Let the function

Question:

Let $\mathrm{S}$ be the set of all integer solutions, $(x, y, z)$, of the system of equations

$x-2 y+5 z=0$

$-2 x+4 y+z=0$

$-7 x+14 y+9 z=0$

such that $15 \leq \mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2} \leq 150$. Then, the number of elements in the set $S$ is equal to__________

Solution:

$\Delta=\left|\begin{array}{ccc}1 & -2 & 5 \\ -2 & 4 & 1 \\ -7 & 14 & 9\end{array}\right|=0$

Let $\quad x=k$

$\Rightarrow \quad$ Put in (1) \& (2)

$k-2 y+5 z=0$

$-2 k+4 y+z=0$'

$z=0, y=\frac{k}{2}$

$\therefore \quad \mathrm{x}, \mathrm{y}, \mathrm{z}$ are integer

$\Rightarrow \quad \mathrm{k}$ is even integer'

Now $\mathrm{x}=\mathrm{k}, \mathrm{y}=\frac{\mathrm{k}}{2}, \mathrm{z}=0$ put in condition

$15 \leq \mathrm{k}^{2}+\left(\frac{\mathrm{k}}{2}\right)^{2}+0 \leq 150$

$12 \leq \mathrm{k}^{2} \leq 120$

$\Rightarrow \quad \mathrm{k}=\pm 4, \pm 6, \pm 8, \pm 10$

$\Rightarrow$ Number of element in $\mathrm{S}=8$

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