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(a-b)^3 + (b-c)^3 + (c-a)^3=? (a)(a+b+c...

`(a-b)^3 + (b-c)^3 + (c-a)^3=?` (a)`(a+b+c)(a^2+b^2+c^2-ab-bc-ac)` (b)`3(a-b)(b-c)(c-a)` (c)`(a-b)(b-c)(c-a)` (d)none of these

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To solve the expression \((a-b)^3 + (b-c)^3 + (c-a)^3\), we can use the identity related to the sum of cubes. ### Step-by-Step Solution: 1. **Recognize the identity**: We know that if \(x + y + z = 0\), then: \[ x^3 + y^3 + z^3 = 3xyz \] ...
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The expression (a-b)^(3)+(b-c)^(3)+(c-a)^(3) can be factorized as (a)(a-b)(b-c)(c-a)(b)3(a-b)(b-c)(c-a)(c)-3(a-b)(b-c)(c-a)(d)(a+b+c)(a^(2)+b^(2)+c^(2)-ab-bc-ca)

The determinant |{:(b^2-ab, b-c, bc-ac), (a b-a^2, a-b, b^2-ab) ,(b c-c a, c-a, a b-a^2):}| equals a b c\ (b-c)(c-a)(a-b) (b) (b-c)(c-a)(a-b) (c) (a+b+c)(b-c)(c-a)(a-b) (d) none of these

Prove: |a^3 2a b^3 2b c^3 2c|=2(a-b)(b-c)(c-a)(a+b+c)

The value of the determinant /_\=|(1,1,1),(a,b,c),(a^3,b^3,c^3)| is (A) (a-b)(b-c)(c-a)(a+b+c) (B) abc(a+b)(b+C)(c+a) (C) (a-b)(b-c)(c-a) (D) none of these

Using a^(3)+b^(3)+c^(3)-3abc=(a+b+c)(a^(2)+b^(2)+c^(2)-ab-bc-ca)

If a statement is true for all the values of the variable, such statements are called as identities. Some basic identities are : (1) (a+b)^(2)=a^(2)+2ab+b^(2)=(a-b)^(2)+4ab (3) a^(2)-b^(2)=(a+b)(a-b) (4) (a+b)^(3)=a^(3)+b^(3)+3ab(a+b) (6) a^(3)+b^(3)=(a+b)^(3)=3ab(a+b)=(a+b) (a^(2)-ab) (8) (a+b+c)^(2)=a^(2)+b^(2)+c^(2)+2ab+2bc+2ca=a^(2)+b^(2)+c^(2)+2abc((1)/(a)+(1)/(b)+(1)/(c)) (10) a^(3)+b^(3)+c^(3)-3abc=(a+b+c)(a^(2)+b^(2)+c^(2)-ab-bc-ca) =1/2(a+b+c)[(a-b)^(2)+(b-c)^(2)+(c-a)^(2)] If a+b+c=0,thena^(3)+b^(3)+c^(3)=3abc If a,b, c are real and distinct numbers, then the value of ((a-b)^(3)+(b-c)^(3)+(c-a)^(3))/((a-b).(b-c).(c-a))is

Show that |[a,b,c],[a^2,b^2,c^2],[a^3,b^3,c^3]|=abc(a-b)(b-c)(c-a)

Show that |abca^(2)b^(2)c^(2)baab|=|111a^(2)b^(2)c^(2)a^(2)b^(2)c^(3)|=(a-b)(b-c)(c-a)(ab+bc+ca)

Prove that |[(b+c)^2, a^2, bc],[(c+a)^2, b^2, ca],[(a+b)^2, c^2, ab]|=(a-b)(b-c)(c-a)(a+b+c)(a^2+b^2+c^2)

The value of the determinant |(1,a,a^2-bc),(1,b,b^2-ca),(1,c,c^2-ab)| is (A) (a+b+c),(a^2+b^2+c^2) (B) a^3+b^3+c^3-3abc (C) (a-b)(b-c)(c-a) (D) 0

RD SHARMA-ALGEBRAIC IDENTITIES-All Questions
  1. If x^3-1/(x^3)=14 , then x-1/x=? (a)5 (b) 4 (d) 3 (d) 2

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  2. If x^4+1/(x^4)=194 , find x^3+1/(x^3),x^2+1/(x^2) and x+1/x

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  3. If x-1/x=(15)/4, then x+1/x\ =? (a)4 (b) (17)/4 (c) (13)/4\ (d...

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  4. If 3x+2/x=7, then (9x^2-4/(x^2))=? (a)25 (b) 35 (c) 49 (d) 30

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  5. If a^2+b^2+c^2-a b-b c-c a=0, then a+b=c b+c=a c+a=b a=b=c

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  6. If a+b+c=0 , then (a^2)/(b c)+(b^2)/(c a)+(c^2)/(a b)=?\ (a)0 (b) ...

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  7. If a^(1/3)+\ b^(1/3)+\ c^(1/3)=0 then (a)a+b+c=0 (b) (a+b+c)^3=27\ a...

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  8. If a+b+c=9 and a b+b c+c a=23 , then a^2+b^2+c^2=? (a)35 (b) 58 (c)...

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  9. If a+b+c=9 and a b+b c+c a=23 , then a^3+b^3+c^3-3a b c=? (a)108 (b) ...

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  10. (a-b)^3 + (b-c)^3 + (c-a)^3=? (a)(a+b+c)(a^2+b^2+c^2-ab-bc-ac) (b)3(...

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  11. Simplify: ((a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3)/((a-b)^3+(b-c)^3+(c-a)...

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  12. The product (a+b)(a-b)(a^2-a b+b^2)(a^2+a b+b^2) is equal to: (a)a^6+b...

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  13. If a/b+b/a=\ -1, then a^3-b^3=? (a)1 (b) -1 (c) 1/2 (d) 0

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  14. The product (x^2-1)(x^4+x^2+1) is equal to: (a)x^8-1 (b) x^8+1 (...

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  15. If a-b=-8\ \ a n d\ \ a b=-12 , then a^3-b^3=? (a)-244 (b) -240 ...

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  16. If the value of a cuboid is 3x^2-27 , then its possible dimensions ar...

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  17. If a/b+b/a=1, then a^3+b^3=? (a)1 (b) -1 (c) 1/2 (d) 0

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  18. 75 \xx\ 75-2 \xx\ 75 \xx\ 25+25 \xx\ 25 is equal to=? (a)10000       ...

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  19. (x−y)(x+y)(x^2+y^2)(x^4+y^4) is equal to=

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  20. If 49 a^2-b=(7a+1/2)\ \ (7a-1/2), then the value of bis =? (a)0 (b) ...

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