(i) Differentiate (a) (b)

(ii) Write down primitive functions of (a) (b)

(iii) (a) Find the equation of the normal N to the curve at A(1, 5).

(b) Find, to the nearest degree, the size of the acute angle between the line N and the line L: .

(i) Two circles cut at B and C. A diameter of one circle is AB while BD is the diameter of the other.

Prove that A, C and D are collinear, giving reason.

(ii) Find the value of x for which

(iii) (a) Differentiate

(b) hence evaluate the integral

(iv) State the domain and range of y =

(i) Find, for , all solutions of the equation sin2x = cosx.

(ii) Find the coefficient of in the expansion

(iii) One fifth of all beans are black. A random sample of 2 black beans is chosen.

(a) What is the probability that this sample contains exactly 2 black beans.

(b) What is the probability that the sample contains fewer than two black beans.

(c) Which is most likely: the sample contains fewer than 2 black beans or

the sample contains more than 2 black beans.

(i) Find the volume of the solid formed when the region bounded by the x-axis and the curve between x=0 and x=2 is rotated about the x-axis. (use u = to evaluate the integral).

(ii) A pebble is projected from the top of a vertical cliff with velcocity 20 m/s at an angle of elevation of 30o. The cliff is 40 m high and overlooks a lake.

(a) Take the origin O to be the point at the base of the cliff immediately below the point of projection. Derive the expressions for the vertical and horizontal component displacement (y and x) of the pebble.

(b) Calculate the time which elapses before the pebble hits the lake and the distance of the point of impact from the foot of the cliff. Taking g=10.

The polynomial has a root at x=2.

(a) Find all roots of f(x) = 0.

(b) Draw a sketch of the polynomial f(x), showing all stationary points and points of intersection between the x-axis and y-axis.

(c) Apply Newton method to find another root of f(x) if the initial approximation is x=1.

(d) Willy chose an initial approximation x=0.49 and use Newton's method a number of times to find other roots. State, giving reasons, the root of f(x) = 0 to which Willy's approximation are getting closer. (It is not necessary to do additional calculation)

(i) The rate at which a body cools in air is assumed to be proportional to the difference between its temperature T and the constant temperature S of surrounding air. This can be expressed by the differential equation , where t is the time in hours and k is a constant.

(a) Show that (where B is another constant) is a solution of the differential equation.

(b) A heated body cools from 80o to 40o in 2 hours. The air temp. S around body is 20o. Find the temp. of the body after one further hour has elapsed.

(ii) Two points and lie on the parabola where a>0, the chord PQ passes through the focus.

(a) Show that pq = -1.

(b) Show that the point of intersection T of the tangents to the parabola at P and Q lies on the line y = -a.

(c) Show that the chord PQ has length

The rectangular piece of paper PQRS shown here is folded along a line AB, where A and B lie on edges PQ and PS, respectively. This line is so positioned that, after folding, P coincides with a point P' which lies on the edge QR. This fold line AB makes an acute angle q with the edge PQ. The length of AB is L and that of PQ is W.

(a) Show that P'AQ = (p - 2q)

(b) Prove that

(c) More than one fold line exists such that P coincides with a point on QR after folding. Find the value of q which corresponds to the fold line of minimum length.

(d) Let CD be the fold line of minimum length, where C lies on PQ and D lies on PS. Calculate the length of CP.