Let S be set of all integer solutions, (x, y, z) , of system of equations x-2 y+5 z=0 -2 x+4 y+z=0 -

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3 and 4 .Determinants and Matrices

hard

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

A

$16$

B

$-8$

C

$-16$

D

$8$

(JEE MAIN-2020)

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 $(1)\;and\;(2)$

$k-2 y+5 z=0$

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

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

$\therefore \quad x , y , z$ are integer

$\Rightarrow \quad k$ is even integer

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

$\begin{array}{l}15 \leq k^{2}+\left(\frac{k}{2}\right)^{2}+0 \leq 150 \\12 \leq k^{2} \leq 120\end{array}$

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

$\Rightarrow$ Number of element in $S =8$

Standard 12

Mathematics

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