May 2014
5th Yr. Maths
Mr. Donnelly(Higher Level)
Attempt all 5 Qs
1. Probability and statistics
(a) Two events A and B are such that P(A) = 0·2 , P(A∩B) = 0·15 and P(A′∩B) = 0·6 .
(i) Represent this information with a Venn diagram
(ii) Find the probability that neither A nor B happens.
(iii) Find the conditional probability P(A|B) .
(iv) State whether A and B are independent events and justify your answer.
(b) What is the probability of choosing 1 red card and 2 black cards from a pack of 52 cards?
(c) The average age of a person getting married for the first time in the U.S. is 26 years.
Assume the ages have a normal distribution with a standard deviation of 4 years.
Use the Empirical rule to write three statements about the percentage of people marrying in
certain age categories.
(d) There are 16 girls and 8 boys in a class. Half of these 24 students study French. The
probability that a randomly selected girl studies French is 1·5 times the probability that a randomly
selected boy studies French. How many of the boys in the class study French?
2. Algebra
(a) Solve the equation:
(b) Scientists can estimate the age of certain ancient items by measuring the proportion of
0.693t
5730
carbon−14, relative to the total carbon content in the item. The formula used is Q = e
where Q is the proportion of carbon−14 remaining and t is the age, in years, of the item.
,
(i) An item is 2000 years old. Use the formula to find the proportion of carbon−14 in the item.
(ii) The proportion of carbon−14 in an item found at Lough Boora, County Offaly, was 0·3402.
Estimate, correct to two significant figures, the age of the item.
3. Coordinate Geometry
Answer question 3 on the question sheet, and hand this page up (make sure to write your name
on it).
4. Sequences and Series
(a) If a sequence begins as follows: 4, 7, 10, 13, …
(i)
the 27th term
(ii)
the sum of the first 19 terms
Find:
(b) A culture of bacteria doubles every hour. If there are 100 bacteria right now, how many
bacteria will there be at the same time tomorrow, assuming none of these bacteria die in the
meantime?
(c) Claire has a starting salary of €20,000 and gets a 5% increase per year.
(i)
How much will she be earning in her 10th year in the job?
(ii)
How much in total will she earn in the first 10 years she works for this firm?
5. Functions
(a) Find the inverse of the following functions
(i) f(x) = 3x + 1
(ii) g(x) =
(c) Identify the quadratic functions labelled a, b and c in the graph below.
The red function f is
6. Trigonometry
y = x2
(a) Explain (in your own words or otherwise) what a radian angle is.
(c) Find
(i) the length of the minor arc PQ
(ii) the area of the corresponding minor sector OPQ
P
8cm
1.2 rad
O
Q
8cm
(d) Solve the equations in Ɵ, for ƟЄR (i.e. the general form of the solution).
(i) tanƟ = -1
(ii) cos2Ɵ = ½
7. Geometry
(a) Construct a triangle with side lengths 6cm, 8cm and 11cm.
Then construct the centroid of this triangle.
(b) Show that the length of steel tubing required to make a sculpture in the shape
of a square-based pyramid, as illustrated below, is given by the equation: