Edexcel A Level Maths C3 · Core Mathematics 3 June 2013 question paper 6665

Given that

find the values of the constants a, b, c, d and e.

Given that f(x) = ln x, x > 0 sketch on separate axes the graphs of

y = f(x),

y = | f(x) |,

y = –f(x – 4). Show, on each diagram, the point where the graph meets or crosses the x-axis. In each case, state the equation of the asymptote.

Given that

Show, without using a calculator, that

Hence solve, for 0 < 360,

giving your answers to 1 decimal place.

f(x) = 25x²e^2x – 16, x

Using calculus, find the exact coordinates of the turning points on the curve with equation y = f(x).

Show that the equation f(x) = 0 can be written as x = ± e^–x The equation f(x) = 0 has a root , where = 0.5 to 1 decimal place.

Starting with x₀ = 0.5, use the iteration formula

to calculate the values of x₁, x₂ and x₃, giving your answers to 3 decimal places.

Give an accurate estimate for to 2 decimal places, and justify your answer.

Given that

Hence show that

Find an expression for in terms of x. Give your answer in its simplest form.

Find algebraically the exact solutions to the equations

ln(4 – 2 x) + ln(9 – 3 x) = 2ln(x + 1), –1 < x < 2

2^x e^3x+1 = 10 Give your answer to (b) in the form where a, b, c and d are integers.

The function f has domain –2 x 6 and is linear from (–2, 10) to (2, 0) and from (2, 0) to (6, 4). A sketch of the graph of y = f(x) is shown in Figure 1.

Write down the range of f.

Find ff(0).

The function g is defined by

Solve the equation gf(x) = 16

Kate crosses a road, of constant width 7 m, in order to take a photograph of a marathon runner, John, approaching at 3 m s^–1. Kate is 24 m ahead of John when she starts to cross the road from the fixed point A. John passes her as she reaches the other side of the road at a variable point B, as shown in Figure 2. Kate’s speed is V m s^–1 and she moves in a straight line, which makes an angle , 0 < < 150, with the edge of the road, as shown in Figure 2. You may assume that V is given by the formula

Express 24sin + 7cos in the form Rcos( – ), where R and are constants and where R > 0 and 0 < < 90, giving the value of to 2 decimal places.

Given that varies,

find the minimum value of V.

Given that Kate’s speed has the value found in part (b),

find the distance AB.

Given instead that Kate’s speed is 1.68 m s^–1,

find the two possible values of the angle , given that 0 < < 150.