Let 's' be the set of all integral solutions of (x, y, z) the system of equations x - 2y - 6z = 0 -2x + 3y + 9z = 0 3x - 5y - 15z = 0 and 10 ≤ x² + y² + z² ≤ 200 then number of element in the set ‘s’ ______.

Clearly all equations can be λ x - 2 y - 6 z = 0 <!-- [if gte msEquation 12]>λx-2y-6z=0

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Let 's' be the set of all integral solutions of (x, y, z) the system of equations
x - 2y - 6z = 0
-2x + 3y + 9z = 0
3x - 5y - 15z = 0
and 10 ≤ x² + y² + z² ≤ 200 then number of element in the set ‘s’ ______.

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Raj Pandey | Contributor-Level 9

a month ago

Clearly all equations can be $λ x - 2 y - 6 z = 0$ form where $λ = 1, \frac{4}{3}, \frac{6}{5}$

So for $x = 0 y = - 3 z$

So $x^{2} + y^{2} + z^{2} = 10 z^{2}$ (as ) $x = 0 y = - 3 z$

ATQ $10 \leq 10 z^{2} \leq 200$

$1 \leq z^{2} \leq 20 \Rightarrow z = \pm 1, \pm 2, \pm 3, \pm 4$

Clearly all equations can be <math> <mi>λ</mi> <mi>x</mi> <mo>-</mo> <mn>2</mn> <mi>y</mi> <mo>-</mo> <mn>6</mn> <mi>z</mi> <mo>=</mo> <mn>0</mn> </math>   form where <math> <mi>λ</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mfrac> <mrow> <mrow> <mn>4</mn> </mrow> </mrow> <mrow> <mrow> <mn>3</mn> </mrow> </mrow> </mfrac> <mo>,</mo> <mfrac> <mrow> <mrow> <mn>6</mn> </mrow> </mrow> <mrow> <mrow> <mn>5</mn> </mrow> </mrow> </mfrac> </math>   So for <math> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mi>y</mi> <mo>=</mo> <mo>-</mo> <mn>3</mn> <mi>z</mi> </math>  So <math> <msup> <mrow> <mrow> <mi>x</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mrow> <mi>y</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mrow> <mi>z</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mo>=</mo> <mn>10</mn> <msup> <mrow> <mrow> <mi>z</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </math>   (as   ) <math> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mrow> <mi mathvariant="normal"> </mi> </mrow> <mi>y</mi> <mo>=</mo> <mo>-</mo> <mn>3</mn> <mi>z</mi> </math> ATQ <math> <mn>10</mn> <mo>≤</mo> <mn>10</mn> <msup> <mrow> <mrow> <mi>z</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mo>≤</mo> <mn>200</mn> </math>   <math> <mn>1</mn> <mo>≤</mo> <msup> <mrow> <mrow> <mi>z</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mo>≤</mo> <mn>20</mn> <mo>⇒</mo> <mi>z</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mn>2</mn> <mo>,</mo> <mo>±</mo> <mn>3</mn> <mo>,</mo> <mo>±</mo> <mn>4</mn> </math>

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Let 's' be the set of all integral solutions of (x, y, z) the system of equations
x - 2y - 6z = 0
-2x + 3y + 9z = 0
3x - 5y - 15z = 0
and 10 ≤ x² + y² + z² ≤ 200 then number of element in the set ‘s’ ______.

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Let 's' be the set of all integral solutions of (x, y, z) the system of equationsx - 2y - 6z = 0-2x + 3y + 9z = 03x - 5y - 15z = 0and 10 ≤ x² + y² + z² ≤ 200 then number of element in the set ‘s’ ______.

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Let 's' be the set of all integral solutions of (x, y, z) the system of equations
x - 2y - 6z = 0
-2x + 3y + 9z = 0
3x - 5y - 15z = 0
and 10 ≤ x² + y² + z² ≤ 200 then number of element in the set ‘s’ ______.