Answer: compliments of the season to you (complements of the C's on 2u)
To find the tangent to a parabola at a point P:
Draw a vertical line through P.
Draw a horizontal line through Q, the vertex of the parabola.
R is the intersection of these two lines.
M is the midpoint of Q and R.
MP is the tangent to the parabola at P.
Can you prove that MP is the tangent to the parabola?
The graph of x^{3} + 3xy + y^{3} = 1 is shown:
Can you prove that the graph is a straight line?
Are there any points that satisfy the equation which aren't on the line?
Item submitted by Dr Jeremy D. King, Tonbridge School
Bill's cat has 8 kittens: 3 female and 5 male. Alex is going to have two of them. Alex decides that she must have both kittens of the same gender. One kitten is chosen at random then the other will be chosen at random from those of the same gender.
What is the probability that she gets two female kittens?
Two possible answers:
Both these answers cannot be correct. Which one is wrong and why?
The diagram shows an equilateral triangle in a rectangle. The two shapes share a corner and the other corners of the triangle lie on the edges of the rectangle.
Prove that the area of the green triangle is equal to the sum of the areas of the blue and red triangles. What is the most elegant proof of this fact?
If you ask students to find an example of something which is impossible their natural reaction is likely to be to ask "Why?". To be fully satisfied that there are no such example most students will want to see a proof.
e.g. (1) "Find a Pythagorean triple where two of the numbers are even and one is odd."
Many mathematical ideas requiring proof can rephrased in such a way.
e.g. (2) "Show that any graph (in discrete mathematics) has an even number of odd nodes" can be rephrased as "Find a graph with an odd number of odd nodes".
e.g. (3) "Show that the difference between the squares of two consecutive numbers is always odd" can be rephrased as "Find two consecutive numbers such that the difference between their squares is even".
It is important not to tell the students that the question is impossible for the question "Why?" to arise naturally.
The circle x^{2} + y^{2} = 5^{2} has 12 points with integer coordinates, as does the circle x^{2} + y^{2} = 13^{2}.
To find a circle with more than 12 points with integer coordinates you could multiply 5^{2} and 13^{2} to obtain x^{2} + y^{2} = 65^{2} or you could multiply 5 and 13 to obtain x^{2} + y^{2} = 65 (as 65 can be written as the sum of two distinct squares in two different ways).
Does this result generalise: can the product of the largest values in two Pythagorean triples always be written as the sum of two distinct squares in two different ways?
MEI has a number of branches around the country. These are local groupings of MEI teachers who meet two or three times a year.
Branch meetings are intended to give teachers an opportunity to discuss issues and help them in their delivery of MEI courses. They are a valuable form of professional development and often include sessions from MEI professional officers as well as discussion of recent examination papers.
Branches are a vital communication link within MEI. There is an annual Branch Chairmen's conference for Chairmen to receive information and for MEI to hear what is going on in the Branches and the concerns of their members.
The first few Fibonacci numbers: F_{1}=1, F_{2}=1, F_{3}=2, F_{4}=3, F_{5}=5, F_{6}=8, F_{7}=13, ...
1, 1, 2, 3, 5, 8, 13, 21, ?
Zeckendorf representations: Let the notation (a_{n}a_{n1}...a_{4}a_{3}a_{2})_{F} where each a_{r} = 0 or 1, represent the number a_{n}F_{n}+a_{n1}F_{n1}+...+a_{4}F_{4}+a_{3}F_{3}+a_{2}F_{2}
Such a Fibonacci representation is called a Zeckendorf representation.
For example, 17=13+3+1 and also 17=8+5+3+1. Therefore there are two Zeckendorf representations of 17: 17 = 11101_{F} = 100101_{F}
Theorem (Edouard Zeckendorf 1972):
Every positive integer n has a unique Zeckendorf representation with no consecutive 1s.
13 
8 
5 
3 
2 
1 

1 
0 
1 
=4 

1 
0 
0 
0 
=5 

1 
0 
0 
1 
=6 

1 
0 
1 
0 
=7 

1 
0 
0 
0 
0 
=8 

1 
0 
0 
0 
1 
=9 

1 
0 
0 
1 
0 
=10 

1 
0 
1 
0 
0 
=11 

1 
0 
1 
0 
1 
=12 

1 
0 
0 
0 
0 
0 
=13 
1 
0 
0 
0 
0 
1 
=14 
1 
0 
0 
0 
1 
0 
=15 
1 
0 
0 
1 
0 
0 
=16 
1 
0 
0 
1 
0 
1 
=17 
1 
0 
1 
0 
0 
0 
=18 
1 
0 
1 
0 
0 
1 
=19 
1 
0 
1 
0 
1 
0 
=20 
For more information see the Zeckendorf Wikipedia page.
Did you know that all of the past papers on this site are provided with the markscheme and the examiner's report as a single file? This is to make it easy for teachers to see the standard expected of their students and identify common mistakes made in examinations.
Hypothesis: An irrational number to the power of an irrational number cannot be rational.
Disproof
First of all consider the number x = .
Is x rational or irrational?
This is a very difficult question to answer but we do know it's one or the other!
So we have two cases to consider:
If x is rational then we have found our counterexample because we know is irrational and so x = is an example of an irrational to the power of an irrational with a rational answer.
If x is irrational then think about the number . By our assumption (that x is irrational) then this is an irrational to the power of an irrational. But we know the value of :
and this is clearly a rational number.
Therefore one of and must be an example of ?irrational to the power of irrational equals rational?, but we don't know which one and nor do we need to know, our counterexample is in there!
(In fact it was proved as recently as 1930 that is irrational and so it is the latter case which is the counterexample.)