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1EDEXCEL C4 June 2012 Q1
f(x) = 1 / (x(3x - 1)^2) = A/x + B/(3x - 1) + C/(3x - 1)^2. (a) Find the values of the constants A, B and C. (4) (b) (i) Hence find the integral of f(x) dx. (ii) Find the integral from 1 to 2 of f(x) dx, leaving your answer in the form a + ln b, where a and b are constants. (6)
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2EDEXCEL C4 January 2011 Q3
(a) Express 5 / ((x - 1)(3x + 2)) in partial fractions. (3) (b) Hence find the integral of 5 / ((x - 1)(3x + 2)) dx, where x > 1. (3) (c) Find the particular solution of the differential equation (x - 1)(3x + 2) dy/dx = 5y, x > 1, for which y = 8 at x = 2. Give your answer in the form y = f(x). (6)
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3EDEXCEL C4 June 2009 Q3
f(x) = (4 - 2x) / ((2x + 1)(x + 1)(x + 3)) = A/(2x + 1) + B/(x + 1) + C/(x + 3). (a) Find the values of the constants A, B and C. (4) (b) (i) Hence find the integral of f(x) dx. (3) (ii) Find the integral from 0 to 2 of f(x) dx in the form ln k, where k is a constant. (3)
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4EDEXCEL C4 January 2008 Q7
Figure 3. The curve C has parametric equations x = ln(t + 2), y = 1/(t + 1), t > -1. The finite region R between the curve C and the x-axis, bounded by the lines with equations x = ln 2 and x = ln 4, is shown shaded in Figure 3. (a) Show that the area of R is given by the integral from 0 to 2 of 1 / ((t + 1)(t + 2)) dt. (4) (b) Hence find an exact value for this area. (6) (c) Find a cartesian equation of the curve C, in the form y = f(x). (4) (d) State the domain of values for x for this curve. (1)
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5Edexcel A Level June 2023 Paper 2 Q10
f(x) = (3kx - 18) / ((x + 4)(x - 2)), where k is a positive constant. (a) Express f(x) in partial fractions in terms of k. (3) (b) Hence find the exact value of k for which the integral from -3 to 1 of f(x) dx = 21. (4)
6Edexcel A Level June 2024 Paper 2 Q12
(a) Express 1 / (V(25 - V)) in partial fractions. (2) The volume, V microlitres, of a plant cell t hours after the plant is watered is modelled by the differential equation dV/dt = (1/10) V(25 - V). The plant cell has an initial volume of 20 microlitres. (b) Find, according to the model, the time taken, in minutes, for the volume of the plant cell to reach 24 microlitres. (5) (c) Show that V = A / (e^(-kt) + B), where A, B and k are constants to be found. (3) The model predicts that there is an upper limit, L microlitres, on the volume of the plant cell. (d) Find the value of L, giving a reason for your answer. (2)
