Among the statements

$(S 1)$ : If $A(5,-1)$ and $B(-2,3)$ are two vertices of a triangle, whose orthocentre is $(0,0)$, then its third vertex is $(-4,-7)$

and

(S2) : If positive numbers $2 a, b, c$ are three consecutive terms of an A.P., then the lines $a x+b y+c=0$ are concurrent at $(2,-2)$,

Let $\alpha, \beta$ be the roots of the quadratic equation $12 x^2-20 x+3 \lambda=0, \lambda \in \mathbf{Z}$. If $\frac{1}{2} \leqslant|\beta-\alpha| \leqslant \frac{3}{2}$, then the sum of all possible values of $\lambda$ is :

The number of elements in the relation $\mathrm{R}=\left\{(x, y): 4 x^2+y^2<52, x, y \in \mathbf{Z}\right\}$ is

Let $f$ and $g$ be functions satisfying $f(x+y)=f(x) f(y), f(1)=7$ and $g(x+y)=g(x y), g(1)=1$, for all $x, y \in \mathbf{N}$. If $\sum\limits_{x=1}^{\mathrm{n}}\left(\frac{f(x)}{\mathrm{g}(x)}\right)=19607$, then n is equal to :