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

Express

as a single fraction in its simplest form.

Show that f(x) = 0 can be rearranged as

The equation f(x) = 0 has one positive root α.

Taking x₁ = 0, find, to 3 decimal places, the values of x₂, x₃ and x₄.

Show that α = 2.057 correct to 3 decimal places.

Express 5 cos x – 3 sin x in the form R cos(x + α), where R > 0 and 0 < α < .

Hence, or otherwise, solve the equation 5 cos x – 3 sin x = 4 for 0 x 2›, giving your answers to 2 decimal places.

Given that x = tan y, show that =

Sketch the graph of y = ln x, stating the coordinates of any points of intersection with the axes.

Figure 1 shows a sketch of the graph of y = f (x). The graph intersects the y-axis at the point (0, 1) and the point A (2, 3) is the maximum turning point. Sketch, on separate axes, the graphs of

y = f(– x) + 1,

y = f(x + 2) + 3,

y = 2f(2 x). On each sketch, show the coordinates of the point at which your graph intersects the y-axis and the coordinates of the point to which A is transformed.

By writing sec x as , show that sec x tan x.

Given that y = e^2x sec 3x,

The curve with equation y = e^2x sec 3x, − x , has a minimum turning point at (a, b).

Find the values of the constants a and b, giving your answers to 3 significant figures.

Solve

Find the exact solutions to the equations

ln (3 x – 7) = 5

The functions f and g are defined by

Find f^–1 and state its domain.

Find fg and state its range.