Edexcel ALevel Maths, Pure Mathematics Paper 2, June 2019
Question Walkthroughs

Pure Paper 2, 2019, Q1Given 2^x * 4^y = 1/2_/2, express y as a function of x.


Pure Paper 2, 2019, Q2The speed of a small jet aircraft was measured every 5 seconds, starting from the time it turned onto a runway, until the time when it left the ground. The results are given in the table below with the time in seconds and the speed in m s^1.

Using all of this information,
(a) estimate the length of runway used by the jet to take off. Given that the jet accelerated smoothly in these 25 seconds,
(b) explain whether your answer to part (a) is an underestimate or an overestimate of the length of runway used by the jet to take off.
(a) estimate the length of runway used by the jet to take off. Given that the jet accelerated smoothly in these 25 seconds,
(b) explain whether your answer to part (a) is an underestimate or an overestimate of the length of runway used by the jet to take off.

Pure Paper 2, 2019, Q3Figure 1 shows a sector AOB of a circle with centre O, radius 5 cm and angle AOB = 40°.
The attempt of a student to find the area of the sector is shown below. Area of sector = 1/2 r^2 theta = 1/2 x 5^2 x 40 = 500 cm^2 (a) Explain the error made by this student. (b) Write out a correct solution. 

Pure Paper 2, 2019, Q4The curve C1 with parametric equations x=10cos t , y = 4 _/2sin t, meets the circle C2 with equation x^2 + y^2 = 66 at four distinct points as shown in Figure 2.
Given that one of these points, S, lies in the 4th quadrant, find the Cartesian coordinates of S. 

Pure Paper 2, 2019, Q5Figure 3 shows a sketch of the curve with equation y = _/x
The point P(x, y) lies on the curve. The rectangle, shown shaded on Figure 3, has height y and width delta x. Calculate lim as delta x tends towards 0 of the sum between 4 and 9 of _/x delta x. 

Pure Paper 2, 2019, Q6Figure 4 shows a sketch of the graph of y=g(x), where ....


Pure Paper 2, 2019, Q7A small factory makes bars of soap. On any day, the total cost to the factory, £y, of making x bars of soap is modelled to be the sum of two separate elements: a fixed cost a cost that is proportional to the number of bars of soap that are made that day
(a) Write down a general equation linking y with x, for this model. 
The bars of soap are sold for £2 each. On a day when 800 bars of soap are made and sold, the factory makes a profit of £500.
n a day when 300 bars of soap are made and sold, the factory makes a loss of £80.
Using the above information, (b) show that y = 0.84x + 428
(c) With reference to the model, interpret the significance of the value 0.84 in the equation.
Assuming that each bar of soap is sold on the day it is made,
(d) find the least number of bars of soap that must be made on any given day for the factory to make a profit that day.
n a day when 300 bars of soap are made and sold, the factory makes a loss of £80.
Using the above information, (b) show that y = 0.84x + 428
(c) With reference to the model, interpret the significance of the value 0.84 in the equation.
Assuming that each bar of soap is sold on the day it is made,
(d) find the least number of bars of soap that must be made on any given day for the factory to make a profit that day.

Pure Paper 2, 2019, Q8Find the value of:
Sum from 4 to infinity of (20 x (1/2)^r) Show that: Sum from 1 to 48 of log5 ((n+2)/(n+1)) = 2 

Pure Paper 2, 2019, Q9A research engineer is testing the effectiveness of the braking system of a car when it is driven in wet conditions. The engineer measures and records the braking distance, d metres, when the brakes are applied from a speed of V km h^1. Graphs of d against V and log10 d against log10 V were plotted. The results are shown below together with a data point from each graph.

(a) Explain how Figure 6 would lead the engineer to believe that the braking distance should be modelled by the formula d = kV n where k and n are constants with k = 0.017
Using the information given in Figure 5, with k = 0.017
b) find a complete equation for the model giving the value of n to 3 significant figures.
Sean is driving this car at 60 km h^1 in wet conditions when he notices a large puddle in the road 100 m ahead. It takes him 0.8 seconds to react before applying the brakes.
(c) Use your formula to find out if Sean will be able to stop before reaching the puddle.
Using the information given in Figure 5, with k = 0.017
b) find a complete equation for the model giving the value of n to 3 significant figures.
Sean is driving this car at 60 km h^1 in wet conditions when he notices a large puddle in the road 100 m ahead. It takes him 0.8 seconds to react before applying the brakes.
(c) Use your formula to find out if Sean will be able to stop before reaching the puddle.

Pure Paper 2, 2019, Q10Figure 7 shows a sketch of triangle OAB. The point C is such that OC = 2OA. The point M is the midpoint of AB. The straight line through C and M cuts OB at the point N. Given OA = a and OB = b
(a) Find CM in terms of a and b (b) Show that ON = (2  3/2 lambda)a + 1/2 lambda b , where lambda is a scalar constant. (c) Hence prove that ON : NB = 2 : 1 

Pure Paper 2, 2019, Q11Figure 8 shows a sketch of the curve C with equation y = x^x , x> 0
(a) Find, by firstly taking logarithms, the x coordinate of the turning point of C. The point P(alpha, 2) lies on C. (b) Show that 1.5 alpha 1.6 A possible iteration formula that could be used in an attempt to find alpha is x(n+1) = xn^(1xn) Using this formula with x1 = 1.5 (c) find x4 to 3 decimal places, (d) describe the longterm behaviour of xn. 

Pure Paper 2, 2019, Q12(a) Prove cos3θ/sinθ + sin3θ/cosθ = 2 cot2θ
(b) Hence solve, for 90° θ 180°, the equation cos3θ/sinθ + sin3θ/cosθ = 4 giving any solutions to one decimal place. 

Pure Paper 2, 2019, Q13A manufacturer produces a storage tank. The tank is modelled in the shape of a hollow circular cylinder closed at one end with a hemispherical shell at the other end as shown in Figure 9. The walls of the tank are assumed to have negligible thickness. The cylinder has radius r metres and height h metres and the hemisphere has radius r metres. The volume of the tank is 6 m3.

(a) Show that, according to the model, the surface area of the tank, in m2, is given by 12/r + 5/3 r^2
The manufacturer needs to minimise the surface area of the tank.
(b) Use calculus to find the radius of the tank for which the surface area is a minimum.
(c) Calculate the minimum surface area of the tank, giving your answer to the nearest integer.
The manufacturer needs to minimise the surface area of the tank.
(b) Use calculus to find the radius of the tank for which the surface area is a minimum.
(c) Calculate the minimum surface area of the tank, giving your answer to the nearest integer.

Pure Paper 2, 2019, Q14Use the substitution u = 4  root h to show that integral
dh/(4  root h) =  8ln (4  root h)  2 root h + k where k is a constant. A team of scientists is studying a species of slow growing tree. The rate of change in height of a tree in this species is modelled by the differential equation dh / dt = t^0.25 (4  root h)/20 where h is the height in metres and t is the time, measured in years, after the tree is planted. 
b) Find, according to the model, the range in heights of trees in this species. One of these trees is one metre high when it is first planted.
According to the model, (c) calculate the time this tree would take to reach a height of 12 metres, giving your answer to 3 significant figures.
According to the model, (c) calculate the time this tree would take to reach a height of 12 metres, giving your answer to 3 significant figures.