SPS SPS SM (SPS SM) 2025 November

3. The equation \(k x ^ { 2 } + 4 x + ( 5 - k ) = 0\), where k is a constant, has 2 different real solutions for \(x\). Find the set of possible values of \(k\).
Write your answer using set notation.

6. A sequence \(t _ { 1 } , t _ { 2 } , t _ { 3 } , t _ { 4 } , t _ { 5 } , \ldots\) is given by $$t _ { n + 1 } = a t _ { n } + 3 n + 2 , \quad t \in \mathbb { N } , \quad t _ { 1 } = - 2 ,$$ where \(a\) is a non zero constant.
a) Given that \(\sum _ { r = 1 } ^ { 3 } \left( r ^ { 3 } + t _ { r } \right) = 12\), determine the possible values of \(a\).
b) Evaluate \(\sum _ { r = 8 } ^ { 31 } \left( t _ { r + 1 } - a t _ { r } \right)\).

8. The circles \(C _ { 1 }\) and \(C _ { 2 }\) have respective equations $$\begin{aligned} & x ^ { 2 } + y ^ { 2 } - 6 x - 2 y = 15
& x ^ { 2 } + y ^ { 2 } - 18 x + 14 y = 95 \end{aligned}$$ a) By considering the coordinates of the centres and the lengths of the radii of \(C _ { 1 }\) and \(C _ { 2 }\), show that \(C _ { 1 }\) and \(C _ { 2 }\) touch internally at some point \(P\).
b) Determine the coordinates of \(P\).
c) Find the equation of the common tangent to the circles at P .