Edexcel A Level Maths C3 · Core Mathematics 3 June 2011 question paper 6665

Differentiate with respect to x

f (x) = 2sinn(x²) + x − 2, 0 x 2

Show that f (x) = 0 has a root between x = 0.75 and x = 0.85

The equation f (x) = 0 can be written as x = arcsin 1− 0 5. x¹².

Use the iterative formula₁

to find the values of x₁, x₂ and x₃, giving your answers to 5 decimal places.

Show that = 0.80157 is correct to 5 decimal places.

Figure 1 shows part of the graph of y = f (x), x . The graph consists of two line segments that meet at the point R (4, – 3), as shown in Figure 1. Sketch, on separate diagrams, the graphs of

y = 2f (x + 4),

y = f (x).

On each diagram, show the coordinates of the point corresponding to R.

The function f is defined by

Find the domain of f ¹.

The function g is defined by

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

Find the range of fg.

The mass, m grams, of a leaf t days after it has been picked from a tree is given by

where k and p are positive constants. When the leaf is picked from the tree, its mass is 7.5 grams and 4 days later its mass is 2.5 grams.

Write down the value of p.

Find the value of t when 0 6. ln 3.

Prove that

Hence, or otherwise,

show that tan 15° = 2 – ¥ 3,

solve, for 0 x 360°, cosec 4 x – cot 4 x = 1

Show that

The curve C has equation y = f (x). The point P −1, − lies on C.

Find an equation of the normal to C at P.

Express 2cos 3x – 3sin 3x in the form R cos (3x + ), where R and are constants, R 0 and 0 α . Give your answers to 3 significant figures.

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

where R and are the constants found in part (a).

Hence, or otherwise, find the smallest positive value of x for which the curve with equation y = f (x) has a turning point.