Question

If the lattice point $P(x, y, z) ; x, y, z > 0$ and $x, y, z \in I$ with least value of $z$ such that the ' $P^{n}$ lies on the planes $7 x+6 y+2 z=272$ and $x-y+z=16$, then the value of $(x+y+z-42)$ is equal to

• A. 0
• B. 3
• C. 2
• D. None of these

Answer 1

Answer: (D)

$7 x+6 y+2 z=272$ and $x-y+z=16$
$\Rightarrow 5 x+8 y=240$
$ \Rightarrow x=48-\frac{8}{5} y$
Let $y=5 \lambda, \lambda \in I $
$\Rightarrow x=48-8 \lambda$
and $z=16+y-x=13 \lambda-32$
But $x > 0, y > 0$ and $z > 0 $
$\Rightarrow 48-8 \lambda > 0 $
$\Rightarrow \lambda > \frac{48}{8}$
$\Rightarrow \lambda \leq 5$ and $13 \lambda-32 > 0$
$ \Rightarrow \lambda > \frac{32}{13}$
$\Rightarrow \lambda \geq 3$
$\therefore \lambda \in[3,5]$
$\therefore Z_{\min }=39-32=7$
$ \Rightarrow x=24, y=15$
$\therefore x+y+z-42=4$