The set of equations λx−$$y+($$$$cos$$⁡θ$$)z=0$$,$$3x+y+2z=0$$,$$($$$$cos$$⁡θ$$)x+y+2z=0$$ ; $$0$$≤θ

Question

The set of equations λx−$$y+($$$$cos$$⁡θ$$)z=0$$,$$3x+y+2z=0$$,$$($$$$cos$$⁡θ$$)x+y+2z=0$$ ; $$0$$≤θ<$$2$$π has $$non-trivial$$ $$solution(s)$$

Infinity Learn · Accepted Answer

The determinant of the coefficient matrix is $$D=$$λ−$$1cos$$⁡θ$$312cos$$⁡θ$$12=$$λ(2$$−$$2)+1(6$$−$$2cos$$⁡θ$$)+$$$$cos$$⁡θ$$(3$$−$$cos$$⁡θ$$)=6$$−$$cos2$$⁡θ$$+$$$$cos$$⁡θ$$=6+14$$−$$cos2$$⁡θ−$$cos$$⁡θ$$+14Converting$$ into a perfect square, we $$get=254$$−$$cos$$⁡θ−$$122$$ Minimum value of D is $$254$$−−$$1$$−$$122=4$$>$$0i$$.eD≠$$0$$∴ Given system has only trivial solutions  For no values of λ and θ system has non trivial $$solution(s)  Therefore, the correct answer is (1).

$The set of equations λ x - y + (\cos θ) z = 0, 3 x + y + 2 z = 0, (\cos θ) x + y + 2 z = 0;$ $0 \leq θ < 2 π has non-trivial solution(s)$

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The set of equations λx−y+(cos⁡θ)z=0,3x+y+2z=0,(cos⁡θ)x+y+2z=0 ; 0≤θ<2π has non-trivial solution(s)

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$The set of equations λ x - y + (\cos θ) z = 0, 3 x + y + 2 z = 0, (\cos θ) x + y + 2 z = 0;$ $0 \leq θ < 2 π has non-trivial solution(s)$