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 expressions and using trigonometric identities to find the values of \theta that satisfy the conditions.
The system of equations is given as:<br>(y + z)cosθ = (xyz)sinθ ... (1)<br>xsinθ = 2cos3θ/y + 2sin3θ/z ... (2)<br>(xyz)sinθ = (y + 2z)cosθ + ysin3θ ... (3)<br>From (1), xyz = (y+z)cotθ. Substituting this into (3):<br>(y+z)cotθ sinθ = (y+2z)cosθ + ysin3θ<br>(y+z)cosθ = (y+2z)cosθ + ysin3θ<br>ycosθ + zcosθ = ycosθ + 2zcosθ + ysin3θ<br>-zcosθ = ysin3θ => y/z = -cosθ/sin3θ.<br>Using (2) and substituting xyz, we can solve for θ in the interval (0, π). The number of such values is 3.
Correct Answer: 3