Edexcel A-Level Maths October 2021 - Question Walkthroughs

Pure Paper 1, 2021, Q1

f(x) = ax^3 + 10x^2 − 3ax − 4
Given that (x − 1) is a factor of f (x), find the value of the constant a.
​You must make your method clear.

Pure Paper 1, 2021, Q2

Given that f(x) = x^2 − 4x + 5
(a) express f(x) in the form (x + a)^2 + b where a and b are integers to be found. The curve with equation y = f(x)
• meets the y-axis at the point P
• has a minimum turning point at the point Q
​(b) Write down (i) the coordinates of P (ii) the coordinates of Q

Pure Paper 1, 2021, Q3

The sequence u1, u2, u3, … is defined by un+1 = k − 24/un
u1 = 2 where k is an integer.
Given that u1 + 2u2 + u3 = 0
(a) show that 3k^2 − 58k + 240 = 0
(b) Find the value of k, giving a reason for your answer.
​(c) Find the value of u3

Pure Paper 1, 2021, Q4

The curve with equation y = f(x) where f(x) = x^2 + ln (2x^2 − 4x + 5) has a single turning point at x = α
(a) Show that α is a solution of the equation 2x^3 − 4x^2 + 7x − 2 = 0
The iterative formula xn+1 =1/7 (2 + 4xn^2 − 2xn^3) is used to find an approximate value for α.
Starting with x1 = 0.3
(b) calculate, giving each answer to 4 decimal places, (i) the value of x2 (ii) the value of x4
Using a suitable interval and a suitable function that should be stated,
(c) show that α is 0.341 to 3 decimal places.

Pure Paper 1, 2021, Q5

A company made a profit of £20 000 in its first year of trading, Year 1.
A model for future trading predicts that the yearly profit will increase by 8% each year, so that the yearly profits will form a geometric sequence. According to the model,
​(a) show that the profit for Year 3 will be £23 328
(b) find the first year when the yearly profit will exceed £65 000
(c) find the total profit for the first 20 years of trading, giving your answer to the nearest £1000.

Pure Paper 1, 2021, Q6

Figure 1 shows a sketch of triangle ABC.
Given that
• AB = −3i − 4j − 5k
• BC = i + j + 4k
(a) find AC
​(b) show that cos ABC = 9/10

Pure Paper 1, 2021, Q7

The circle C has equation x^2 + y^2 − 10x + 4y + 11 = 0
(a) Find (i) the coordinates of the centre of C,
(ii) the exact radius of C, giving your answer as a simplified surd.
The line l has equation y = 3x + k where k is a constant. Given that l is a tangent to C,
​(b) find the possible values of k, giving your answers as simplified surds.

Pure Paper 1, 2021, Q8

A scientist is studying the growth of two different populations of bacteria. The number of bacteria, N, in the first population is modelled by the equation N = Ae^kt where A and k are positive constants and t is the time in hours from the start of the study. Given that
• there were 1000 bacteria in this population at the start of the study
​• it took exactly 5 hours from the start of the study for this population to double 
(a) find a complete equation for the model.
(b) Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures.
​The number of bacteria, M, in the second population is modelled by the equation M = 500e^1.4kt where k has the value found in part (a) and t is the time in hours from the start of the study. Given that T hours after the start of the study, the number of bacteria in the two different populations was the same, (c) find the value of T.

Pure Paper 1, 2021, Q9

f(x) = (50x^2 + 38x + 9) / (5x + 2)^2 (1 - 2x)
Given that f(x) can be expressed in the form A/(5x + 2) + B/(5x + 2)^2 + C/(1 - 2x) where A, B and C are constants
(a) (i) find the value of B and the value of C
(ii) show that A = 0
(b) (i) Use binomial expansions to show that, in ascending powers of x f(x) = p + qx + rx^2 + … where p, q and r are simplified fractions to be found.
​(ii) Find the range of values of x for which this expansion is valid.

Pure Paper 1, 2021, Q10

(a) Given that 1 + cos 2θ + sin 2θ ≠ 0 prove that 1- cos 2θ + sin 2θ / 1+ cos 2θ + sin 2θ = tan θ
​(b) Hence solve 1- cos 4x + sin 4x / 1+ cos 4x + sin 4x = 3 sin 2x giving your answers to one decimal place where appropriate.