Edexcel A Level Maths C4 · Core Mathematics 4 June 2014 question paper 6666

A curve C has the equation

Find an equation of the tangent to C at the point (3, –2), giving your answer in the form ax + by + c = 0, where a, b and c are integers.

Given that the binomial expansion of (1 + kx)^–4, kx < 1, is

find the value of the constant k,

find the value of the constant A, giving your answer in its simplest form.

The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis, and the lines with equations x = 1 and x = 4

Complete the table above by giving the missing value of y to 5 decimal places.

Use the trapezium rule, with all the values of y in the completed table, to find an estimate for the area of R, giving your answer to 4 decimal places.

By reference to the curve in Figure 1, state, giving a reason, whether your estimate in part (b) is an overestimate or an underestimate for the area of R.

Use the substitution u = ¥ x, or otherwise, to find the exact value of

A vase with a circular cross-section is shown in Figure 2. Water is flowing into the vase. When the depth of the water is h cm, the volume of water V cm³ is given by

Water flows into the vase at a constant rate of 80 cm³s^–1 Find the rate of change of the depth of the water, in cms^–1, when h = 6

Figure 3 shows a sketch of the curve C with parametric equations

Show that

Show that a cartesian equation of C is

where a and b are integers to be determined.

Find

Find

Given that y = at x = 0, solve the differential equation

___________________________________________________________________________

Figure 4 shows a sketch of part of the curve C with parametric equations

The point P lies on C and has coordinates (3, 2). The line l is the normal to C at P. The normal cuts the x-axis at the point Q.

Find the x coordinate of the point Q.

The finite region S, shown shaded in Figure 4, is bounded by the curve C, the x-axis, the y-axis and the line l. This shaded region is rotated 2 radians about the x-axis to form a solid of revolution.

Find the exact value of the volume of the solid of revolution, giving your answer in the form + q², where p and q are rational numbers to be determined. [You may use the formula V πr²h for the volume of a cone.]

Relative to a fixed origin O, the point A has position vector 4

and the point B has position vector 3

The line l₁ passes through the points A and B.

Find the vector AB.

Hence find a vector equation for the line l₁

The point P has position vector 2

Given that angle PBA is ,

The line l₂ passes through the point P and is parallel to the line l₁

Find a vector equation for the line l₂ The points C and D both lie on the line l₂ Given that AB = PC = DP and the x coordinate of C is positive,

find the coordinates of C and the coordinates of D.

find the exact area of the trapezium ABCD, giving your answer as a simplified surd.