Edexcel May June 2021 IAL Pure Maths WMA13/01

Edexcel May/June 2021 IAL Pure Maths WMA13/01 question paper

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1. The curve C has equation
   The curve has a stationary point at the point P.
   (a) Show, using calculus, that the x coordinate of P is a solution of the equation
   Using the iteration formula
2. (a) Show that
   where k is a constant to be found.
   (b) Hence solve, for 0 < θ < 90°
   giving your answers to one decimal place.
3. (i) Find
   giving your answer in simplest form.
4. The functions f and g are defined by
   (a) Solve the equation
   fg(x) = 3
   (b) Find f⁻¹
   (c) Sketch and label, on the same axes, the curve with equation y = g(x) and the curve with equation y = g⁻¹(x). Show on your sketch the coordinates of the points where each curve meets or cuts the coordinate axes.
5. The growth of duckweed on a pond is being studied.
   The surface area of the pond covered by duckweed, Am², at a time t days after the start of the study is modelled by the equation
   A = pq^t
   where p and q are positive constants
   Figure 1 shows the linear relationship between log10A and t.
   The points (0, 0.32) and (8, 0.56) lie on the line as shown.
   (a) Find, to 3 decimal places, the value of p and the value of q.
   Using the model with the values of p and q found in part (a),
   (b) find the rate of increase of the surface area of the pond covered by duckweed, in m²/day, exactly 6 days after the start of the study.
   Give your answer to 2 decimal places.

6. Given that k is a positive constant,
   (a) on separate diagrams, sketch the graph with equation
   (i) y = k − 2|x|
   (ii) y = |2x − k/3| Show on each sketch the coordinates, in terms of k, of each point where the graph meets or cuts the axes.
   (b) Hence find, in terms of k, the values of x for which
7. Given that
8. A scientist is studying a population of fish in a lake. The number of fish, N, in the population, t years after the start of the study, is modelled by the equation
   Use the equation of the model to answer parts (a), (b), (c), (d) and (e).
   (a) Find the number of fish in the lake at the start of the study.
   (b) Find the upper limit to the number of fish in the lake.
   (c) Find the time, after the start of the study, when there are predicted to be 500 fish in the lake. Give your answer in years and months to the nearest month.
9. (a) Express 12 sin x − 5cos x in the form Rsin (x − α), where R and α are constants, R > 0 and 0 < α < π/2. Give the exact value of R and give the value of α in radians, to 3 decimal places.

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