 # Edexcel GCE Core Mathematics C4 Advanced January 2012

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C4 Edexcel Core Mathematics January 2012 Question 1

1. The curve C has the equation 2x + 3y2 + 3x2y = 4x2
The point P on the curve has coordinates (–1, 1).

(a) Find the gradient of the curve at P.

(b) Hence find the equation of the normal to C at P, giving your answer in the form ax + by + c = 0, where a, b and c are integers.

1 (a) Implicit differentiation 1 (b) C4 Edexcel Core Mathematics January 2012 Question 2

2. (a) Use integration by parts to find ∫x sin 3x dx

(b) Using your answer to part (a), find ∫x2 cos 3x dx

2 (a) Integration by Parts 2 (b) Integration by Parts

C4 Edexcel Core Mathematics January 2012 Question 3

3. (a) Expand

1/(2 - 5x)2, |x| < 2/5

in ascending powers of x, up to and including the term in x2, giving each term as a simplified fraction.

Given that the binomial expansion of (2 + kx)/(2 - 5x)2, |x| < 2/5 is

1/2 + 7/4x + Ax2 + ...

(b) find the value of the constant k,

(c) find the value of the constant A.

3 (a) Binomial Expansion 3 (b)(c)

C4 Edexcel Core Mathematics January 2012 Question 4

Figure 1 shows the curve with equation

y = [2x/(3x2 + 4)], x ≥ 0

The finite region S, shown shaded in Figure 1, is bounded by the curve, the x-axis and the line x = 2

The region S is rotated 360° about the x-axis.

Use integration to find the exact value of the volume of the solid generated, giving your answer in the form k ln a, where k and a are constants.

Volume of Revolution
C4 Edexcel Core Mathematics January 2012 Question 5

Figure 2 shows a sketch of the curve C with parametric equations

x = 4 sin (t + π/6), y = cos 2t, 0 ≤ t < 2π

(a) Find an expression for dy/dx in terms of t.

(b) Find the coordinates of all the points on C where dy/dx = 0

5 (a) Parametric Curves 5 (b)

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