A circle has equation
A. Find the centre of the circle (1 mark)
B. Find the equation of the tangent to the circle at the point (4 marks)
C. Calculate the value of (1 mark)
Solve the simultaneous equations:
Solve for y first so you can eliminate the need for negative square roots.
Therefore or 1 (1 mark)
Subbing values of y into the original equation gives and 1 (2 marks)
Award full marks of any alternative method
You are given 2 odd integers with a difference of 4.
Prove that the difference between the squares of the integers is 8 times the mean of the integers. (5 marks)
Odd integers are and . (1 mark for selecting 2 odd integers)
Doubling any number and adding 1 will always be odd, and adding 4 to that number will always be odd.
The difference between the squares of the integers will be (1 mark)
The mean of the original integers is the integers added together divided by the number of integers:
Therefore, the difference between the squares of the integers is 8 times the mean.
Award full marks for any alternative method including using different odd integers in terms of n.
Simplify the fraction in the form where a,b,c and d are integers to be found (4 marks)
Where n is a positive integer and y is a positive integer.
Use this statement to prove that x is odd. (4 marks)
As n is a positive integer. is also positive. The right side can simply be referred to as m, where m is a positive even integer. (1 mark for recognising that is even)
since (x + 1) and (x + 2y +1) must multiply to make an even number, this can only be true when either both terms are even, or 1 term is odd, and the other term is even.
Scenario 1: Both terms are even.
x + 1 is even, therefore x is odd. Then x + 2y + 1 is even, so this scenario is possible. (1 mark)
Scenario 2: First term is odd and the second term is even.
x + 1 is odd, therefore X is even. Then x + 2y + 1 is odd. This is impossible as both x and y are even, thus x + 2y + 1 must be odd. This scenario is not possible as both terms are odd. (1 mark)
Scenario 3: the second term is odd and the first term is even.
If x +1 is even, x is odd. This will give the same answer as scenario 1 where x + 2y + 1 is even, so this scenario doesn’t exist. (1 mark)
Therefore x must be odd.
Award full marks for any alternative method that concludes x is odd.