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The number of values of $\theta \in \left( {0,2\pi } \right)$ for which the system of linear equations
$
  x + 3y + 7z = 0 \\
  x + 4y + 7z = 0 \\
  \left( {\sin 3\theta } \right)x + \left( {\cos 2\theta } \right)y + 2z = 0 \\
$
has a non-trivial solution is:
A. One
B. Three
C. Four
D. Two

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Last updated date: 13th Jun 2024
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Answer
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Hint: In the system of linear equations, write the system in the form of
$Ax = b$ and then use the method of determinants to see for which values of the variable the system has a non-trivial solution. Put the determinant of A to be equal to 0 and find the value of the required variable.

Complete step-by-step solution
Let us consider the given system first,
$
  x + 3y + 7z = 0 \\
  x + 4y + 7z = 0 \\
  \left( {\sin 3\theta } \right)x + \left( {\cos 2\theta } \right)y + 2z = 0 \\
$
Write it in the form of $Ax = b$, where the matrix A contains the coefficients of x, y and z. The matrix x is a column matrix of entries x, y and z and the matrix b is a column matrix containing the entries from the right side of the equation.
\[ \Rightarrow \left[ {\begin{array}{*{20}{c}}
  1&3&7 \\
  1&4&7 \\
  {\sin 3\theta }&{\cos 2\theta }&2
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  x \\
  y \\
  z
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  0 \\
  0 \\
  0
\end{array}} \right]\]

Let us find the determinant of the matrix A first.
\[
  \Delta = \left| {\begin{array}{*{20}{c}}
  1&3&7 \\
  1&4&7 \\
  {\sin 3\theta }&{\cos 2\theta }&2
\end{array}} \right| \\
   = \left( {8 - 7\cos 2\theta } \right) - 3\left( {2 - 7\sin 3\theta } \right) + 7\left( {\cos 2\theta - 4\sin 3\theta } \right) \\
   = 8 - 6 - 7\cos 2\theta + 21\sin 3\theta + 7\cos 2\theta - 28\sin 3\theta \\
   = 2 + - 7\sin 3\theta \\
 \]

Put the determinant equals to 0.
$
   \Rightarrow 2 - 7\sin 3\theta = 0 \\
   \Rightarrow \sin 3\theta = \dfrac{2}{7} \\
   \Rightarrow \theta = \dfrac{1}{3}{\sin ^{ - 1}}\left( {\dfrac{2}{7}} \right) \\
$

Hence, the number of values of $\theta \in \left( {0,2\pi } \right)$ for which the system of linear equations
$
  x + 3y + 7z = 0 \\
  x + 4y + 7z = 0 \\
  \left( {\sin 3\theta } \right)x + \left( {\cos 2\theta } \right)y + 2z = 0 \\
$
has a non-trivial solution, is one.

Hence, option (A) is the correct option.

Note: Wherever you come across a system of linear equations asking for the number of values for which the given system has a non-trivial solution, simply begin by writing the system in the matrix form, $Ax = b$. But the most important thing where you can go wrong is you need to first analyze the order of all the matrices A, b and x, so that the matrix properties hold true for them. Also while finding the determinant, be careful of the calculation error.