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Prove that : |{:(1,x,x^(3)),(1,y,y^(3)),...

Prove that : `|{:(1,x,x^(3)),(1,y,y^(3)),(1,z,z^(3)):}|`

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Prove that : |{:(1,1,1),(x,y,z),(x^(3),y^(3),z^(3)):}|=(x-y)(y-z)(x+y+z)

Without expanding as far as possible, prove that |{:(1,1,1),(x,y,z),(x^(3),y^(3),z^(3)):}| = (x-y)(y-z)(z-x)(x+y+z) .

Prove that : =2|{:(1,1,1),(x,y,rz),(x^(2),y^(2),z^(2)):}|=(x-y)(y-z)(z-x)

Prove the following : |{:(x,y,z),(x^(2),y^(2),z^(2)),(x^(3),y^(3),z^(3)):}|=|{:(x,x^(2),x^(3)),(y,y^(2),y^(3)),(z,z^(2),z^(3)):}|=xyz(x-y)(y-z)(z-x)

Prove that |{:(x,y,z),(x^2,y^2,z^2),(yz,zy,xy):}|=|{:(1,1,1),(x^2,y^2,z^2),(x^3,y^3,z^3):}|=(y-z)(z-x)(x-y)(yz+zy+xy)

If |{:(x,x^2,1+x^3),(y,y^2,1+y^3),(z, z^2,1+z^3):}|=0 then relation of x,y and z is

For any scalar p prove that =|x^(2)1+px^(3)yy^(2)1+py^(3)zz^(2)1+pz^(3)|=(1+pxyz)(x-y)(y-z)(z-x)

Given that xyz = -1 , the value of the determinant |(x,x^(2),1 +x^(3)),(y,y^(2),1 + y^(3)),(z,z^(2),1 +z^(3))| is

|[yz,x,x^(2)],[zx,y,y^(2)],[xy,z,z^(2)]|=|[1,x^(2),x^(3)],[1,y^(2),y^(3)],[1,z^(2),z^(3)]|

MODERN PUBLICATION-DETERMINANTS-Exercise 4(b) (LONG ANSWER TYPE QUESTIONS (II))
  1. Prove that |(a,b-c,c+b),(a+c,b,c-a),(a-b,b+a,c)|=(a+b+c)(a^(2)+b^(2)...

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  2. Prove the following : |{:(1,a,a),(a,1,a),(a,a,1):}|=(2a+1)(1-a)^(2)

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  3. Prove that : |{:(1,x,x^(3)),(1,y,y^(3)),(1,z,z^(3)):}|

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

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  5. |[1,a, bc] ,[1, b, ca], [1, c, ab]| =(a-b)(b-c)(c-a)

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

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

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

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

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

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

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

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

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  17. [[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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  18. 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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  19. 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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  20. Using properties of determinants, prove the following: |xx+y x+2y\...

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