Edexcel ALevel Maths, Pure Paper 1, June 2019 Exam Question Walkthroughs

Pure Paper 1, 2019, Q1f(x) = 3x^3 + 2ax^2 – 4x + 5a
Given that (x + 3) is a factor of f(x), find the value of the constant a. 

Pure Paper 1, 2019, Q2Figure 1 shows a plot of part of the curve with equation y = cos x where x is measured in radians. Diagram 1, on the opposite page, is a copy of Figure 1.
(a) Use Diagram 1 to show why the equation cos x − 2x − 1/2 = 0 has only one real root, giving a reason for your answer. Given that the root of the equation is alpha, and that alpha is small, b) use the small angle approximation for cos x to estimate the value of alpha to 3 decimal places. 

Pure Paper 1, 2019, Q3y = 5x^2+10x / (x+1)^2
(a) Show that dy/dx = A / (x+q)^n where A and n are constants to be found. (b) Hence deduce the range of values for x for which dy/dx is less than 0. 

Pure Paper 1, 2019, Q4Find the first three terms, in ascending powers of x, of the binomial expansion of 1/ _/(4 − x) giving each coefficient in its simplest form.
The expansion can be used to find an approximation to 2. Possible values of x that could be substituted into this expansion are: x = –14, x = 2, x = − 1 b) Without evaluating your expansion, (i) state, giving a reason, which of the three values of x should not be used (ii) state, giving a reason, which of the three values of x would lead to the most accurate approximation to 2. 

Pure Paper 1, 2019, Q5f(x) = 2x^2 + 4x + 9
(a) Write f(x) in the form a(x + b)^2 + c, where a, b and c are integers to be found. (b) Sketch the curve with equation y = f(x) showing any points of intersection with the coordinate axes and the coordinates of any turning point. (c) (i) Describe fully the transformation that maps the curve with equation y = f(x) onto the curve with equation y = g(x) where g(x) = 2(x – 2)^2 + 4x – 3 (ii) Find the range of the function h(x)= 21/(2x^2 + 4x + 9) 

Pure Paper 1, 2019, Q6Solve, for –180° < theta < 180°, the equation
5 sin 2theta= 9 tan theta giving your answers, where necessary, to one decimal place. (b) Deduce the smallest positive solution to the equation 5 sin (2x – 50°) = 9 tan (x – 25°) 

Pure Paper 1, 2019, Q7In a simple model, the value, £V, of a car depends on its age, t, in years. The following information is available for car A its value when new is £20 000 its value after one year is £16 000.
(a) Use an exponential model to form, for car A, a possible equation linking V with t. The value of car A is monitored over a 10year period. Its value after 10 years is £2 000. 
(b) Evaluate the reliability of your model in light of this information.
The following information is available for car B: it has the same value, when new, as car A; its value depreciates more slowly than that of car A.
(c) Explain how you would adapt the equation found in (a) so that it could be used to model the value of car B.
The following information is available for car B: it has the same value, when new, as car A; its value depreciates more slowly than that of car A.
(c) Explain how you would adapt the equation found in (a) so that it could be used to model the value of car B.

Pure Paper 1, 2019, Q8Figure 2 shows a sketch of part of the curve with equation y = x(x + 2)(x – 4). The region R1 shown shaded in Figure 2 is bounded by the curve and the negative xaxis.
(a) Show that the exact area of R1 is 20/3 The region R2 also shown shaded in Figure 2 is bounded by the curve, the positive xaxis and the line with equation x = b, where b is a positive constant between 0 and 4. Given that the area of R1 is equal to the area of R2; 
b) verify that b satisfies the equation (b + 2)^2 (3b^2 – 20b + 20) = 0
The roots of the equation 3b^2 – 20b + 20 = 0 are 1.225 and 5.442 to 3 decimal places. The value of b is therefore 1.225 to 3 decimal places.
(c) Explain, with the aid of a diagram, the significance of the root 5.442
The roots of the equation 3b^2 – 20b + 20 = 0 are 1.225 and 5.442 to 3 decimal places. The value of b is therefore 1.225 to 3 decimal places.
(c) Explain, with the aid of a diagram, the significance of the root 5.442

Pure Paper 1, 2019, Q9Given that a > b > 0 and that a and b satisfy the equation log a – log b = log(a – b)
(a) show that a = b^2 / b  1 (b) Write down the full restriction on the value of b, explaining the reason for this restriction. 

