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Using properties of determinants. Prove ...

Using properties of determinants. Prove that`|(x, x^2, 1+p x^3),( y, y^2, 1+p y^3),(z, z^2, 1+p z^3)|=(1+p x y z)(x-y)(y-z)(z-x)`, where `p` is any scalar.

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To prove that \[ |(x, x^2, 1 + p x^3), (y, y^2, 1 + p y^3), (z, z^2, 1 + p z^3)| = (1 + pxyz)(x - y)(y - z)(z - x), \] where \( p \) is any scalar, we will use properties of determinants step by step. ...
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NCERT-DETERMINANTS-MISCELLANEOUS EXERCISE
  1. Let A=[[1,sintheta,1],[-sintheta,1,sintheta],[-1,-sintheta,1]], where ...

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  2. If x, y, z are non-zero real numbers, then the inverse of matrix A=[(...

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  3. If a, b, c, are in A.P, then the determinant |(x+2,x+3,x+2a),( x+3,x+4...

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  4. Solve the system of equations 2/x+3/y+(10)/z=4 4/x-6/y+5/z=1 6/x+9/y-(...

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  5. Using properties of determinants. Prove that |(sinalpha,cosalpha,cos(a...

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  6. Using properties of determinants. Prove that|[1 ,1+p,1+p+q],[2, 3+2p,4...

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  7. Using properties of determinants. Prove that|[3a,-a+b,-a+c],[-b+a,3b,-...

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  8. Using properties of determinants. Prove that|(x, x^2, 1+p x^3),( y, y^...

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  9. Using properties of determinants. Prove that|(alpha,alpha^2,beta+gamma...

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  10. Evaluate |[1,x, y],[1,x+y, y],[1,x,x+y]|

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  11. Let A=[{:(1,-2,1),(-2,3,1),(1,1,5):}]. Verify that ltbtgt (i) [adjA]^...

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  12. Evaluate |(x, y, x+y),( y, x+y, x),( x+y, x, y)|.

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  13. Without expanding the determinant, prove that |(a,a^2,bc),(b,b^2,ca),(...

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  14. Evaluate |(cosalphacosbeta,cosalphasinbeta,-sinalpha),(-sinbeta,cosbet...

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  15. 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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  16. Solve the equation |(x+a, x,x),(x,x+a, x),(x,x,x+a)|=0, a!= 0

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  17. Prove that |[a^2,bc,ac+c^2],[a^2+ab,b^2,ac],[ab,b^2+bc,c^2]|=4a^2b^2c^...

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  18. Prove that the determinant [(x,sintheta,costheta),(-sintheta,-x,1),(co...

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  19. If A^(-1)=[(3,-1, 1),(-15, 6,-5),( 5,-2, 2)]and B=[(1 ,2,-2),(-1, 3, 0...

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