Edexcel GCE Core Maths C4 Advanced June 2012

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C4 Edexcel Core Mathematics June 2012 Question 6

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

x = (√3) sin 2t, y = 4cos²t, 0 ≤ t ≤ π

(a) Show that dy/dx = k(√3) tan 2t, where k is a constant to be determined.

(b) Find an equation of the tangent to C at the point where t = π3

Give your answer in the form y = ax + b, where a and b are constants.

(c) Find a Cartesian equation of C.

6 (a) Parametric differentiation 6 (b) Tangent to Parametric Curve 6 (c) Parametric to Cartesian Form C4 Edexcel Core Mathematics June 2012 Question 7

Figure 3 shows a sketch of part of the curve with equation y = x^1/2ln2x

The finite region R, shown shaded in Figure 3, is bounded by the curve, the x-axis and the lines x = 1 and x = 4

(a) Use the trapezium rule, with 3 strips of equal width, to find an estimate for the area of R, giving your answer to 2 decimal places.

(b) Find ∫x^1/2ln2x dx

(c) Hence find the exact area of R, giving your answer in the form aln2 + b where a and b are exact constants.

7 (a) Trapezium Rule

7 (b) Integration 7 (c)

C4 Edexcel Core Mathematics June 2012 Question 8

8. Relative to a fixed origin O, the point A has position vector (10i + 2j + 3k) and the point B has position vector (8i + 3j + 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.
The point C has position vector (3i + 12j + 3k).

The point P lies on l. Given that the vector CP is perpendicular to l,
(c) find the position vector of the point P.

8 (a)(b) Vectors

8 (c)