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Given : a^(2)+b^(2)+c^(2) =0 Prove the...

Given : ` a^(2)+b^(2)+c^(2) =0 `
Prove the following :
` |{:(b^(2)+c^(2),ab,ca),(ab,c^(2)+a^(2),bc),(ca,bc,a^(2)+b^(2)):}|=4a^(2)b^(2)c^(2)`

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If a+b+c=0 then prove the following. (bc+ca+ab)^(2) =b^(2)c^(2) +c^(2)a^(2) +a^(2)b^(2)=(1)/(4) (a^(2) +b^(2) +c^(2))^(2)

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MODERN PUBLICATION-DETERMINANTS-Exercise 4(b) (LONG ANSWER TYPE QUESTIONS (II))
  1. |[a,b,c],[a-b,b-c,c-a],[b+c,c+a,a+b]|=a^3+b^3+c^3-3abc

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  2. Evaluate the following: |[a^2+2a, 2a+1, 1],[2a+1, a+2, 1],[3,3,1]|

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  3. Given : a^(2)+b^(2)+c^(2) =0 Prove the following : |{:(b^(2)+c^(2...

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  4. |[1+a^2-b^2,2ab,-2b],[2ab,1-a^2+b^2,2a],[2b,-2a,1-a^2-b^2]|=(1+a^2+b^2...

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  5. Prove the following : |{:(x,y,z),(x^(2),y^(2),z^(2)),(x^(3),y^(3),z^...

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  6. [[x, x^2, yz],[y, y^2, zx],[z, z^2, xy]]=(x-y)(y-z)(z-x)(xy+yz+zx)

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

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

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  9. Using properties of determinants, prove the following: |xx+y x+2y\...

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  10. |[b+c, a,a] , [b,c+a,b] , [c,c,a+b]|=4abc

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  11. Prove the following : |{:(2ab,a^(2),b^(2)),(a^(2),b^(2),2ab),(b^(2),...

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  12. Prove the following : |(1,x,x^(2)-yz),(1,y,y^(2)-zx),(1,z,z^(2)-xy)|=...

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  13. For any scalar p prove that =|xx^2 1+p x^3y y^2 1+p y^3z z^2 1+p z^3|=...

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  14. |(x+y+z,-z,-y),(-z,x+y+z,-x),(-y,-x,x+y+z)|=2(x+y)(y+z)(z+x)

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  15. Prove: |2y y-z-x2y2z2z z-x-y x-y-z2x2x|=(x+y+z)^3

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  16. |(a-b-c,2a,2a),(2b,b-c-a,2b),(2c,2c,c-a-b)|

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  17. Show that: |3a-a+b-a+c-b+a3b-b+c-c+a-c+b3c|=3(a+b+c)(a b+b c+c a)dot

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  18. Using properties of determinants. Find the value of 'x' |(4-x,4+x,4+...

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  19. Solve: |a+x a-x a-x a-x a+x a-x a-x a-x a+x|=0

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  20. Prove that |[x, sintheta, costheta],[-sintheta, -x, 1],[costheta, 1, ...

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