WMA12 Pure Mathematics 2 (P2): May/June 2021 Paper 01

Adina is saving money to buy a new computer. She saves £5 in week 1, £5.25 in week 2, £5.50 in week 3 and so on until she has enough money, in total, to buy the computer.

She decides to model her savings using either an arithmetic series or a geometric series.

Using the information given,

(a)(i)state with a reason whether an arithmetic series or a geometric series should be used,

(ii)write down an expression, in terms of n, for the amount, in pounds (£), saved in week n.[3]

Given that the computer Adina wants to buy costs £350

(b)find the number of weeks it will take for Adina to save enough money to buy the computer.[4]

Figure 1 shows a sketch of the curve with equation y = 4^x

A copy of Figure 1, labelled Diagram 1, is shown on the next page.

(a)On Diagram 1, sketch the curve with equation

(i)y = 2^x

(ii)y = 4^x – 6

Label clearly the coordinates of any points of intersection with the coordinate axes.[4]

The curve with equation y = 2^x meets the curve with equation y = 4^x – 6 at the point P.

(b)Using algebra, find the exact coordinates of P.[4]

(i)Prove that for all single digit prime numbers, p,

[2]

(ii)Show, using algebra, that for n  ¥

[3]

(a)Find, in ascending powers of x, up to and including the term in x³, the binomial expansion of

fully simplifying each coefficient.[4]

(b)Use the answer to part (a) to find an approximation for 2.0125¹³

Give your answer to 3 decimal places.[3]

Without calculating 2.0125¹³

(c)state, with a reason, whether the answer to part (b) is an overestimate or an underestimate.[1]

Figure 2 shows a sketch of part of the graph of the curves C₁ and C₂

The curves intersect when x = 2.5 and when x = 4

A table of values for some points on the curve C₁ is shown below, with y values given to 3 decimal places as appropriate.

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

(a)find, to 2 decimal places, an estimate for the area bounded by the curve C₁, the line with equation x = 2.5, the x-axis and the line with equation x = 4[4]

The curve C₂ has equation

(b)Find  x − 3x + 9 dx

[3]

The region R, shown shaded in Figure 2, is bounded by the curves C₁ and C₂

(c)Use the answers to part (a) and part (b) to find, to one decimal place, an estimate for the area of the region R.[3]

A circle has equation

where k is a positive constant.

Given that the x-axis is a tangent to this circle,

(a)find the value of k.[3]

The circle meets the coordinate axes at the points R, S and T.

(b)Find the exact area of the triangle RST.[4]

(a)Given that

[4]

(b)Show that –4 is a root of this cubic equation.[2]

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

[4]

(i)Solve, for 0 < θ < 360°, the equation

giving your answers to one decimal place.[4]

(ii)(a)Show that the equation

can be written in the form

where a, b and c are constants to be found.

(b)Hence solve for − π2  x  π2 the equation

[6]

Figure 3 shows a sketch of a square based, open top box.

The height of the box is h cm, and the base edges each have length l cm.

Given that the volume of the box is 250 000 cm³

(a)show that the external surface area, S cm², of the box is given by

[3]

(b)Use algebraic differentiation to show that S has a stationary point when h = 250 where k is a rational constant to be found.[5]

(c)Justify by further differentiation that this value of h gives the minimum external surface area of the box.[2]