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 $7x+6y+2z=272$ 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)

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