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JEE Advanced 2009Paper 2
Exam
JEE Advanced
Year
2009
Paper
Paper 2
Subject
Mathematics
Chapter
Determinants

JEE Advanced 2009 · Paper 2 · Mathematics · Determinants

Let (x, y, z) be points with integer coordinates satisfying the...

Let (x,y,z)(x, y, z) be points with integer coordinates satisfying the system of homogeneous equations 3x−y−z=0,−3x+z=0,−3x+2y+z=03 x-y-z=0,-3 x+z=0,-3 x+2 y+z=0. Then, the number of such points for which x2+y2+z2≤100x^2+y^2+z^2 \leq 100 is

Correct answer

7

Step-by-step solution

Given, 3x−y−z=0\quad 3 x-y-z=0 and −3x+2y+z=0−3x+z=0\begin{array}{r} -3 x+2 y+z=0 \\ -3 x+z=0 \end{array} On adding Eqs. (i) and (ii), we get y=0y=0 So, 3x=z\quad 3 x=z Now, x2+y2+z2≤100\quad x^2+y^2+z^2 \leq 100 ⇒x2+(3x)2+0≤100⇒10x2≤100⇒x2≤10∴x=−3,−2,−1,0,1,2,3\begin{array}{lc} \Rightarrow & x²+(3 x)^2+0 \leq 100 \\ \Rightarrow & 10 x² \leq 100 \\ \Rightarrow & x² \leq 10 \\ \therefore & x=-3,-2,-1,0,1,2,3 \end{array} So, number of such 7 points are possible.
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