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The following videos will give you the worked solutions and answers for the Edexcel GCE Core Mathematics C4 Advanced January 2012. Try out the Past paper for Edexcel C4 January 2012 and check out the video solutions if you need any help.

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C4 Edexcel Core Mathematics January 2012 Question 6

6. Figure 3 shows a sketch of the curve with equation y = 2 sin2x/(1 + cos x), 0 ≤ x ≤ π/2

The finite region R, shown shaded in Figure 3, is bounded by the curve and the x-axis.

The table below shows corresponding values of x and y for y = 2 sin2x/(1 + cos x)

(a) Complete the table above giving the missing value of y to 5 decimal places.

(b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to 4 decimal places.

(c) Using the substitution u = 1 + cos x , or otherwise, show that

∫2 sin2x/(1 + cos x) dx = 4ln(1 + cos x) - 4cosx + k

where k is a constant.

(d) Hence calculate the error of the estimate in part (b), giving your answer to 2 significant figures.

6 (a)(b) Trapezium Rule7. Relative to a fixed origin O, the point A has position vector ( 2i – j + 5k ), the point B has position vector ( 5i + 2j + 10k ), and the point D has position vector ( –i + j + 4k ).

The line l passes through the points A and B.

(a) Find the vector AB.

(b) Find a vector equation for the line l.

(c) Show that the size of the angle BAD is 109°, to the nearest degree.

The points A, B and D, together with a point C, are the vertices of the parallelogram ABCD, where AB = DC.

(d) Find the position vector of C.

(e) Find the area of the parallelogram ABCD, giving your answer to 3 significant figures.

(f) Find the shortest distance from the point D to the line l, giving your answer to 3 significant figures.

7 (a) Vectors7 (d)

8. (a) Express 1/[P(5-P)] in partial fractions.

A team of conservationists is studying the population of meerkats on a nature reserve. The population is modelled by the differential equation

dP/dt = 1/15 P(5 - P), t ≥ 0

where P, in thousands, is the population of meerkats and t is the time measured in years since the study began.

Given that when t = 0, P = 1,

(b) solve the differential equation, giving your answer in the form,

P = /(b + ce^{-1/3t}) where a, b and c are integers.

(c) Hence show that the population cannot exceed 5000.

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