Edexcel ALevel Maths, Pure Paper 2, June 2018
Question Walkthroughs

Pure Paper 2, 2018, Q11. g(x) = 2x+5 / x3
a. Find gg(5) b. State the range of g c. Find g1(x), stating its domain. 

Pure Paper 2, 2018, Q21. Relative to a fixed origin O.
the point A has position vector (2i + 3j  4k), the point B has position vector (4i  2j + 3k), and the point C has position vector (ai + 5j  2k), where a is a constant and a < 0 D is the point such that AB = BD. a. Find the position vector of D. Given AC=4 b. find the value of a. 

Pure Paper 2, 2018, Q33a. 'If m and n are irrational numbers, where m=/n, then mn is also irrational.' Disprove this statement by means of a counter example.
bi. Sketch the graph of y = x + 3 ii. Explain why x + 3 is larger than or equal to x + 3 for all real values of x. 

Pure Paper 2, 2018, Q44.i. Show that sum(3 + 5r + 2^r) = 131 798.
ii. A sequence U1, U2, U3, .... is defined by Un+1 = 1/Un, U1 = 2/3 Find the exact value of sum(Ur), r from 1 to 100. 

Pure Paper 2, 2018, Q5The equation 2x^3 + x^2 1 = 0 has exactly one real root.
a. Show that, for this equation, the NewtonRaphson formula can be written x = (4x^3 + x^2 +1)/(6x^2 + 2x) Using the formula give in part (a) with x1 = 1 b. find the values of x2 and x3 c. Explain why, for this question, the NewtonRaphson method cannot be used with x1=0. 

Pure Paper 2, 2018, Q6f(x) = 3x^3 + 8x^2  9x + 10.
ai. Calculate f(2) aii. Write f(x) as a product of two algebraic factors. Using the answers to aii. b. prove that there are exactly two real solutions to the equation 3y^6 + 8y^4  9y^2 + 10 = 0 c. deduce the number of real solutions, for theta between 7pi and 10pi, to the equation, 3tan^3 theta  8tan^2 theta + 9 theta  10 = 0. 

Pure Paper 2, 2018, Q7i. Solve, for x between 0 and pi/2, the equation 4 sin x = sec x
ii. Solve, for x between 0 and 360, the equation 5 sin theta  5 cos theta = 2 giving your answers to one decimal place. 

Pure Paper 2, 2018, Q8Figure 1 is a graph showing the trajectory of a rugby ball. The height above the ground, H metres, has been plotted against the horizontal distance, x metres, measured from the point where the ball was kicked. The ball travels in a vertical plane. The ball reaches a maximum height of 12 metres and hits the ground 40 metres from where it was kicked.
a. Find a quadratic equation linking H with x that models this situation. The ball passes over the horizontal bar of a set of rugby posts that is perpendicular to the path of the ball. The bar is 3 metres above the ground. b. Use your equation to find the greatest horizontal distance of the bar from O. c. Give one limitation of the model. 

Pure Paper 2, 2018, Q9Given that theta is measured in radians, prove, from first principles, that d/dtheta(cos theta) =  sin theta.


Pure Paper 2, 2018, Q10A spherical mint of radius 5 mm is placed in the mouth and sucked. Four minutes later, the radius of the mint is 3 mm. In a simple model, the rate of decrease of the radius of the mint is inversely proportional to the square of the radius. Using this model and all the information given,
a. find an equation linking the radius of the mint and the time. b. Hence find the total time taken for the mint to completely dissolve. Give your answer in minutes and seconds to the nearest second. c. Suggest a limitation of the model. 

Pure Paper 2, 2018, Q11 (1 + 11x  6x^2)/(x  3)(1  2x) = A + B/(x  3) + C/(1  2x)
a. Find the values of the constants A, B and C. b. Prove that f(x) is a decreasing function. 

Pure Paper 2, 2018, Q12a. Prove that 1  cos 2theta = tan theta sin 2theta
b. Hence solve the equation (sec^2 x  5)(1  cos 2x) = 3tan^2 x sin 2x Give any nonexact answer to 3 decimal places where appropriate. 

Pure Paper 2, 2018, Q13Figure 2 shows a sketch of part of the curve C with equation y = xlnx.
The line l is the normal to C at the point P (e, e). The region R, shown shaded in figure 2, is bounded by the curve C, the line l and the xaxis. Show that the exact area of R is Ae^2 + B where A and B are rational numbers to be found. 

Pure Paper 2, 2018, Q14A scientist is studying a population of mice on an island. The number of mice, N, in the population, t months after the start of the study, is modelled by the equation N =900/(3+7e^0.25t).
a. Find the number of mice in the population at the start of the study. b. Show that the rate of growth dN/dt is given by dN/dt = N(300N)/1200. The rate of growth is a maximum after T months. c. Find, according to the model, the value of T. According to the model, the maximum number of mice on the island is P. d. State the value of P. 