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Prove that: (i) |{:(,1,1,1),(,a,b,c),(...

Prove that:
(i) `|{:(,1,1,1),(,a,b,c),(,a^(3),b^(3),c^(3)):}|=(a-b)(b-c)(c-a)(a+b+c)`
(ii) `|{:(,a,b+c,a^(2)),(,b,c+a,b^(2)),(,c,a+b,c^(2)):}|=-(a+b+c)(a-b)(b-c)(c-a)`
(iii) `|{:(,b+c,a,a),(,b,c+a,b),(,c,c,a+b):}|=4abc`
(iv) If `|{:(,1,a^(2),a^(4)),(,1,b^(2),b^(4)),(,1,c^(2),c^(4)):}|=(a+b)(b+c)(c+a)=|{:(,1,1,1),(,a,b,c),(,a^(2),b^(2),c^(2)):}|`

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OSWAAL PUBLICATION-DETERMINANTS-TOPIC -1 DETERMINANTS, MINORS & COFACTORS (LONG ANSWER TYPE QUESTIONS-I)
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  2. Prove that |[1,a,a^3],[1,b,b^3],[1,c,c^3]|=(a-b)(b-c)(a+b+c)

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  3. Prove that: (i) |{:(,1,1,1),(,a,b,c),(,a^(3),b^(3),c^(3)):}|=(a-b)(b...

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

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  5. Find the equation of the line joining A( 1,3) and B (0,0) using det...

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

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  7. Using properties of determinants, prove the following: |alphabetag...

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  8. 15. Using properties of determinants, prove the following |[a,b,c],[a-...

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  9. [[b+c,a-b,a],[c+a,b-c,b],[a+b,c-a,c]] = 3abc - a^3 - b^3 - c^3

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  10. Prove that |[a^2, a^2-(b-c)^2, bc],[b^2, b^2-(c-a)^2, ca],[c^2, c^2-(...

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

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  12. Using properties of determinants, prove that |b+c q+r y+z c+a r+p ...

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  13. Using the Properties of determinants, prove that following: {:|(-a^2...

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  14. Using the Properties of determinants, prove the following: {:|(1,1...

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  15. If |{:(x,x^2,1+x^3),(y,y^2,1+y^3),(z, z^2,1+z^3):}|=0 and x, y, z are ...

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  16. Using properties of determinants, solve the following for x: |x-2 ...

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  17. Using properties of determinants, solve for x:|a+x a-x a-x a-x a+x a...

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

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  19. Prove, using properties of determinants: |y+k y y y y+k y y y y+k|=k^...

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  20. Prove |[-bc, b^2+bc, c^2+bc] , [a^2+ac, -ac, c^2+ac] , [a^2+ab, b^2+ab...

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