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using the properties of determinants prove that
`|{:(1,x+y,x^2+y^2),(1,y+z,y^2+z^2),(1,z+x,z^2+x^2):}|=(x-y)(y-z)(z-x)`

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Using properties of determinants, prove that |{:(y^2z^2,yz,y+z),(z^2x^2,zx,z+x),(x^2y^2,xy,x+y):}|=0

Using properties of determinants prove that |{:(a+x,y,z),(x,a+y,z),(x,y,a+z):}|=a^2 (a+x+y+z)

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

Using properties of the determinants, prove that: |[2y, y-z-x, 2y],[2z, 2z, z-x-y], [x-y-z, 2x, 2x]| = (x+y+z)^3

Using the properties of determinants prove that |{:(a+b+2c,a,b),(c,b+c+2a,b),(c,a,c+a+2b):}|=2(a+b+c)^3 or |{:(x+y+2z,x,y),(z,y+z+2x,y),(z,x,z+x+2y):}|=2(x+y+z)^3

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

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

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

Using coafactors of the elements of third row, evaluate Delta=|{:(1,x,y+z),(1,y,z+x),(1,z,x+y):}|

ARIHANT PUBLICATION-DETERMINANTS -PART -1 QUESTION FOR PRACTICE
  1. IF cos 2 theta=0 then simplify |{:(0,cos theta, sin theta),(cos theta,...

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

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  3. using the properties of determinants prove that |{:(1,x+y,x^2+y^2),(...

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  4. Using properties of determinants prove that |{:(a^2+2a,2a+1,1),(2a+1...

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  5. Prove that the following. [[(a+1)(a+2),a+2,1],[(a+2)(a+3),a+3,1],[(a+3...

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  6. Using properties of determinants solve the following for x, |{:(x+a,x,...

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  7. Solve [[a+x,a-x,a-x],[a-x,a+x,a-x],[a-x,a-x,a+x]]=0

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  8. IF |{:(4-x,4+x,4+x),(4+x,4-x,4+x),(4+x,4+x,4-x):}|=0 then find the val...

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  9. Prove the following : [[1,x,x^2],[x^2,1,x],[x,x^2,1]]=(1-x^3)^2

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  10. Using properties of determinant prove that |{:(alpha,beta,gamma),(alph...

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  11. IF a,b and c are real numbers and Delta=|{:(b+c,c+a,a+b),(c+a,a+b,b+c)...

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  12. Using the properties of determinants prove that |{:(a+b+2c,a,b),(c,b...

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  13. Find the value of x if |(2x,3),(1,x)|=|(3,4),(-1,2)|

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  14. Prove that |[x+y, x, x],[5x+4y, 4x, 2x], [10x+8y, 8x, 3x]|=x^3

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  15. If x,y and z are different and Delta=|{:(x,x^2,1+x^3),(y,y^2,1+y^3),...

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  16. Prove that abs[[b+c,c+a,a+b],[q+r,r+p,p+q],[y+z,z+x,x+y]]=2abs[[a,b,c]...

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  17. Show that Delta =Delta1 where Delta=|{:(Ax,x^2,1),(By,y^2,1),(Cz,z^2...

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  18. IFDelta=|{:(1,x,x^2),(1,y,y^2),(1,z,z^2):}| and Delta1=|{:(1,1,1),(yz,...

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  19. Using properties of determinants, prove that |{:(y^2z^2,yz,y+z),(z^2x^...

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  20. If f(x)=|{:(a,-1,0),(ax,a,-1),(ax^2,ax,a):}| then using properties of...

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