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Prove that =|[a,c,c-a,a+c],[c,b,b-c,b+c]...

Prove that `=|[a,c,c-a,a+c],[c,b,b-c,b+c],[a-b,b-c,0,a-c],[x,y,z,1+x+y]|=0` implies that `a ,b ,c` are in A.P. or `a ,c ,b` are in G.P.

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CENGAGE PUBLICATION-DETERMINANTS-All Questions
  1. If a1, a2, a3,54,a6,a7, a8, a9 are in H.P., and D=|[a1,a2,a3],[5, 4,a6...

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  2. If f(x) is a polynomial of degree <3, then, f(x)/((x−a)(x−b)(x−c))​is ...

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  3. Prove that =|[a,c,c-a,a+c],[c,b,b-c,b+c],[a-b,b-c,0,a-c],[x,y,z,1+x+y]...

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  4. Solve for x in R : |((x+a)(x-a),(x+b)(x-b),(x+c)(x-c)),((x-a)^3,(x-b)...

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  5. Absolute value of sum of roots of the equation |[x+2,2x+3, 3x+4],[ 2x+...

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  6. Let alpha1,alpha2 and beta1, beta2 be the roots of the equation ax^2+b...

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  7. The product of all values of t , for which the system of equations (a-...

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  8. "If " a(1) ,a(2), a(3)….." are in A.P, then find the value of the fol...

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  9. Let omega=-1/2+i(sqrt(3))/2. Then the value of the determinant |(1,1,1...

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  10. Factorize: a x+b x+a y+b y +a x^2+b y^2+b x^2+a y^2 a^2+b c+a b+...

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  11. If g(x)=|[a^(-x),e^(x loge a),x^2],[a^(-3x),e^(3x loge a),x^4],[a^(-5x...

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  12. The sum of values of p for which the equations x+y+z=1, x+2y +4z...

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  13. The value of |alpha| for which the system of equation alphax+y+z=...

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  14. Let D1=|[a, b, a+b], [c, d, c+d], [a, b, a-b]| and D2=|[a, c, a+c], [b...

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  15. Three distinct points P(3u^(2),2u^(3));Q(3v^(2),2v^(3)) and R(3w^(2),2...

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  16. If |[x,x+y, x+y+z],[2x,3x+2y,4x+3y+2z],[3x,6x+3y, 10 x+6y+3z]|=64 , th...

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  17. Prove that : (1+ sin theta - cos theta) / (1+ sin theta + cos theta ) ...

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  18. If |[x^n,x^(n+2),x^(n+4)],[y^n,y^(n+2),y^(n+4)],[z^n, z^(n+2),z^(n+4)]...

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  19. Let Delta1=|{:(a1,a2,a3),(b1,b2,b3),(c1,c2,c3):}| , Delta2=|{:(6a1,2a2...

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  20. The value of |[2x1y1, x1y2+x2y1, x1y3+x3y1], [x1y2+x2y1, 2x2y2, x2y3+...

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