Edexcel GCE Core Mathematics C4 Advanced January 2013
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C4 Edexcel Core Mathematics January 2013 Question 1
Binomial Expansion
1. Given
f(x) = (2 + 3x)–3, |x| < 2/3
find the binomial expansion of f(x), in ascending powers of x, up to and including the term in x3.
Give each coefficient as a simplified fraction.
Integration
2. (a) Use integration to find
∫ 1/x3 ln x dx
(b) Hence calculate
3. Express
(9x2 + 20x -10)/[(x + 2)(3x -1)]
in partial fractions.
Trapezium Rule
Figure 1 shows a sketch of part of the curve with equation y = x/(1 + √x). The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis, the line with equation x = 1 and the line with equation x = 4.
(a) Complete the table with the value of y corresponding to x = 3, giving your answer to 4 decimal places.
(b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate of the area of the region R, giving your answer to 3 decimal places.
(c) Use the substitution u = 1 + √x, to find, by integrating, the exact area of R.
4 (c) Integration (substitution)
Parametric Equations
Figure 2 shows a sketch of part of the curve C with parametric equations
x = 1 - ½t, y = 2t -1
The curve crosses the y-axis at the point A and crosses the x-axis at the point B.
(a) Show that A has coordinates (0, 3).
(b) Find the x coordinate of the point B.
(c) Find an equation of the normal to C at the point A.
The region R, as shown shaded in Figure 2, is bounded by the curve C, the line x = –1 and the x-axis.
(d) Use integration to find the exact area of R.
5 (c) Normal to Parametric Curve
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