WMA12 Pure Mathematics 2 (P2): Oct/Nov 2020 Paper 01

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

giving each term in its simplest form.[4]

(b)Hence find the constant term in the series expansion of

[3]

(a)Complete the table below, giving the values of y to 3 decimal places.

[1]

(b)Use the trapezium rule, with all the values of y from the completed table, to find an approximate value for

[3]

where a and b are constants.

When f(x) is divided by (x + 4), the remainder is –108

(a)Use the remainder theorem to show that

[2]

Given also that (2x – 1) is a factor of f(x),

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

(c)Find f′(x).[1]

(d)Hence find the exact coordinates of the stationary points of the curve with equation y = f(x).[4]

The points P and Q have coordinates (–11, 6) and (–3, 12) respectively.

Given that PQ is a diameter of the circle C,

(a)(i)find the coordinates of the centre of C,

(ii)find the radius of C.[4]

(b)Hence find an equation of C.[2]

(c)Find an equation of the tangent to C at the point Q giving your answer in the form ax + by + c = 0 where a, b and c are integers to be found.[3]

Ben is saving for the deposit for a house over a period of 60 months.

Ben saves £100 in the first month and in each subsequent month, he saves £5 more than the previous month, so that he saves £105 in the second month, £110 in the third month, and so on, forming an arithmetic sequence.

(a)Find the amount Ben saves in the 40th month.[2]

(b)Find the total amount Ben saves over the 60‑month period.[3]

Lina is also saving for a deposit for a house.

Lina saves £600 in the first month and in each subsequent month, she saves £10 less than the previous month, so that she saves £590 in the second month, £580 in the third month, and so on, forming an arithmetic sequence.

Given that, after n months, Lina will have saved exactly £18 200 for her deposit,

(c)form an equation in n and show that it can be written as

[3]

(d)Solve the equation in part (c).[2]

(e)State, with a reason, which of the solutions to the equation in part (c) is not a sensible value for n.[1]

Figure 1 shows a sketch of part of the curves C₁ and C₂ with equations

The curves C₁ and C₂ intersect at the points A and B as shown in Figure 1.

The point A has coordinates (1, 4).

Using algebra and showing all steps of your working,

(a)find the coordinates of the point B.[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.[5]

(i)Show that

[3]

(ii)Solve, for 0  x < 90°, the equation

giving your answers in degrees to one decimal place.

[4]

A geometric series has first term a and common ratio r.

(a)Prove that the sum of the first n terms of this series is given by

[3]

The second term of a geometric series is –320 and the fifth term is 51225

(b)Find the value of the common ratio.[2]

(c)Hence find the sum of the first 13 terms of the series, giving your answer to 2 decimal places.[3]

(i)Find the exact value of x for which

[4]

(ii)Given that

(a)show that

[4]

(b)Hence find the value of y, giving your answer to 3 decimal places.[2]