Matrices & Determinants
System of linear equations
Grade Class 12
Question:
<p>The number of all possible values of θ, where 0 < θ < π, for which the system of equations</p><p>(y + z)cosθ = (xyz)sinθ</p><p>xsinθ = 2cos3θ/y + 2sin3θ/z</p><p>(xyz)sinθ = (y + 2z)cosθ + ysin3θ</p><p>have a solution (x<sub>0</sub>, y<sub>0</sub>, z<sub>0</sub>) with y<sub>0</sub>z<sub>0</sub> ≠ 0, is</p>
Step-by-Step Solution
Key Concept: The system of equations can be simplified by substituting the given relations. By analyzing the equations, we can express the variables in terms of trigonometric functions of \theta and solve for \theta within the given interval (0, \pi).
<p>Given the system of equations:</p><p>1) (y + z)cosθ = (xyz)sinθ</p><p>2) xsinθ = 2cos3θ/y + 2sin3θ/z</p><p>3) (xyz)sinθ = (y + 2z)cosθ + ysin3θ</p><p>From (1) and (3), we have (y + z)cosθ = (y + 2z)cosθ + ysin3θ, which implies zcosθ + ysin3θ = 0. Substituting this into the equations and solving for θ in (0, π) yields 3 distinct values.</p>
Correct Answer: 3