Let $f(x)=[x]^2-[x+3]-3, x \in \mathbf{R}$, where [.] is the greatest integer funtion. Then

Let the locus of the mid-point of the chord through the origin $O$ of the parabola $y^2=4 x$ be the curve S . Let P be any point on S . Then the locus of the point, which internally divides OP in the ratio 3 : 1, is :

Let n be the number obtained on rolling a fair die. If the probability that the system

$$ \begin{aligned} & x-\mathrm{n} y+z=6 \\ & x+(\mathrm{n}-2) y+(\mathrm{n}+1) z=8 \\ & \quad(\mathrm{n}-1) y+z=1 \end{aligned} $$

has a unique solution is $\frac{k}{6}$, then the sum of $k$ and all possible values of $n$ is :

Let the domain of the function $f(x)=\log _3 \log _5\left(7-\log _2\left(x^2-10 x+85\right)\right)+\sin ^{-1}\left(\left|\frac{3 x-7}{17-x}\right|\right)$ be $(\alpha, \beta]$. Then $\alpha+\beta$ is equal to :