The number of
has no solution, is
(A) 6
(B) 7
(C) 8
(D) 9
Solution:
Tip for solving this question:
- By converting linear equations into matrix form, find the determinant of the matrix. Also, check out What is Determinant of Matrix in detail here.
- Then using properties of linear equations, we find the number for no solutions.
Step 1:
Matrix Form of Linear Equations:
![\left[ {\begin{array}{*{20}{c}} {3(\sin 3\theta )}&{ - 1}&1 \\ {3(\cos 2\theta )}&4&3 \\ 6&7&7 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 2 \\ 3 \\ 9 \end{array}} \right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+%7B3%28%5Csin+3%5Ctheta+%29%7D%26%7B+-+1%7D%261+%5C%5C+%7B3%28%5Ccos+2%5Ctheta+%29%7D%264%263+%5C%5C+6%267%267+%5Cend%7Barray%7D%7D+%5Cright%5D%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+x+%5C%5C+y+%5C%5C+z+%5Cend%7Barray%7D%7D+%5Cright%5D+%3D+%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+2+%5C%5C+3+%5C%5C+9+%5Cend%7Barray%7D%7D+%5Cright%5D+&bg=ffffff&fg=000&s=0&c=20201002)
Now, the determinant of a matrix is
![\begin{gathered} \Delta = \left[ {\begin{array}{*{20}{c}} {3(\sin 3\theta )}&{ - 1}&1 \\ {3(\cos 2\theta )}&4&3 \\ 6&7&7 \end{array}} \right] \hfill \\ = 3(\sin 3\theta )(28 - 21) + 1(21\cos 2\theta - 18) + 1(21\cos 2\theta - 24) \hfill \\ = 21\sin 3\theta + 42\cos 2\theta - 42 \hfill \\ \end{gathered}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bgathered%7D+%5CDelta+%3D+%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+%7B3%28%5Csin+3%5Ctheta+%29%7D%26%7B+-+1%7D%261+%5C%5C+%7B3%28%5Ccos+2%5Ctheta+%29%7D%264%263+%5C%5C+6%267%267+%5Cend%7Barray%7D%7D+%5Cright%5D+%5Chfill+%5C%5C+%3D+3%28%5Csin+3%5Ctheta+%29%2828+-+21%29+%2B+1%2821%5Ccos+2%5Ctheta+-+18%29+%2B+1%2821%5Ccos+2%5Ctheta+-+24%29+%5Chfill+%5C%5C+%3D+21%5Csin+3%5Ctheta+%2B+42%5Ccos+2%5Ctheta+-+42+%5Chfill+%5C%5C+%5Cend%7Bgathered%7D&bg=ffffff&fg=000&s=0&c=20201002)
Step 2:
We have given that system of linear equations has no solution
For no solution, = 0
and
No solution is there for
Therefore, from (1) and (2)
Number of solutions = 7
Final Answer:
Option (B) is correct.