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Questions and Worked Solutions for C3 Edexcel Core Mathematics June 2010.

Edexcel Core Mathematics C3 June 2010 Past Paper

C3 Mathematics Edexcel June 2010 Question 1

- (a) Show that

sin2θ/(1 + cos2θ) = tanθ

(b) Hence find, for –180° ≤ θ < 180°, all the solutions of

2sin2θ/(1 + cos2θ) + 1 = tanθ

Give your answers to 1 decimal place.

1 (a)

1 (b)

C3 Mathematics Edexcel June 2010 Question 2

2. A curve C has equation

y = 3/(5 - 3x)^{2}, x ≠ 5/3

The point P on C has x-coordinate 2. Find an equation of the normal to C at P in the form ax + by + c = 0, where a, b and c are integers.

C3 Mathematics Edexcel June 2010 Question 3

3. f(x) = 4cosec x - 4x + 1, where x is in radians.

(a) Show that there is a root α of f(x) = 0 in the interval [1.2, 1.3].

(b) Show that the equation f(x) = 0 can be written in the form

x = 1/sin x + 1/4

(c) Use the iterative formula

x_{n+1} = 1/sin x_{n} + 1/4, x_{0} = 1.25

to calculate the values of x_{1}, x_{2} and x_{3} , giving your answers to 4 decimal places.

(d) By considering the change of sign of f(x) in a suitable interval, verify that α = 1.291 correct to 3 decimal places.

3 (a)

3 (b)

3 (c)

3 (d)

C3 Mathematics Edexcel June 2010 Question 4

4. The function f is defined by

f: x ↦ |2x -5|, x ∈ ℝ

(a) Sketch the graph with equation y = f(x), showing the coordinates of the points where the graph cuts or meets the axes.

(b) Solve f(x) = 15 + x

The function g is defined by

g: x ↦ x^{2} - 4x + 1, x ∈ ℝ, 0 ≤ x ≤ 5

(c) Find fg(2).

(d) Find the range of g.

4 (a)

4 (b)

4 (c)

4 (d)

C3 Mathematics Edexcel June 2010 Question 5

Figure 1 shows a sketch of the curve C with the equation y = (2x^{2} - 5x + 2)e^{-x}

(a) Find the coordinates of the point where C crosses the y-axis.

(b) Show that C crosses the x-axis at x = 2 and find the x-coordinate of the other point where C crosses the x-axis.

(c) Find dy/dx

(d) Hence find the exact coordinates of the turning points of C.

5 (a)(b)

5 (c)

5 (d)

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