4 The line $y = 2 x + 1$ is an asymptote of the curve $C$ with equation

$$y = \frac { x ^ { 2 } + 1 } { a x + b }$$

(i) Find the values of the constants $a$ and $b$.\\

(ii) State the equation of the other asymptote of $C$.\\

(iii) Sketch C. [Your sketch should indicate the coordinates of any points of intersection with the $y$-axis. You do not need to find the coordinates of any stationary points.]\\
$5 \quad$ Let $S _ { N } = \sum _ { r = 1 } ^ { N } ( 5 r + 1 ) ( 5 r + 6 )$ and $T _ { N } = \sum _ { r = 1 } ^ { N } \frac { 1 } { ( 5 r + 1 ) ( 5 r + 6 ) }$.\\
(i) Use standard results from the List of Formulae (MF10) to show that

$$S _ { N } = \frac { 1 } { 3 } N \left( 25 N ^ { 2 } + 90 N + 83 \right)$$

(ii) Use the method of differences to express $T _ { N }$ in terms of $N$.\\

(iii) Find $\lim _ { N \rightarrow \infty } \left( N ^ { - 3 } S _ { N } T _ { N } \right)$.\\

\hfill \mbox{\textit{CAIE FP1 2019 Q4}}