Edexcel Jan 2021 IAL Pure Maths WMA13/01

Edexcel Jan 2021 IAL Pure Maths WMA13/01 question paper

Edexcel Jan 2021 IAL Pure Maths WMA13/01 mark scheme

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1. Find
   giving your answer in simplest form.
2. Figure 1 shows a sketch of the curve with equation y = f(x), where x ∈ R and f(x) is a polynomial.
   The curve passes through the origin and touches the x-axis at the point (3, 0)
   There is a maximum turning point at (1, 2) and a minimum turning point at (3, 0)
   On separate diagrams, sketch the curve with equation
   (i) y = 3f(2x)
   (ii) y = f(−x) − 1
   On each sketch, show clearly the coordinates of

• the point where the curve crosses the y-axis
• any maximum or minimum turning points

3. (a) Show that
   where a, b, c and d are integers to be found.
   (b) Hence find f⁻¹(x)
   (c) Find the domain of f⁻¹
4. Figure 2 shows a sketch of the graph with equation y = f(x), where f(x) = |3x + a| + a
   and where a is a positive constant.
   The graph has a vertex at the point P, as shown in Figure 2.
   (a) Find, in terms of a, the coordinates of P.
   (b) Sketch the graph with equation y = g(x), where
   g(x) = |x + 5a|
   On your sketch, show the coordinates, in terms of a, of each point where the graph cuts or meets the coordinate axes.
   The graph with equation y = g(x) intersects the graph with equation y = f(x) at two points.
   (c) Find, in terms of a, the coordinates of the two points
5. The temperature, θ °C, inside an oven, t minutes after the oven is switched on, is given by
   θ = A – 180e^–kt
   where A and k are positive constants.
   Given that the temperature inside the oven is initially 18°C,
   (a) find the value of A.
   The temperature inside the oven, 5 minutes after the oven is switched on, is 90°C.
   (b) Show that k = plnq where p and q are rational numbers to be found.
   Hence find
   (c) the temperature inside the oven 9 minutes after the oven is switched on, giving your answer to 3 significant figures,
   (d) the rate of increase of the temperature inside the oven 9 minutes after the oven is switched on. Give your answer in °C min^–1 to 3 significant figures.

6. f(x) = x cos(x/3), x > 0
   (a) Find fʹ(x)
   (b) Show that the equation fʹ(x) = 0 can be written as
7. (a) Prove that
8. The percentage, P, of the population of a small country who have access to the internet, is modelled by the equation
   P = ab^t
   where a and b are constants and t is the number of years after the start of 2005
   Using the data for the years between the start of 2005 and the start of 2010, a graph is plotted of log₁₀ P against t.
   The points are found to lie approximately on a straight line with gradient 0.09 and intercept 0.68 on the log₁₀ P axis.
   (a) Find, according to the model, the value of a and the value of b, giving your answers to 2 decimal places.
   (b) In the context of the model, give a practical interpretation of the constant a.
   (c) Use the model to estimate the percentage of the population who had access to the internet at the start of 2015
9. Find
10. The curve C has equation
    x = 3 sec² 2y, x > 3, 0 < y < π/4
    (a) Find dy/dx in terms of y.
    (b) Hence show that

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