WMA12 Pure Mathematics 2 (P2): Oct/Nov 2024 Paper 01

A continuous curve has equation y = f (x).

A table of values of x and y for y = f (x) is shown below.

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

(a)find an estimate for

giving your answer to one decimal place.[3]

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

[2]

A sequence of numbers u₁, u₂, u₃, ... is defined by

where k is a constant.

(a)Show that u₅= 7

[3]

Given that u_r  30 

(b)find the value of k.

[2]

(c)Hence find the value of u_r

[2]

where A and B are integers.

Given that when f (x) is divided by (x + 3) the remainder is 55

(a)show that

[2]

Given also that (2x – 5) is a factor of f (x),

(b)find the value of A and the value of B.[3]

(c)Hence find the quotient when f (x) is divided by (x – 7)[2]

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

Solutions relying entirely on calculator technology are not acceptable.

The curve C has equation

(a)Find dydx giving each term in simplest form.[2]

(b)Hence find the x coordinate of the stationary point of C.[2]

(c)(i)Find d ydx₂ giving each term in simplest form.

(ii)Hence determine the nature of the stationary point of C, giving a reason for your answer.[2]

(d)State the range of values of x for which y is decreasing.[1]

(a)Find, in terms of a, the first 3 terms, in ascending powers of x, of the binomial expansion of

where a is a non‑zero constant. Give each term in simplest form.[3]

Given that the constant term in the expansion of f (x) is 576

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

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

Solutions relying entirely on calculator technology are not acceptable.

Using the laws of logarithms, solve

[5]

Jem pays money into a savings scheme, A, over a period of 300 months.

Jem pays £20 into scheme A in month 1, £20.50 in month 2, £21 in month 3 and so on, so that the amounts Jem pays each month form an arithmetic sequence.

(a)Show that Jem pays £69.50 into scheme A in month 100[1]

(b)Find the total amount that Jem pays into scheme A over the period of 300 months.[2]

Kim pays money into a different savings scheme, B, over the same period of 300 months.

In a model, the amounts Kim pays into scheme B increase by the same percentage each month, so that the amounts Kim pays each month form a geometric sequence.

Given that Kim pays

• £20 into scheme B in month 1

• £250 into scheme B in month 300

(c)use the model to calculate, to the nearest £10, the difference between the total amount paid into scheme A and the total amount paid into scheme B over the period of 300 months.[3]

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 C₁ with equation

and part of the curve C₂ with equation

The curves C₁ and C₂ intersect at the points P and Q as shown in Figure 1.

(a)Use algebra to find the x coordinate of P and the x coordinate of Q.[4]

The finite region R, shown shaded in Figure 1, is bounded by C₁ and C₂

(b)Use algebraic integration to find the exact area of R.[4]

Solutions relying entirely on calculator technology are not acceptable.

(a)Show that the equation

can be written as

[3]

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

giving your answers to 3 significant figures.[4]

The circle C has equation

(a)Find

(i)the coordinates of the centre of C,

(ii)the exact value of the radius of C.[3]

The line L has equation y = mx + 1, where m is a constant.

Given that L is the tangent to C at the point P,

(b)show that

[3]

(c)Hence find the possible pairs of coordinates of P.[4]

(i)Prove by counter example that the statement

is false.[2]

(ii)Use proof by exhaustion to prove that if m is an integer that is not divisible by 3, then

is divisible by 3[4]