Pure Paper 1, 2019, Q10(i) Prove that for all n in the natural numbers, n^2 + 2 is not divisible by 4
(ii) “Given x in the real numbers, the value of 3x – 28 is greater than or equal to the value of (x – 9).” State, giving a reason, if the above statement is always true, sometimes true or never true. 

Pure Paper 1, 2019, Q11A competitor is running a 20 kilometre race. She runs each of the first 4 kilometres at a steady pace of 6 minutes per kilometre. After the first 4 kilometres, she begins to slow down. In order to estimate her finishing time, the time that she will take to complete each subsequent kilometre is modelled to be 5% greater than the time that she took to complete the previous kilometre.

Using the model, (a) show that her time to run the first 6 kilometres is estimated to be 36 minutes 55 seconds,
(b) show that her estimated time, in minutes, to run the rth kilometre, for r between 5 and 20 is 6 x 1.05^(r–4)
(c) estimate the total time, in minutes and seconds, that she will take to complete the race.
(b) show that her estimated time, in minutes, to run the rth kilometre, for r between 5 and 20 is 6 x 1.05^(r–4)
(c) estimate the total time, in minutes and seconds, that she will take to complete the race.

Pure Paper 1, 2019, Q12f(x) = 10e^0.25x sin x, x larger than or equal to 0
(a) Show that the x coordinates of the turning points of the curve with equation y = f(x) satisfy the equation tan x = 4 Figure 3 shows a sketch of part of the curve with equation y = f(x). 
(b) Sketch the graph of H against t where H(t)=10e^0.25t sin t showing the longterm behaviour of this curve.
The function H(t) is used to model the height, in metres, of a ball above the ground t seconds after it has been kicked. Using this model, find (c) the maximum height of the ball above the ground between the first and second bounce.
(d) Explain why this model should not be used to predict the time of each bounce.
The function H(t) is used to model the height, in metres, of a ball above the ground t seconds after it has been kicked. Using this model, find (c) the maximum height of the ball above the ground between the first and second bounce.
(d) Explain why this model should not be used to predict the time of each bounce.

Pure Paper 1, 2019, Q13The curve C with equation y = (p  x) / (2x  q)(x + 3), x ≠ −3, x ≠ 2 where p and q are constants, passes through the point (3, 1/2) and has two vertical asymptotes with equations x = 2 and x = –3.
(a) (i) Explain why you can deduce that q = 4 (ii) Show that p = 15 Figure 4 shows a sketch of part of the curve C. The region R, shown 
shaded in Figure 4, is bounded by the curve C, the xaxis and the line with equation x = 3.
(b) Show that the exact value of the area of R is a ln 2 + b ln 3, where a and b are rational constants to be found.
(b) Show that the exact value of the area of R is a ln 2 + b ln 3, where a and b are rational constants to be found.

Pure Paper 1, 2019, Q14The curve C, in the standard Cartesian plane, is defined by the equation x = 4 sin 2y. The curve C passes through the origin O
(a) Find the value of dy/dx at the origin. (b) (i) Use the small angle approximation for sin 2y to find an equation linking x and y for points close to the origin. 
(ii) Explain the relationship between the answers to (a) and (b)
(i). (c) Show that, for all points (x, y) lying on C, dy/dx = 1/ a_/(bx^2) where a and b are constants to be found.
(i). (c) Show that, for all points (x, y) lying on C, dy/dx = 1/ a_/(bx^2) where a and b are constants to be found.