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Edexcel Core Mathematics C3 January 2011




 

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Edexcel Core Mathematics C3 January 2011 Past Paper

C3 Mathematics Edexcel January 2011 Question 1

1. (a) Express 7 cos x - 24 sin x in the form R cos (x + α) where R > 0 and 0 < α < π/2

Give the value of α to 3 decimal places.

(b) Hence write down the minimum value of 7 cos x – 24 sin x.

(c) Solve, for 0 ≤ x < 2π, the equation

7 cos x – 24 sin x = 10

giving your answers to 2 decimal places.
1 (a) Trigonometry

1(b)



1 (c)
C3 Mathematics Edexcel January 2011 Question 2

2. (a) Express
(4x -1)/[2(x -1)] - 3/[2(x - 1)(2x - 1)]

as a single fraction in its simplest form.

Given that

f(x) = (4x -1)/[2(x -1)] - 3/[2(x - 1)(2x - 1)] - 2, x > 1

(b) show that f(x) = 3/(2x -1)

(c) Hence differentiate f(x) and find f'(2).

2 (a)


 
2(b)
2 (c)


C3 Mathematics Edexcel January 2011 Question 3

3. Find all the solutions of

2 cos 2θ = 1 – 2 sin θ

in the interval 0 ≤ θ < 360°.

C3 Mathematics Edexcel January 2011 Question 4

4. Joan brings a cup of hot tea into a room and places the cup on a table. At time t minutes after Joan places the cup on the table, the temperature, θ °C, of the tea is modelled by the equation

θ = 20 + Ae-kt ,

where A and k are positive constants.

Given that the initial temperature of the tea was 90°C,

(a) find the value of A.

The tea takes 5 minutes to decrease in temperature from 90°C to 55°C.

(b) Show that k = 1/5 ln2.

(c) Find the rate at which the temperature of the tea is decreasing at the instant when t = 10. Give your answer, in °C per minute, to 3 decimal places.

4(a)


 
4 (b)
4 (c)


C3 Mathematics Edexcel January 2011 Question 5

Figure 1 shows a sketch of part of the curve with equation y = f(x) , where

f(x) = (8 - x)ln x , x > 0

The curve cuts the x-axis at the points A and B and has a maximum turning point at Q, as shown in Figure 1.

(a) Write down the coordinates of A and the coordinates of B.

(b) Find f'(x)

(c) Show that the x-coordinate of Q lies between 3.5 and 3.6

(d) Show that the x-coordinate of Q is the solution of

x = 8/(1 + ln x)

To find an approximation for the x-coordinate of Q, the iteration formula

xn+1 = 8/(1 + ln xn)

is used.

(e) Taking x0 = 3.55, find the values of x1, x2 and x3.

Give your answers to 3 decimal places. 5(a)

5 (b)


 
5 (c)
5 (d)

5 (e)
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