Edexcel ALevel Maths, Pure Mathematics Paper 1, October 2020 Question Walkthroughs
Full walkthroughs with explanations and working out for questions from the Edexcel ALevel Maths, Pure Mathematics Paper 1 from October 2020.

Pure Paper 1, October 2020, Q1a. Find the first four terms, in ascending powers of x, of the binomial expansion of (1 + 8x)^(1/2) giving each term in simplest form.
b. Explain how you could use x = 1/32 in the expansion to find an approximation for the square root of 5. There is no need to carry out the calculation. 

Pure Paper 1, October 2020, Q2By taking logarithms of both sides, solve the equation 4^(3p − 1) = 5^210 giving the value of p to one decimal place.


Pure Paper 1, October 2020, Q3Relative to a fixed origin O
● point A has position vector 2i + 5j − 6k ● point B has position vector 3i − 3j − 4k ● point C has position vector 2i − 16j + 4k a. Find AB b. Show that quadrilateral OABC is a trapezium, giving reasons for your answer. 

Pure Paper 1, October 2020, Q4The function f is defined by f(x) = (3x−7)/(x−2)
a. Find f^(−1)(7) b. Show that ff(x) = (ax+b)/(x−3) where a and b are integers to be found. 

Pure Paper 1, October 2020, Q5A car has six forward gears. The fastest speed of the car
● in 1st gear is 28 km h^–1 ● in 6th gear is 115 km h^–1 Given that the fastest speed of the car in successive gears is modelled by an arithmetic sequence, (a) find the fastest speed of the car in 3rd gear. Given that the fastest speed of the car in successive gears is modelled by a geometric sequence, (b) find the fastest speed of the car in 5th gear. 

Pure Paper 1, October 2020, Q6a. Express sin x + 2 cos x in the form R sin (x + α) where R and α are constants. Give the exact value of R and give the value of α in radians to 3 decimal places. The temperature, θ °C , inside a room on a given day is modelled by the equation θ = 5 + sin (pi t/12  3) + 2 cos (pi t/12  3) where t is the number of hours after midnight. Using the equation of the model and your answer to part (a),

b. deduce the maximum temperature of the room during this day,
c. find the time of day when the maximum temperature occurs, giving your answer to the nearest minute.
c. find the time of day when the maximum temperature occurs, giving your answer to the nearest minute.

Pure Paper 1, October 2020, Q7Figure 1 shows a sketch of a curve C with equation y = f (x) and a straight line l. The curve C meets l at the points (−2, 13) and (0, 25) as shown. The shaded region R is bounded by C and l as shown in Figure 1. Given that
● f (x) is a quadratic function in x ● (−2, 13) is the minimum turning point of y = f (x) use inequalities to define R. 

Pure Paper 1, October 2020, Q8A new smartphone was released by a company. The company monitored the total number of phones sold, n, at time t days after the phone was released. The company observed that, during this time, the rate of increase of n was proportional to n.
Use this information to write down a suitable equation for n in terms of t. (You do not need to evaluate any unknown constants in your equation). 

Pure Paper 1, October 2020, Q9Figure 2 shows a sketch of the curve C with equation y = f(x) where f(x) = 4(x^2 − 2)e^−2x
(a) Show that fʹ(x) = 8(2 + x − x^2)e^−2x (b) Hence find, in simplest form, the exact coordinates of the stationary points of C. The function g and the function h are defined by g(x) = 2f(x), h(x) = 2f(x) − 3, x larger than or equal to 0 (c) Find (i) the range of g (ii) the range of h 

Pure Paper 1, October 2020, Q10Use the substitution x = u^2 + 1 to show that the integral of 3 dx / (x  1)(3 + 2root(x  1) = integral 6 du / u (3 + 2u) where p and q are positive constants to be found.
(b) Hence, using algebraic integration, show that the integral = ln a where a is a rational constant to be found. 

Pure Paper 1, October 2020, Q11Circle C1 has equation x^2 + y^2 = 100
Circle C2 has equation (x − 15)^2 + y^2 = 40 The circles meet at points A and B as shown in Figure 3. (a) Show that angle AOB = 0.635 radians to 3 significant figures, where O is the origin. The region shown shaded in Figure 3 is bounded by C1 and C2 (b) Find the perimeter of the shaded region, giving your answer to one decimal place. 

Pure Paper 1, October 2020, Q12(a) Show that cosec θ − sin θ ≡ cos θ cot θ, θ ≠ (180n)
(b) Hence, or otherwise, solve for x between 0° and 180° cosec x − sin x = cos x cot (3x − 50°) 

Pure Paper 1, October 2020, Q13A sequence of numbers a1, a2, a3, … is defined by a n+1 = k (a n + 2)/ a n where k is a constant.
Given that ● the sequence is a periodic sequence of order 3 ● a1 = 2 (a) show that k^2 + k − 2 = 0 (b) For this sequence explain why k ≠ 1 (c) Find the value of the sum r from 0 to 80 of a r. 

Pure Paper 1, October 2020, Q14A large spherical balloon is deflating. At time t seconds the balloon has radius r cm and volume V cm^3 The volume of the balloon is modelled as decreasing at a constant rate.
(a) Using this model, show that dr/dt = k / r^2 where k is a positive constant. Given that ● the initial radius of the balloon is 40 cm ● after 5 seconds the radius of the balloon is 20 cm 
● the volume of the balloon continues to decrease at a constant rate until the balloon is empty
(b) solve the differential equation to find a complete equation linking r and t.
(c) Find the limitation on the values of t for which the equation in part (b) is valid.
(b) solve the differential equation to find a complete equation linking r and t.
(c) Find the limitation on the values of t for which the equation in part (b) is valid.

Pure Paper 1, October 2020, Q15The curve C has equation x^2 tan y = 9
(a) Show that dy/dx = 18x / (x^4 + 81) (b) Prove that C has a point of inflection at x = 27 

Pure Paper 1, October 2020, Q16Prove by contradiction that there are no positive integers p and q such that 4 p^2 − q^2 = 25
