for 7ii - I used the method that since Q lies on both lines $P_1P_3$ and $ P_2P_4 $

$ q= p_1 + t(p_1 - p_3)$
$ q=p_2 + s(p_4 - p_2) $

hence equating the above two we find that
$ p_1 (1-t) + p_2 ( s-1) + tp_3 - sp_4 = 0 $

since there must exist real numbers t and s such that the above holds and as the sum of coefficients on the vectors is 0 it implies that
$ a_1 = 1-t$
$a_2 = s-1 $
$a_3 = t $
$a_4 = -s $

I reached the stage where you have to prove the expression for $ \lambda_1 $

I realised the use of $ x.x = 1 $ and I came to the below expression

$ \lambda^2 (3-2\gamma -2\beta + \alpha) + \lambda(2\alpha - \beta-\gamma) + \alpha + \beta + \gamma - 3 = 0 $

For the last part of the question, I replicated the earlier parts and got to the below stage when calculating the root locus:

$ y=0 \implies p = \frac{-x^2 \pm \sqrt (x^4 - 8x) } {2x} $

The question on the question paper does not match the solution given. As shown in the link below:

http://postimg.org/image/p85p25oyn/

I think that the question is not the correct version. Firstly, it is way too easy, you just need to set a=1, b=c=2. Secondly, both the solution they give and people in TSR gives the answer x=7. Thirdly, it is completely unrelated with part ii, a huge massive jump from i to ii.

In the second part of the question, I reached the below stage:

$ kz = z^4 - 3z^2 x + 3x^2 $

The mark scheme says that $ z^3 > k $ - how do I go about proving this?

My main attempts so far have been trying to replace the two variables with $ x $ or $y$ and using the fact that these are positive integers.

Also I am not sure when to use the equation $ x+y = z^2 $

The third part of the question asks you to integrate using the formula given to find the area between the curve, the x axis and limits of $ \pi $ and $ \frac{\pi}{2} $

The mark scheme uses $ \frac{\pi}{2} $ as the upper limit while $ \pi $ as the lower limit - how would you know to do this?

As I used the upper limit as $ \pi $ the answers I obtained were the same as the mark scheme except negative.

I have used another method to the one listed in the mark scheme for (ii) and (iii) - this method was hinted as being used by candidates in the examiners report. I would be grateful if someone could check this method for (iii) (similar method used in (ii) ):

(iii) From the stem of the question, we know that if there is a rational root then it must be an integer.
Assume there exists an integer $ p $ such that it is a root and hence $ p^n - 5p + 7=0 $
Rearranging we get $ p = \frac {-7}{p^{n-1} - 5} $

for 7i, the mark scheme says that both turning points are below axis hence the curve crosses x axis once only.

What is the need to consider both turning points?

Surely proving that the turning point at $ x= - \sqrt q $ is less than zero would be sufficient in showing that the given cubic has only one real root?

Since it is a positive cubic, the maximum is at $ x= - \sqrt q $ and the minimum must be at $ x= \sqrt q $ . The minimum y value must be less than the maximum y value so if the maximum is less than zero so is the minimum.

I am struggling with understanding the geometric interpretation of this question and hence the last part 'describe the surface...'

5i - If L1 is inclined equally to OA and to OB then surely it is already an angle bisector of AOB for all values of m,n,p? - if it is inclined equally it must make an angle of $ \alpha $ to OA and to OB

How can you deduce that the plane equation in the last part describes a double cone? and how can a plane equation describe a set of lines - surely the equation of a plane describes a set of points?

Why is it I cannot multiply the first equation by 25 and the second by 18 and then equate the RHS of each equation to find the equation of the points of intersection?

$ 25(x+2)^2 + 50y^2 = 162(x-1)^2 + 288y^2 $

Simplifying the above leads to

$ 0 = 137x^2 - 424x + 238y^2 + 62 $

The above is not in the form stated in the question. Why does eliminating y work but equating constants does not work in solving this problem?

NB: The problem is also Q17 in Advanced problems in core mathematics