Calculus
Question 1 Use a SEPARATE book clearly marked Question 1
(i) Find
(ii) Simplify
(iii) Find how many ways 3 boys and 4 girls can be selected from a group of 8 boys and 6 girls.
(iv) In a box of 8 light globes two are defective. If I select 3 light globes at random, find the probability that none are defective.
(v) Two dice are tossed and the numbers on the two uppermost faces are multiplied together.
(a) List out the Sample Space in table form.
(b) Find the probability that the result is a perfect square.
(vi) Find the odds against rolling a sum of 9 in a single roll of two fair dice.
(vii) If find the value of k.
Question 2 Use a SEPARATE book clearly marked Question 2
(i) Find the indefinite integral of each of the following:
(a)
(b)
(ii) Sketch the graph of the curve and find:
(a) the area of the region enclosed by the curve and the x-axis.
(b) the volume of the solid of revolution formed when the region in part (a) is rotated about the x-axis.
(iii) Consider the curve
(a) Write down the equations of all the asymptotes of the curve.
(b) Find the stationary point on the curve and determine its nature.
(c) Find the exact x-intercept of the curve.
(d) Sketch the curve showing all of the above features.
(e) Find the values of x for which the curve is concave down.
Question 3 Use a SEPARATE book clearly marked Question 3
(i) Three adults and five children stand in a line to have their photo taken. In how many different ways can they arrange themselves if:
(a) there is no restriction.
(b) the children must all be together.
(c) there must be an adult at the start and end of the line.
(ii) The probability that a jackpot prize will be won in any lottery draw is 0.012.
(a) Find correct to five decimal places the probability the jackpot prize will be won exactly three times in 10 independent lottery draws.
(b) The jackpot prize is initially $10 000 and increases by $10 000 each time it is not won so that for example if it is not won in the first lottery draw then the value of the prize increases to $20 000 for the second lottery draw. Find the probability the jackpot prize will exceed $ 200 000 when it is finally won.
(iii) A rare violin worth $10 000 is insured against theft for an annual premium of $600. The probability that the violin will be stolen during any one year is estimated to be 0.05 .
(a) Copy and complete the payoff table below for the probability distribution of X, the value of the insurance policy to the policy holder.
(b) Hence find the expected value of the insurance policy to the policy holder.