Edexcel A Level Maths C3 · Core Mathematics 3 June 2012 question paper 6665

Express

as a single fraction in its simplest form.

f (x) = x³ + 3x² + 4x −12

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

The equation x³ + 3x² + 4x −12 = 0 has a single root which is between 1 and 2

Use the iteration formula

with x₀ = 1 to find, to 2 decimal places, the value of x₁, x₂ and x₃.

The root of f (x) = 0 is .

By choosing a suitable interval, prove that α 1 272. to 3 decimal places.

Figure 1 shows a sketch of the curve C which has equation

Find the x coordinate of the turning point P on C, for which x 0 Give your answer as a multiple of .

Find an equation of the normal to C at the point where x 0

Figure 2 shows part of the curve with equation y = f (x) The curve passes through the points P(1 5., 0) and Q(0, 5) as shown. On separate diagrams, sketch the curve with equation

y = 2f (3x)

Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

Express 4cosec² 2 − cosec² in terms of sin and cos .

Hence show that

Hence or otherwise solve, for 0 < <,

giving your answers in terms of .

The functions f and g are defined by

State the range of f.

Find fg(x), giving your answer in its simplest form.

Find the exact value of x for which f (2x 3) 6

Find f ¹, the inverse function of f, stating its domain.

On the same axes sketch the curves with equation y = f (x) and y = f ¹(x), giving the coordinates of all the points where the curves cross the axes.

(a) Differentiate with respect to x,

, giving your answer in its simplest form.

Given that x = 3tan 2y find in terms of x.

f (x) = 7 cos 2x − 24sin 2x Given that f (x) = R cos(2x + α), where R 0 and 0 α 90,

find the value of R and the value of .

Hence solve the equation

for 0 ≤ x180, giving your answers to 1 decimal place.

Express 14cos² x 48sin x cos x in the form a cos 2x + bsin 2x + c, where a, b, and c are constants to be found.

Hence, using your answers to parts (a) and (c), deduce the maximum value of