Question

determinant x sin theta cos theta minus sin theta minus 1 cos theta 1 x is equal to 8 ​

Answer 1

∣∣xsinθcosθ−sinθ−x1cosθ1x∣∣

∣∣=8|xsin⁡θcos⁡θ−sin⁡θ−x1cos⁡θ1x|=8

Expanding along R1, we get

x(−x2−1)−sinθ(−xsinθ−cosθ)+cosθ(−sinθ+xcosθ)=8x(−x2−1)−sin⁡θ(−xsin⁡θ−cos⁡θ)+cos⁡θ(−sin⁡θ+xcos⁡θ)=8

⇒−x3−x+xsin2θ+sinθcosθ−sinθcosθ+xcos2θ=8⇒−x3−x+xsin2⁡θ+sin⁡θcos⁡θ−sin⁡θcos⁡θ+xcos2⁡θ=8

⇒−x3−x+x(sin2θ+cos2θ)=8⇒−x3−x+x(sin2⁡θ+cos2⁡θ)=8

⇒−x3−x+x=8⇒−x3−x+x=8

⇒x3+8=0⇒x3+8=0

⇒(x+2)(x2−2x+4)=0⇒(x+2)(x2−2x+4)=0

⇒x+2=0[∵x2−2x+4>0∀x]⇒x+2=0[∵x2−2x+4>0∀x]

⇒x=−2⇒x=−2