Question 1:

(i) Find a primitive of .

(ii) Find the second derivative of

(iii) Evaluate

(iv) Use Simpson's rule with three points to approximately evaluate

Question 2:

(i) Find the equation of the tangent to the curve at the point x=1,

(ii) Find the acute angle between the lines 2y - x - 1 = 0 and y - 3x + 2 = 0.

(iii) If p is parallel to q , find a and b.

(iv) Find a pair of values of a and b which would instead make p perpendicular to q.

Question 3:

(i) Find the area under the curve between x = 0 and x = 1.

(ii) Find the x-coordinates of all stationary points of .

(iii) Find the set of values of x for which the expression is positive.

Question 4:

(i) Find the equation of the curve in the new (X, Y) coordinate system in which the origin is at x = -2 and y = -1, and the direction of the axes are unchanged.

(ii) Find the focus and vertex of the parabola .

(iii) Prove by math. induction that .

Question 5:

(i) Find the expression for sin(x+y).

(ii) Deduce that sin(x + p/2) = cos(x).

(iii) Express V in the form C.sin(x + a) where C is positive.

(iv) Find all solutions in the interval of the equation

Question 6:

(i) Sketch the graph of . State the domain and range of the function.

(ii) If show that , and sketch the graph of f'(x). Describe the behaviour of this graph near x = +1 and x = -1.

(iii) State the remainder theorem for polynomials. Hence or otherwise, factorise into linear factors.

Question 7:

(i) Two balls are drawn in succession (without replacement) from a bag containing 18 red, 14 blue and 4 white balls. What is the probability that

(a) the first ball is white

(b) both balls are red.

(ii) A biased coin has a probability p=1/3 of giving the result "heads".

(a) If the coin is thrown n times, state without proof the expected number of heads.

(b) If the coin is thrown 3 times, find the probability of obtaining each of the results:

(1) 3 heads (2) 2 heads (3) 1 head

Hence determine the expected number of heads for 3 throws. Verify that this result for 3 agrees with your answer to part (a)

Question 8:

A particle moves on a line so that its distance from the origin at time t is x and its velocity is v:

(i) Prove that

(ii) If the acceleration satisfies where n is a constant, and if the particle is released from rest at x = 0, show that . Hence show that the particle never moves outside a certain interval.

(iii) Show that the expression for the acceleration given in (ii) can be simplified by an approximate change of origin. Hence state without proof the period of the motion.

Question 10:

A can fruit producer wishes to minimise the area of sheet metal used in manufacturing cans of a given volume. Find the ratio of radius to height for the desired can. (Treat the can as a circular cylinder with closed ends).