IIT-JEE 2009 · Paper 2 · Mathematics Q32

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MathematicsInteger+4 / -1Question 32 of 57

Let $(x,y,z)$ be points with integer coordinates satisfying the system of homogeneous equations:
$3x-y-z=0,\qquad-3x+z=0,\qquad-3x+2y+z=0.$
Then the number of such points for which $x^2+y^2+z^2\le100$ is

Your answer

Answer and worked solution

Official answer: 7

Solve. The second equation gives $z=3x$ ; the first then gives $y=0$ ; the third, $-3x+3x=0$ , holds. The solutions are $(x,0,3x)$ .

Count. $x^2+9x^2\le100$ means $|x|\le\sqrt{10}$ , so $x\in\{-3,\ldots,3\}$ : 7 points.

Answer: 7

Topics:Systems of linear equations

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IIT-JEE 2009 · Paper 2, Mathematics question 32