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|[x-3, x-4, x-alpha], [x-2, x-3, x-beta]...

`|[x-3, x-4, x-alpha], [x-2, x-3, x-beta], [x-1, x-2, x-gamma]| =0,"where" alpha, beta, gamma "are in AP"`

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Apply `R_(1) to (R_(1) + R_(2) + R_(3)) "and use"alpha + gamma = 2beta.`
Then, `R_(1) "and"R_(2) "are proportional and hence"Delta = 0.`
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RS AGGARWAL-DETERMINANTS-Exercise 6B
  1. Using properties of determinants, prove that: |[b^2+c^2,a^2,a^2],[b^2,...

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  2. |[1+a^2-b^2,2ab,-2b],[2ab,1-a^2+b^2,2a],[2b,-2a,1-a^2-b^2]|=(1+a^2+b^2...

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  3. Prove that |[a,b-c,c+b],[a+c,b,c-a],[a-b,a+b,c]|=(a+b+c)(a^2+b^2+c^2)

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  4. If a,b,c are non-zero real numbers then D=|[b^2 c^2, bc, b+c] , [c^2a^...

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  5. Using properties of determinants, show the following: |(a+C)^2a b c a...

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  6. The determinat Delta=|(b^2-ab,b-c,bc-ac),(ab-a^2,a-b,b^2-ab),(bc-ac,c-...

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  7. Show that | (-a(b^2 + c^2 - a^2), 2b^3, 2c^3), (2a^3, -b(c^2 + a...

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  8. |[x-3, x-4, x-alpha], [x-2, x-3, x-beta], [x-1, x-2, x-gamma]| =0,"whe...

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  9. |[(a+1)(a+2), a+2, 1], [(a+2)(a+3), a+3, 1], [(a+3)(a+4), a+4, 1]| =-2

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  10. Prove that |(1,a^2+bc,a^3),(1,b^2+ca,b^3),(1,c^2+ca,c^3)|=-(a-b)(b-c)(...

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  11. Prove that |[1,a,bc] , [1,b,ca], [1,c,ab]|=|[1,a,a^2] , [1,b,b^2] , [1...

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  12. |[1,bc,b+c],[1,ca,c+a],[1,ab,a+b]|=|[1,a,a^2],[1,b,b^2],[1,c,c^2]|

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  13. Show that x = 2 is a root of the equation |[x, -6, -1], [2, -3x, x-3],...

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  14. |[1,x,x^3],[1,b,b^3],[1,c,c^3]|=0; b!=c

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  15. |[x+a, b, c], [a, x+b, c], [b, b, x+c]|=0

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  16. One root of the equation |(3x-8, 3, 3),(3,3x-8, 3),(3,3,3x-8)|=0 si (A...

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  17. |[x+1, 3, 5], [2, x+2, 5], [2, 3, x+4]|=0

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  18. The solution set of the equation |[x, 3, 7], [2, x, 2], [7, 6, x]|=0 i...

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  19. The number of real roots of the equation | (x,-6,-1), (2,-3x,x-3), (-...

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  20. Prove that |[a,b-c,c+b],[a+c,b,c-a],[a-b,a+b,c]|=(a+b+c)(a^2+b^2+c^2)

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