Question 1:

(i) Find the vertex and the axis of symmetry of

(ii) Find the derivative of

(iii) Find the equation of the tangent to at x = 1.

(iv) Sketch the curve and find the area above the x-axis and under the curve.

Question 2:

(i) The first two terms of an AP sequence are -17 , -14. Write down the sum of the first n terms. What is the least value of n for which the sum of the first n terms is positive?

(ii) Evaluate

(iii) In how many ways can a team of 4 be chosen from 7 players.

Question 3:

(i) Write down the equation of the line through (-4, -3) perpendicular to 2x + y = 5.

(ii) Use one application of Newton's method to find approximate the root of

cos(x) + sin(x) = x near x = 1.2 (note that sin 1.2 = 0.932 and cos 1.2 = 0.362).

(iii) The position at time t of a particle moving along the x-axis is given by . When and where the particle comes to rest ?

Question 4:

(i) Find the sum to infinity of the GP sequence

(ii) Prove by math induction that ().

Question 5:

P is on the parabola and L is the tangent at P.

(i) Prove that the equation of L is

(ii) If L cuts the x-axis at A and the y-axis at B, find the coordinates of A and B.

(iii) In what ratio does P divides AB ?

(iv) What is the slope of the line joining P to the focus S ?

(v) Show that L makes equal angles with the y-axis and with PS.

Question 6:

(i) Sketch the curve and with the same coordinate axes for . Find the area enclosed between the two curves on this sketch.

(ii) Differentiate

(iii) Show that

Question 7:

(i) Indicate by shading the region . Find the volume of the solid of revolution obtained by rotating this region about the y-axis.

(ii) Two dice (each with faces labelled 1,2,3,4,5,6) are thrown together

(a) what is the probability of a double 6 in a throw ?

(b) what is the probability of NO double 6 in a throw ?

(c) for what values of n will the probability of (at least) one double 6 in n throws be greater than 1/2. ?

Question 8:

(i) Use the remainder theorem to find one factor of x(x+1) - a(a+1). By division, or otherwise, find the other factor.

Question 8:

A particle moves on a line so that its distance from the origin at time t is x.

(i) Prove that where v denotes velocity.

(ii) If and v = 0 at t = 1, find v in terms of x.

(iii) Describe the motion. Is it simple harmonic ?

Question 10:

(i) Sketch the graph of log[x(1 - x)].

(ii) When the temp. of a body is T degrees above the temp (assuming constant) of its surroundings, its temp falls at a rate proportional to T.

(a) Write down the equation satisfies by T and, by integration, show that where A, k are constants and t denotes time.

(b) If the body is initially at 95oC, the surrounding at 25oC, and the body cools to 75oC after 15 minutes, what will its temp be after a further 20 minutes,