harbour at 2 am is just over 4 metres.[1]

(b)Find, to the nearest minute, the first time after midday when the depth of water in the harbour is exactly 6 metres.[5]

where p and q are constants.

Given that

• (2x + 3) is a factor of f (x)

• f ( x) has a remainder of –5 when divided by (x + 2)

(a)(i)show that p = 10

(ii)find the value of q.[5]

(b)Hence find the range of values of x for which f ( x) is decreasing.

(Solutions based entirely on calculator technology are not acceptable.)[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 < 2π , the equation

giving the answers in radians to 3 decimal places.[5]

(ii)Given that

• 0 < θ < 90°

• cos θ = 13

find, in simplest form, the exact value of tan θ[2]

(i)(a)In an arithmetic series the first term is a and the common difference is d.

Show that

[3]

(b)Hence find

[3]

(ii)Given that the first three terms of a geometric series are

where k is a constant,

(a)show that

[3]

Given also that this series is convergent,

(b)find the value of S_∞[4]

Given that

(a)verify that t = 4 is a solution of the above equation,[2]

(b)show that

[3]

(c)Hence, using algebra and showing your working, solve

giving each answer in simplest form.

(Solutions based entirely on calculator technology are not acceptable.)[4]

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 part of the curve C with equation

The point P is the stationary point on C.

(a)Find, using calculus, the x coordinate of P.[4]

The finite region R, shown shaded in Figure 1, is bounded by the curve C, the x-axis and the line with equation x = 1

(b)Using calculus, calculate the exact area of R.[5]