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Prove the following : |{:(a,b,c),(a^(2...

Prove the following :
`|{:(a,b,c),(a^(2),b^(2),c^(2)),(bc,ca,ab):}|=|{:(a,a^(2),bc),(b,b^(2),ca),(c,c^(2),ab):}|=(ab+bc+ca)(a-b)(b-c)(c-a)`.

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MODERN PUBLICATION-DETERMINANTS-Exercise 4(b) (LONG ANSWER TYPE QUESTIONS (II))
  1. |[1,a, bc] ,[1, b, ca], [1, c, ab]| =(a-b)(b-c)(c-a)

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  2. Prove the following : |{:(bc,a,1),(ca,b,1),(ab,c,1):}|=(a-b)(b-c)(a-...

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  3. Prove the following : |{:(a,b,c),(a^(2),b^(2),c^(2)),(bc,ca,ab):}|=|...

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  4. Prove the following : |{:(a^(2),a,b+c),(b^(2),b,c+a),(c^(2),c,a+b):}...

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

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  6. Prove the following : |{:(x,x^(2),y+z),(y,y^(2),z+x),(z,z^(2),x+y):}...

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  7. Prove the following : |{:(alpha,alpha^(2),beta+gamma),(beta,beta^(2)...

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  8. |[a,b,c],[a-b,b-c,c-a],[b+c,c+a,a+b]|=a^3+b^3+c^3-3abc

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  9. Evaluate the following: |[a^2+2a, 2a+1, 1],[2a+1, a+2, 1],[3,3,1]|

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  10. Given : a^(2)+b^(2)+c^(2) =0 Prove the following : |{:(b^(2)+c^(2...

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  11. |[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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  12. Prove the following : |{:(x,y,z),(x^(2),y^(2),z^(2)),(x^(3),y^(3),z^...

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  13. [[x, x^2, yz],[y, y^2, zx],[z, z^2, xy]]=(x-y)(y-z)(z-x)(xy+yz+zx)

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  14. Prove that : |{:(x+y+2z,x,y),(z,y+z+2x,y),(z,x,x+a+2y):}|=2(x+y+)^(3)

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  15. Prove that |{:(b+c, c+a, a+b),(c+a, a+b,b+c),(a+b, b+c, c+a):}| =2(...

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  16. Using properties of determinants, prove the following: |xx+y x+2y\...

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  17. |[b+c, a,a] , [b,c+a,b] , [c,c,a+b]|=4abc

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  18. Prove the following : |{:(2ab,a^(2),b^(2)),(a^(2),b^(2),2ab),(b^(2),...

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  19. Prove the following : |(1,x,x^(2)-yz),(1,y,y^(2)-zx),(1,z,z^(2)-xy)|=...

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  20. For any scalar p prove that =|xx^2 1+p x^3y y^2 1+p y^3z z^2 1+p z^3|=...

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