Edexcel GCE Core Mathematics C3 January 2012

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C3 Edexcel Core Mathematics January 2012 Question 6

6. f(x) = x² - 3x + 2cos(½ x) , 0 ≤ x ≤ π

(a) Show that the equation f (x) = 0 has a solution in the interval 0.8 <x< 0.9

The curve with equation y = f (x) has a minimum point P.

(b) Show that the x-coordinate of P is the solution of the equation

x = [3 + sin(½ x)]/2

(c) Using the iteration formula
x_n+1 = [3 + sin(½ x_n)]/2 , x₀ = 2

find the values of x₁, x₂ and x₃ , giving your answers to 3 decimal places.
(d) By choosing a suitable interval, show that the x-coordinate of P is 1.9078 correct to 4 decimal places.
6 (a) Roots - Change of sign method.

6 (b) 6 (c) Iteration 6 (d) Change of sign.

C3 Edexcel Core Mathematics January 2012 Question 7

7. The function f is defined by

(a) Show that f(x) = 1/(2x - 1)

(b) Find f^-1(x)

(c) Find the domain of f⁻¹

g(x) = ln(x + 1)

(d) Find the solution of fg(x) = 1/7, giving your answer in terms of e.

7 (a) Simplifying algebraic fractions. 7 (b) Inverse functions

7 (c) domain 7 (d) combining functions C3 Edexcel Core Mathematics January 2012 Question 8

8. (a) Starting from the formulae for sin ( A + B ) and cos ( A + B ), prove that

tan (A + B) = (tan A + tan B)/(1 - tanA tanB)

(b) Deduce that

(c) Hence, or otherwise, solve, for 0 ≤ θ ≤ π ,

1 + √3 tanθ = (√3 - tanθ)tan(π - θ)

Give your answers as multiples of π

8 (a) Trig. Identities 8 (b) Trigonometry

8 (c) Trig. equations