WMA13 Pure Mathematics 3 (P3): May/June 2021 Paper 01

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

(b)find the value of x₂ and the value of x₆ , giving your answers to 3 decimal places.

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(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.

(Solutions based entirely on graphical or numerical methods are not acceptable.)

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(i)Find

giving your answer in simplest form.

(ii)(a)Write 4x + 3x + 2 in the form

(b)Hence find, using algebraic integration, the exact value of

giving your answer in simplest form.

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The functions f and g are defined by

(a)Solve the equation

(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.

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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

Figure 1 shows the linear relationship between log₁₀ A 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.

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Given that k is a positive constant,

(a)on separate diagrams, sketch the graph with equation

(i)y = k − 2| x |

(ii)y = x − k3

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

giving your answers in simplest form.

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Given that

where A and B are integers to be found.

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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

(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.

(d)Show that

where A is a constant to be found.

Given that when t = T, dNdt = 8

(e)find the value of T to one decimal place.

(Solutions relying entirely on calculator technology are not acceptable.)

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(a)Express 12 sin x − 5 cos x in the form Rsin (x − α), where R and α are constants,

R > 0 and 0 < α < π2 . Give the exact value of R and give the value of α in radians,

to 3 decimal places.

The function g is defined by

(b)(i)the minimum value of g(θ)

(ii)the smallest value of θ at which the minimum value occurs.

The function h is defined by

(c)Find the range of h.

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