Questions & Answers

Question

Answers

A. $ > 2$

B. $ > 3$

C. $ < 3$

D. $ < 2$

Answer
Verified

Hint- In this question, provided that system is a homogeneous equation in \[3\] variables, its determinant will be zero because it has non-zero value which implies infinite solutions, hence the determinant of the matrix coefficient is zero after we get the correct answer by simplifying the equation.

__Complete step-by-step answer:__

Provided the equations of the system

\[

x{\text{ }} = {\text{ }}cy{\text{ }} + {\text{ }}bz \\

y{\text{ }} = {\text{ }}az{\text{ }} + {\text{ }}cx \\

z = bx + ay \\

\]

Since we know that if the system of homogeneous equations in \[3\] variables has non-zero value or infinite solution then matrix determinant is zero

Now we are going to write above \[3\] equation in the form of a determinant matrix and make it equal to zero, so that we get

$\left| {\begin{array}{*{20}{c}}

1&{ - c}&{ - b} \\

{ - c}&1&{ - a} \\

{ - b}&{ - a}&1

\end{array}} \right| = 0$

Now we're going to solve it with the help of first raw so we're getting

\[

\Rightarrow 1(1 - {a^2}) + c( - c - ab) - b(ac + b) = 0 \\

\Rightarrow 1 - 2abc - {a^2} - {b^2} - {c^2} = 0 \\

\]

Further simplifying above equation we get

\[{a^2} + {b^2} + {c^2} + 2abc = 1\]

By adding \[{b^2}{c^2}\]in both the sides of equation

\[

{a^2} + {b^2}{c^2} + 2ab = 1 - {b^2} - {c^2} + {b^2}{c^2} = (1 - {b^2})(1 - {c^2}) \\

\Rightarrow {(a + bc)^2} = (1 - {b^2})(1 - {c^2}) \\

\]

Similarly,

\[

{(b + ac)^2} = (1 - {a^2})(1 - {b^2}) \\

{(c + ab)^2} = (1 - {a^2})(1 - {b^2}) \\

\]

Hence, \[\left( {1{\text{ }} - {\text{ }}{a^2}} \right),{\text{ }}\left( {1{\text{ }} - {\text{ }}{b^2}} \right)\]and \[\left( {1{\text{ }} - {\text{ }}{c^2}} \right)\]all have same sign. Since at least one of them is a proper fraction, it implies all of them are positive.

So \[1{\text{ }} - {\text{ }}{a^2}{\text{ }} > {\text{ }}0,{\text{ }}1{\text{ }} - {\text{ }}{b^2}{\text{ }} > {\text{ }}0,{\text{ }}1{\text{ }} - {\text{ }}{c^2}{\text{ }} > {\text{ }}0\]

By adding above \[3\] equation we get

\[ \Rightarrow {a^2} + {b^2} + {c^2} < 3\]\[ \Rightarrow {a^2} + {b^2} + {c^2} < 3\]

Hence the correct answer is option C.

Note-When matrix has non zero solution this means that the lines are either equivalent (the number of solutions is endless) or parallel (there are no solutions). The slopes must be different if the determinant is non-zero, and the lines must intersect in exactly one point.

Provided the equations of the system

\[

x{\text{ }} = {\text{ }}cy{\text{ }} + {\text{ }}bz \\

y{\text{ }} = {\text{ }}az{\text{ }} + {\text{ }}cx \\

z = bx + ay \\

\]

Since we know that if the system of homogeneous equations in \[3\] variables has non-zero value or infinite solution then matrix determinant is zero

Now we are going to write above \[3\] equation in the form of a determinant matrix and make it equal to zero, so that we get

$\left| {\begin{array}{*{20}{c}}

1&{ - c}&{ - b} \\

{ - c}&1&{ - a} \\

{ - b}&{ - a}&1

\end{array}} \right| = 0$

Now we're going to solve it with the help of first raw so we're getting

\[

\Rightarrow 1(1 - {a^2}) + c( - c - ab) - b(ac + b) = 0 \\

\Rightarrow 1 - 2abc - {a^2} - {b^2} - {c^2} = 0 \\

\]

Further simplifying above equation we get

\[{a^2} + {b^2} + {c^2} + 2abc = 1\]

By adding \[{b^2}{c^2}\]in both the sides of equation

\[

{a^2} + {b^2}{c^2} + 2ab = 1 - {b^2} - {c^2} + {b^2}{c^2} = (1 - {b^2})(1 - {c^2}) \\

\Rightarrow {(a + bc)^2} = (1 - {b^2})(1 - {c^2}) \\

\]

Similarly,

\[

{(b + ac)^2} = (1 - {a^2})(1 - {b^2}) \\

{(c + ab)^2} = (1 - {a^2})(1 - {b^2}) \\

\]

Hence, \[\left( {1{\text{ }} - {\text{ }}{a^2}} \right),{\text{ }}\left( {1{\text{ }} - {\text{ }}{b^2}} \right)\]and \[\left( {1{\text{ }} - {\text{ }}{c^2}} \right)\]all have same sign. Since at least one of them is a proper fraction, it implies all of them are positive.

So \[1{\text{ }} - {\text{ }}{a^2}{\text{ }} > {\text{ }}0,{\text{ }}1{\text{ }} - {\text{ }}{b^2}{\text{ }} > {\text{ }}0,{\text{ }}1{\text{ }} - {\text{ }}{c^2}{\text{ }} > {\text{ }}0\]

By adding above \[3\] equation we get

\[ \Rightarrow {a^2} + {b^2} + {c^2} < 3\]\[ \Rightarrow {a^2} + {b^2} + {c^2} < 3\]

Hence the correct answer is option C.

Note-When matrix has non zero solution this means that the lines are either equivalent (the number of solutions is endless) or parallel (there are no solutions). The slopes must be different if the determinant is non-zero, and the lines must intersect in exactly one point.

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