Edexcel A Level Maths C3 · Core Mathematics 3 June 2014 question paper 6665

The curve C has equation y = f (x) where

Show that

Given that P is a point on C such that f (x) = –1,

find the coordinates of P.

Find the exact solutions, in their simplest form, to the equations

2 ln(2 x + 1) – 10 = 0

The curve C has equation x = 8 y tan 2 y

Verify that P lies on C.

Find the equation of the tangent to C at P in the form ay = x + b, where the constants a and b are to be found in terms of .

Figure 1 shows part of the graph with equation y = f (x), x . The graph consists of two line segments that meet at the point Q (6, –1). The graph crosses the y-axis at the point P (0, 11). Sketch, on separate diagrams, the graphs of

y = | f (x) |

y = 2f (– x) + 3

On each diagram, show the coordinates of the points corresponding to P and Q. Given that f (x) = a | x – b | – 1, where a and b are constants,

state the value of a and the value of b.

Find the range of g.

Find the exact value of a for which g(a) = g^–1(a).

Figure 2 shows a sketch of part of the curve with equation

The curve crosses the x-axis at the point Q and has a minimum turning point at R.

Show that the x coordinate of Q lies between 2.1 and 2.2

Show that the x coordinate of R is a solution of the equation

Using the iterative formula

find the values of x₁ and x₂ to 3 decimal places.

Show that cosec 2x + cot 2x = cot x, x  90n°, n

Hence, or otherwise, solve, for 0 θ 180°,

You must show your working. (Solutions based entirely on graphical or numerical methods are not acceptable.)

A rare species of primrose is being studied. The population, P, of primroses at time t years after the study started is modelled by the equation

Calculate the number of primroses at the start of the study.

Find the exact value of t when P = 250, giving your answer in the form a ln(b) where a and b are integers.

Find the exact value of when t =10. Give your answer in its simplest form.

Explain why the population of primroses can never be 270

Express 2 sin – 4 cos in the form R sin( ± ), where R and are constants, R 0

Give the value of to 3 decimal places.

Find

the maximum value of H(),

the smallest value of , for 0 ≤ θ , at which this maximum value occurs.

Find

the minimum value of H(),

the largest value of , for 0 ≤ , at which this minimum value occurs.