Edexcel A Level Maths C3 · Core Mathematics 3 June 2010 question paper 6665

Show that

Hence find, for –180° θ 180°, all the solutions of

Give your answers to 1 decimal place.

A curve C has equation

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.

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

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

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

Use the iterative formula

to calculate the values of x₁, x₂ and x₃, giving your answers to 4 decimal places.

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

The function f is defined by

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

Solve f (x) =15 + x.

The function g is defined by

Find fg(2).

Find the range of g.

Figure 1 shows a sketch of the curve C with the equation y (2x² 5x 2)e^x.

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

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.

Hence find the exact coordinates of the turning points of C.

Figure 2 shows a sketch of the curve with the equation y = f (x), The curve has a turning point at A (3, 4) and also passes through the point (0, 5).

Write down the coordinates of the point to which A is transformed on the curve with equation

Sketch the curve with equation

On your sketch show the coordinates of all turning points and the coordinates of the point at which the curve cuts the y-axis.

The curve with equation y f (x) is a translation of the curve with equation y = x².

Find f (x).

Explain why the function f does not have an inverse.

Express 2 sin θ – 1.5 cos θ in the form R sin (θ – α), where R 0 and 0 . Give the value of α to 4 decimal places.

Find the maximum value of 2 sin θ – 1.5 cos θ.

Find the value of θ, for 0 θ π, at which this maximum occurs.

Tom models the height of sea water, H metres, on a particular day by the equation

where t hours is the number of hours after midday.

Calculate the maximum value of H predicted by this model and the value of t, to 2 decimal places, when this maximum occurs.

Calculate, to the nearest minute, the times when the height of sea water is predicted, by this model, to be 7 metres.

Simplify fully

Given that ln(2x² + 9x− 5) = 1+ ln(x² + 2x 15), x 5,

find x in terms of e.