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Without expanding, prove the following ...

Without expanding, prove the following
`|(a,b,c),(a^2,b^2,c^2),(bc,ca,ab)|=(a-b)(b-c)(c-a)(ab+bc+ca)`

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Prove that: {:|(a^2,a,b+c),(b^2,b,c+a),(c^2,c,ab)| = -(a+b+c)(a-b)(b-c)(c-a)

Without expanding, prove that the following determinants vanish: {:|(a,b,a+b),(b,c,b+c),(c,a,c+a)|

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BETTER CHOICE PUBLICATION-DETERMINANTS -ASSIGNMENT (MOST IMPORTANT QUESTIONS FOR PRACTICE) (SECTION III)
  1. Without expanding, prove the following |(a,a+b,a+2b),(a+2b,a,a+b),(...

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  2. Prove that: |[1,1+p,1+p+q],[2,3+2p,4+3p+2q],[3,6+3p,10+6p+3q]|=1

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  3. Without expanding, prove the following |(1+a^2-b^2,2ab,-2b),(2ab,1-...

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  4. Without expanding, prove the following |(x,x+y,x+2y),(x+2y,x,x+y),(...

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  5. Show that: |[x-y-z,2x,2x],[2y,y-z-x,2y],[2z,2z,z-x-y]|=(x+y+z)^3

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  6. Without expanding, prove the following |(1,a,a^3),(1,b,b^3),(1,c,c^...

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  7. Without expanding, prove the following |(x,y,z),(x^2,y^2,z^2),(x^3,...

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  8. Without expanding, prove the following |(1,1,1),(alpha,beta,gamma),...

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  9. Without expanding, prove the following |(a,b,c),(a^2,b^2,c^2),(bc,c...

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  10. Without expanding, prove the following |(a^3+1,a^2,a),(b^3+1,b^2,b)...

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  11. Without expanding, prove the following |(b+c,c+a,a+b),(q+r,r+p,p+q)...

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  12. Without expanding, prove the following |(b+c,c+a,a+b),(c+a,a+b,b+c)...

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  13. Without expanding, prove the following |(a^2,2ab,b^2),(b^2,a^2,2ab)...

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  14. Without expanding, prove the following |(a,b-c,c-b),(a-c,b,c-a),(a-...

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  15. Without expanding, prove the following |(a,b,c),(a-b,b-c,c-a),(b+c,...

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  16. Solve the equation |(3x-8,3,3),(3,3x-8,3),(3,3,3x-8)|=0

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  17. Solve the equation |(x-2,2x-3,3x-4),(x-4,2x-9,3x-16),(x-8,2x-27,3x-64)...

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  18. Solve for x: |(x+9,x,x),(x,x+9,x),(x,x,x+9)|=0

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  19. Solve for x":"|(a+x,a-x,a-x),(a-x,a+x,a-x),(a-x,a-x,a+x)|=0

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  20. Using properties of determinants , prove that |(x^2+1,xy,zx),(xy,y^2...

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