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Prove [[a^3-x^3,a^2,a],[b^3-x^3,b^2,b],[...

Prove `[[a^3-x^3,a^2,a],[b^3-x^3,b^2,b],[c^3-x^3,c^2,c]] = (a-b)(a-c)(b-c)(abc-x^3)`

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

Prove the following: [[(b+c)^2,a^2,bc],[(c+a)^2,b^2,ca],[(a+b)^2,c^2,ab]] = (a^2+b^2+c^2)(a+b+c)(b-c)(c-a)(a-b)

Factorise the determinant |[x^3-a^3,x^2,x],[b^2-a^3,b^2,b],[c^3-a^3,c^2,c]| without expanding.

Show that: abs((a,a^2,a^3),(b,b^2,b^3),(c,c^2,c^3))=abc(a-b)(b-c)(c-a)

Prove the following: [[1,1,1],[a,b,c],[a^3,b^3,c^3]] =(b-c)(c-a)(a-b)(a+b+c)

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

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

Show that [[3a,-a+b,-a+c],[-b+a,3b,-b+c],[-c+a,-c+b,3c]] = 3(a+b+c)(ab+bc+ca)

Prove that: |[1, 1, 1],[a, b, c],[a^2, b^2, c^2]|=(a-b)(b-c)(c-a)

Prove the following: [[b+c,a+b,a],[c+a,b+c,b],[a+b,c+a,c]] = a^3+b^3+c^3-3abc

SHARAM PUBLICATION-DETERMINANT-EXAMPLE
  1. Using properties of the determinants, prove that: |[2y, y-z-x, 2y],[2z...

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

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  3. Show that: |[x, y ,z],[x^2, y^2, z^2], [yz, zx, xy ]|=(x-y)(y-z)(z-x)....

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  4. Prove [[a^3-x^3,a^2,a],[b^3-x^3,b^2,b],[c^3-x^3,c^2,c]] = (a-b)(a-c)(b...

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  5. Prove that: 1/(bc+ca+ab)|[a, b, c],[a^2, b^2, c^2], [bc, ca, ab]|=(b-c...

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  6. Prove that: |[y+z, z, y],[z, z+x, x], [y, x, x+y]|= 4 xyz

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  7. Prove that: |[a+3b, a+5b, a+7b],[a+4b, a+6b, a+8b], [a+5b, a+7b, a+9b]...

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  8. Prove the following: [[a^2+1,ab,ac],[ab,b^2+1,bc],[ac,bc,c^2+1]] =1+...

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  9. Prove the following: [[-a^2,ab,ac],[ab,-b^2,bc],[ac,bc,-c^2]]=4a^2b^...

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  10. Prove the following: [[b^2-ab,b-c,bc-ac],[ab-a^2,a-b,b^2-ab],[bc-ac,...

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  11. Show that: |[(y+z)^2, xy, zx],[xy, (x+z)^2, yz], [xz, yz, (x+y)^2]|=2x...

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  12. If a, b, c, are in A.P. find the value of |[2y+4, 5y+7, 8y+a],[3y+5, 6...

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  13. Prove the following: [[1,1,1],[a,b,c],[a^3,b^3,c^3]] =(b-c)(c-a)(a...

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  14. If [[x,x^2,x^3-1],[y,y^2,y^3-1],[z,z^2,z^3-1]]=0 then prove that xyz...

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  15. Prove the following: [[a+b+c,-c,-b],[-c,a+b+c,-a],[-b,-a,a+b+c]] =...

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  16. Prove the following: [[b+c,a+b,a],[c+a,b+c,b],[a+b,c+a,c]] =a^3+b^...

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  17. If ax+hy+g=0, hx+by+f=0 and gx+fy+c=lamda, find the value of lamda in ...

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  18. Prove that abs((-2a,a+b,c+a),(a+b,-2b,b+c),(c+a,c+b,-2c))=4(b+c)(c+a)(...

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  19. If 2s=a+b+c show that [[a^2,(s-a)^2,(s-a)^2],[(s-b)^2,b^2,(s-b)^2],[...

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  20. If A+B+C = pi, prove that [[sin^2A,cotA,1],[sin^2B,cotB,1],[sin^2C,c...

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