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Show that : |[x, y, z ],[x^2,y^2,z^2],[x...

Show that : `|[x, y, z ],[x^2,y^2,z^2],[x^3,y^3,z^3]|=x y z(x-y)(y-z)(z-x)dot`

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`|(x,y,z),(x^2,y^2,z^2),(x^3 , y^3,z^3)|`
`= xyz|(1,1,1),(x,y,z),(x^2,y^2,z^2)|`
`c_2 -> c_2 - c_1`
`c_3 -> c_3 - c_1`
`=> |(1,0,0),(x, y-x, z-x),(x^2, y^2-x^2, z^2-x^2)|`
`= 1((y-x)(z^2-x^2) - (z-x)(y^2 - x^2))`
`= xyz(y-x)(z-x)(z+x-y-x)`
`= xyz(y-x)(z-x)(z-y)`= RHS
...
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RD SHARMA ENGLISH-DETERMINANTS-All Questions
  1. Using properties of determinants, show that triangle ABC is isosceles,...

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  2. In a triangleABC, if |[1,1,1][1+sinA,1+sinB,1+sinC],[sinA+sin^2A, sinB...

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

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  4. Without expanding or evaluating show that |[0 , b-a , c-a] , [a-b , 0...

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  5. If A is a skew-symmetric matrix of odd order n , then |A|=0

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  6. Using properties of determinants, show that |{:(x, p, q), ( p, x, q)...

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  7. If f(x)=|[a,-1 ,0],[ax,a,-1],[a x^2,a x, a]|, using properties of det...

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  8. If a , b , c are distinct real numbers and the system of equations a x...

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  9. If x , y , z are not all zero such that a x+y+z=0, x+b y+z=0, ...

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  10. Find the value of lambda for which the homogeneous system of equations...

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  11. If the system of equations x=c y+b z y=a z+c x z=b x+a y has...

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  12. A matrix A of order 3x3 has determinant 5. What is the value of |3A|?

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  13. If A is a square matrix such that |A|=2 , write the value of |AA^T|

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  14. Find the real values of lambda for which the following system of linea...

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  15. If a , b , c are non-zero real numbers and if the system of equations...

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  16. Which of the following is not correct in a given determinant of A , wh...

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  17. Let |x2xx^2x6xx6|=a x^4+b x^3+c x^2+dx+edot Then, the value of 5a+4b+3...

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  18. Let delta=|A x x^2 1 B y y^2 1 C z z^2 1|a n d1=|A B C x y z y z z xx ...

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  19. If Delta1=|[a, b, c],[ x, y, z],[ p, q, r]| and Delta2=|[q,-b, y],[-p,...

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  20. Without expanding show that: =|(cose c^2theta,cot^2theta,1),(cot^2thet...

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