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Prove that : Det[[x,x^2,x^3],[y,y^2,y^3]...

Prove that : `Det[[x,x^2,x^3],[y,y^2,y^3],[z,z^2,z^3]]=xyz(x-y)(y-z)(z-x)`

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NAGEEN PRAKASHAN ENGLISH-DETERMINANTS-Exercise 4b
  1. Prove that : |{:(x+a,x+2a,x+3a),(x+2a,x+3a,x+4a),(x+4a,x+5a,x+6a):}|=0

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  2. Prove that : |{:(1,x,yz),(1,y,zx),(1,z,xy):}|=(x-y)(y-z)(z-x)

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  3. Prove that : Det[[x,x^2,x^3],[y,y^2,y^3],[z,z^2,z^3]]=xyz(x-y)(y-z)(z-...

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

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

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

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  7. Prove that : |{:(x-y-z ,2x, 2x),(2y,y-z-x,2y),(2z,2z,z-x-y):}|=(x+y+z)...

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

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  9. Using the properties of determinants, prove that |{:((y+z)^(2)," "x...

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

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  11. Prove that : |{:(a^(2),b^(2)+c^(2),bc),(b^(2),c^(2)+a^(2),ca),(c^(2),a...

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  12. Prove that : (i) |{:(a,c,a+c),(a+b,b,a),(b,b+c,c):}|=2 abc (ii) Pr...

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  13. Prove that : |{:(a,b,c),(a^(2),b^(2),c^(2)),(bc,ca,ab):}|=(a-b)(b-c)(c...

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  14. Prove that : |{:(0,a-b,a-c),(b-a,0,b-c),(c-a,c-b,0):}|=0

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  15. |[x,-6,-1],[2,-3x,x-3],[-3,2x,x+2]|=0

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  16. Solve the equation |{:(x+a,x+b,x+c),(x+b,x+c,x+a),(x+c,x+a,x+b):}|=0

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  17. One root of the equation |(3x-8, 3, 3),(3,3x-8, 3),(3,3,3x-8)|=0 is ...

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  18. If 2s=a+b+c and A=|[a^2,(s-a)^2,(s-a)^2],[(s-b)^2,b^2,(s-b)^2],[(s-c)^...

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  19. If the sides of a DeltaABC are a, b, c and |{:(a^(2),b^(2),c^(2)),((a+...

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  20. If the pth, qth and rth terms of a G.P, are x,y and z repectively, the...

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