Edexcel A Level Maths C4 · Core Mathematics 4 June 2008 question paper 6666

Figure 1 shows part of the curve with equation y = e^{0 5.x2}. The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis, the y-axis and the line x = 2.

Complete the table with the values of y corresponding to x = 0.8 and x = 1.6.

Use the trapezium rule with all the values in the table to find an approximate value for the area of R, giving your answer to 4 significant figures.

2._x (a) Use integration by parts to find. x e dx

Figure 2 shows a right circular cylindrical metal rod which is expanding as it is heated. After t seconds the radius of the rod is x cm and the length of the rod is 5 x cm. The cross-sectional area of the rod is increasing at the constant rate of 0.032 cm² s^–1.

Find when the radius of the rod is 2 cm, giving your answer to 3 significant figures.

Find the rate of increase of the volume of the rod when x = 2.

A curve has equation 3x² – y² + xy = 4. The points P and Q lie on the curve. The gradient of the tangent to the curve is at P and at Q.

Use implicit differentiation to show that y – 2 x = 0 at P and at Q.

Find the coordinates of P and Q.

With respect to a fixed origin O, the lines l₁ and l₂ are given by the equations

where λ and μ are scalar parameters.

Show that l₁ and l₂ meet and find the position vector of their point of intersection.

Show that l₁ and l₂ are perpendicular to each other.

The point A has position vector 5 i + 7 j + 3 k.

Show that A lies on l₁.

The point B is the image of A after reflection in the line l₂.

Find the position vector of B.

Hence obtain the solution of

for which y = 0 at x =x giving your answer in the form sec² x = g(y).

Figure 3 shows the curve C with parametric equations

The point P lies on C and has coordinates (4, 2√3).

Find the value of t at the point P.

The line l is a normal to C at P.

Show that an equation for l is y = –x√3 + 6√3.

The finite region R is enclosed by the curve C, the x-axis and the line x = 4, as shown shaded in Figure 3.

Show that the area of R is given by the integral.²64 sin² t cost dt

Use this integral to find the area of R, giving your answer in the form a + b√3, where a and b are constants to be determined.