Edexcel 2020 Pure Maths Paper 1, 9MA0/01

Edexcel 2020 Pure Maths Paper 1 (Question Paper)

Edexcel 2020 Pure Maths Paper 1 (Mark Scheme)

1. (a) 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 √5
   There is no need to carry out the calculation.
2. By taking logarithms of both sides, solve the equation
   4^3p−1 = 5²¹⁰
   giving the value of p to one decimal place.
3. Relative 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.

4. The function f is defined by
   f(x) = (3x - 7)/(x - 2), 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.
5. A 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 3^rd 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 5^th gear.

6. (a) Express sin x + 2 cos x in the form Rsin (x + α) where R and α are constants, R > 0 and 0 < α < 2π
   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 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.
7. Figure 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.

8. A 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.)
9. Figure 2 shows a sketch of the curve C with equation y = f(x) where
10. (a) Use the substitution x = u² + 1 to show that
11. Circle C1 has equation x² + y² = 100
    Circle C2 has equation (x − 15)² + y² = 40
12. (a) Show that
    cosec θ − sin θ ≡ cos θ cot θ
13. A sequence of numbers a1, a2, a3, … is defined by
14. A large spherical balloon is deflating.
    At time t seconds the balloon has radius r cm and volume Vcm³
    The volume of the balloon is modelled as decreasing at a constant rate.
    (a) Using this model, show that
    
15. The curve C has equation
16. Prove by contradiction that there are no positive integers p and q such that

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