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Prove that |{:(b+c, c+a, a+b),(c+a, a...

Prove that
`|{:(b+c, c+a, a+b),(c+a, a+b,b+c),(a+b, b+c, c+a):}| =2(a+b+c)(ab+bc+ca-a^(2)-b^(2)-c^(2)).`

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MODERN PUBLICATION-DETERMINANTS-Exercise 4(b) (LONG ANSWER TYPE QUESTIONS (II))
  1. [[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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  2. 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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  3. 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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  4. Using properties of determinants, prove the following: |xx+y x+2y\...

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

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

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

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  8. 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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  9. |(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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  10. Prove: |2y y-z-x2y2z2z z-x-y x-y-z2x2x|=(x+y+z)^3

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

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

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

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

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  16. Prove that |{:(1+a,1,1),(1,1+b,1),(1,1,1+c):}| =abc (1+(1)/(a)+(1)/...

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  17. Prove that : |{:((y+z)^(2),x^(2),x^(2)),(y^(2),(x+z)^(2),y^(2)),(z^(2)...

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  18. Prove that | ((b+c)^2, a^2,a^2),(b^2,(c+a)^2,b^2),(c^2,c^2,(a+b)^2)|=2...

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  19. |[(b+c)^2,ab,ca],[ab,(a+c)^2,bc],[ac,bc,(a+b)^2]|=2abc(a+b+c)^3

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  20. Show that Delta=|(y+z)^2x y z xx y(x+z)^2y z x z y z(x+y)^2|=2x y z(x+...

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