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  • Solve this problem - The number of values of for which the system of linear equations has non-trivalsolution, is: - Matrices and Determinants - JEE Main

The number of values of \theta\in(0,\pi)  for which the system of linear equations 

x + 3y + 7z = 0

-x + 4y + 7z = 0

(\sin3\theta)x + (\cos2\theta)y + 2z = 0 has non-trival solution, is: 

  • Option 1)

    three

  • Option 2)

    four

  • Option 3)

    two

  • Option 4)

    one

Answers (1)

best_answer

 

Homogeneous system of linear equation -

b=0

- wherein

As we have learnt from the concept for non-trivial solution 

\Delta =\begin{vmatrix} 1 &3 &7 \\ -1 & 4 &7 \\ \sin 3\theta &\cos 2\theta & 2 \end{vmatrix}=0

\Delta =(8-7\cos 2\theta )-3(-2-7\sin 3\theta )+7(-\cos 2\theta -4\sin 3\theta )

     =14-7\cos 2\theta+21\sin 3\theta-7\cos 2\theta -28\sin 3\theta

    =14-14\cos 2\theta-7\sin 3\theta

   =14-14(1-2\sin ^{2}\theta )-7(3\sin \theta -4\sin ^{3}\theta )

   =-21\sin \theta+28\sin ^{3}\theta+28\sin ^{2}\theta

   =7\sin \theta[-3+4\sin ^{2}\theta+4\sin \theta]

\sin \theta=0 \: or\: \sin \theta=\frac{1}{2}\: or\: \sin \theta=\frac{-3}{2}

for\: \: \theta \epsilon (0,\pi)

\theta =\frac{\pi}{6} \: \: and\: \: \frac{5\pi}{6}

 

 

 

 


Option 1)

three

Option 2)

four

Option 3)

two

Option 4)

one

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