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

Prove that `|[b+c,a,a],[b,c+a,b],[c,c,a+b]| = 4abc`

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

Prove that: |[a+b, b+c, c+a],[b+c,c+a,a+b],[c+a,a+b,b+c]|=2|[a,b,c],[b,c,a],[c,a,b]|

Using the property of determinants and without expanding , prove that: |[a-b,b-c,c-a],[b-c,c-a,a-b],[c-a,a-b,b-c]| = 0

Prove that: {:|(b+c,a-b,a),(c+a,b-c,b),(a+b,c-a,c)|=3abc -a^3-b^3-c^3

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

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BETTER CHOICE PUBLICATION-DETERMINANTS -ASSIGNMENT (MOST IMPORTANT QUESTIONS FOR PRACTICE) (SECTION III)
  1. Without expanding, prove the following |(x+4,x,x),(x,x+4,x),(x,x,x+...

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  2. Evaluate |[1,x,y],[1,x+y,y],[1,x,x+y]|

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  3. Prove that |[b+c,a,a],[b,c+a,b],[c,c,a+b]| = 4abc

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  4. Prove that: |[-a^2, ab,ac],[ba,-b^2,bc],[ca,cb,-c^2]|=4a^2b^2c^2

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  5. Without expanding, prove the following |(0,ab^2,ac^2),(a^2b,0,bc^2)...

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  6. Without expanding, prove the following |(y+z,z,y),(z,z+x,x),(y,x,x+...

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  7. Without expanding, prove the following |(a,a+b,a+2b),(a+2b,a,a+b),(...

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

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

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

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

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

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

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

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

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

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

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

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