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Prove the identities: |[z, x, y],[ z^...

Prove the identities: `|[z, x, y],[ z^2,x^2,y^2],[z^4,x^4,y^4]|=|[x, y, z],[ x^2,y^2,z^2],[x^4,y^4,z^4]|=|[x^2,y^2,z^2],[x^4,y^4,z^4],[x, y, z]|` =x y z (x-y)(y-z)(z-x)(x+y+z)

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Prove: |(z, x, y),( z^2,x^2,y^2),(z^4,x^4,y^4)|=|(x, y, z),( x^2,y^2,z^2),(x^4,y^4,z^4)|=|(x^2,y^2,z^2),(x^4,y^4,z^4),(x, y, z)|=x y z(x-y)(y-z)(z-x)(x+y+z) .

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RD SHARMA ENGLISH-DETERMINANTS-All Questions
  1. If f(x)=|0x-a x-b x+a0x-c x+b x+c0|,t h e n f(x)=0 (b) f(b)=0 (c) f...

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  2. Prove the identities: |[a^2,a^2-(b-c)^2,b c], [b^2,b^2-(c-a)^2,c a],[...

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  3. Prove the identities: |[z, x, y],[ z^2,x^2,y^2],[z^4,x^4,y^4]|=|[x,...

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  4. Without expanding, show that the value of each of the determinants is ...

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  5. Without expanding, show that the value of each of the determinants is ...

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  6. Without expanding, show that the value of each of the determinants is ...

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  7. Without expanding, show that the value of each of the determinants is ...

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  8. Using the properties of determinants, prove that following |(a-b-c,2a,...

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  9. Prove the identities: |[a, b, c],[ a-b,b-c,c-a],[ b+c,c+a, a+b]|=a^3+...

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  10. For any triangleABC, the value of determinant |[sin^2A,cotA,1],[sin^2B...

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  11. Without expanding, show that the value of each of the determinants is ...

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  12. If x , y in R , then the determinant =|(cos x , -sin x,1),(sin x, cos...

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  13. The maximum value of Delta=|(1,1,1),(1,1+sintheta,1),(1+costheta,1,1)...

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  14. Prove that: |[-2a, a+b,a+c],[ b+a,-2b,b+c],[c+a, c+b,-2c]|=4(a+b)(b+c...

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  15. Without expanding, show that the value of each of the determinants is ...

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  16. If m is a positive integer and Dr=|2r-1\ ^m Cr1m^2-1 2^m m+1s in^2(...

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  17. Without expanding, show that "Delta"=|(a-x)^2(a-y)^2(a-z)^2(b-x)^2(b-y...

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  18. Let Deltar=|[r , x , (n(n+1))/2] , [2r-1 , y , n^2] , [3r-2 , z , (n(3...

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  19. If Deltar=|[2^(r-1) , 2.3^(r-1) , 4. 5^(r-1)] , [x , y , z] , [2^n-1 ,...

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  20. Find a quadratic polynomial phi(x) whose zeros are the maximum and mi...

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