Edexcel Jan 2020 IAL Pure Maths WMA13/01

Edexcel Jan 2020 IAL Pure Maths WMA13/01 (pdf)

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1. A population of a rare species of toad is being studied.
   The number of toads, N, in the population, t years after the start of the study, is modelled by the equation
   According to this model,
   (a) calculate the number of toads in the population at the start of the study,
   (b) find the value of t when there are 420 toads in the population, giving your answer to 2 decimal places.
   (c) Explain why, according to this model, the number of toads in the population can never reach 500
2. The function f and the function g are defined by
3. Figure 1 shows a linear relationship between log₁₀ y and log₁₀ x
   The line passes through the points (0, 4) and (6, 0) as shown.
   (a) Find an equation linking log₁₀ y with log₁₀ x
4. (a) Find f′(x) in the form
   
5. (a) Use the substitution t = tanx to show that the equation
   12tan 2x + 5cot x sec² x = 0
   can be written in the form
   5t⁴ – 24t² – 5 = 0
   (b) Hence solve, for 0 ≤ x < 360°, the equation
   12tan 2x + 5 cot x sec2 x = 0
   Show each stage of your working and give your answers to one decimal place

6. Figure 2 shows part of the graph with equation y = f(x), where
   f(x) = 2|2x – 5| + 3, x ≥ 0
   The vertex of the graph is at point P as shown.
   (a) State the coordinates of P.
   (b) Solve the equation f(x) = 3x – 2
   Given that the equation
   f(x) = kx + 2
   where k is a constant, has exactly two roots,
   (c) find the range of values of k.
7. Figure 3 shows a sketch of part of the curve with equation
   y = 2 cos 3x – 3x + 4, x > 0
   where x is measured in radians.
   The curve crosses the x‑axis at the point P, as shown in Figure 3.
   Given that the x coordinate of P is α,
   (a) show that α lies between 0.8 and 0.9
8. (i) Find, using algebraic integration, the exact value of
9. f(θ) = 5cos θ – 4sinθ, θ ∈ R
   (a) Express f(θ) in the form Rcos(θ + 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.
   The curve with equation y = cos θ is transformed onto the curve with equation y = f(θ) by a sequence of two transformations.
   Given that the first transformation is a stretch and the second a translation,
   (b) (i) describe fully the transformation that is a stretch,
   (ii) describe fully the transformation that is a translation

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