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Factorize the following |[3,a+b+c,a^3+b^...

Factorize the following `|[3,a+b+c,a^3+b^3+c^3],[a+b+c,a^2+b^2+c^2,a^4+b^4+c^4],[a^2+b^2+c^2,a^3+b^3+c^3,a^5+b^5+c^5]|`

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Prove without expanding |[a,a^2,bc],[b,b^2,ca],[c,c^2,ab]|=|[1,a^2,a^3],[1,b^2,b^3],[1,c^2,c^3]|

Find the value of a^(3)+b^(3)+c^(3)-3abc, when a + b + c = 5 , a^(2) + b^(2) + c^(2) = 11

Using properties of determinants , prove that, |{:(a,b,c),(b,c,a),(c,a,b):}|=-(a^3+b^3+c^3-3abc) and hence show that, |{:(2bc-a^2," "c^2," "b^2),(" "c^2,2ca-b^2," "a^2),(" "b^2," "a^2,2ab-c^2):}|=(a^3+b^3+c^3-3abc)^2

|{:(a,b,c),(a^2,b^2,c^2),(a^3,b^3,c^3):}|=abc(a-b)(b-c)(c-a)

If a, b, c be in continued proportional, then prove that a^2 b^2 c^2 (1/a^3 + 1/b^3+1/c^3) =a^3 +b^3+c^3

Which of the following has/have value equal to zero? |[8, 2, 7],[ 12, 3, 5],[ 16, 4, 3]| b. |[1//a, a^2,b c],[1//b,b^2,a c],[1//c,c^2,a b]| c. |[a+b,2a+b,3a+b],[2a+b,3a+b,4a+b],[4a+b,5a+b,6a+b]| d. |[2, 43, 3],[ 7, 35, 4],[ 3, 17, 2]|

|{:(-a(b^2+c^2-a^2)," "2b^3," "2c^3),(" "2a^3,-b(c^2+a^2-b^2)," "2c^3),(" "2a^3," "2b^3,-c(a^2+b^2-c^2)):}|=abc(a^2+b^2+c^2)^3

CENGAGE PUBLICATION-DETERMINANTS-All Questions
  1. Prove that |{:((b+x)(c+x),,(c+x)(a+x),,(a+x)(b+x)),((b+y)(c+y) ,,(c+y)...

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  2. If a+b+c=0, one root of [[a-x, c, b],[c, b-x, a],[ b, a, c-x]]=0 is ...

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  3. Factorize the following |[3,a+b+c,a^3+b^3+c^3],[a+b+c,a^2+b^2+c^2,a^4+...

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  4. Let {D1, D2, D3 ,Dn} be the set of third order determinant that can be...

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  5. If g(x)=(f(x))/((x-a)(x-b)(x-c)),w h e r ef(x) is a polynomial of de...

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  6. If the equation 2x+3y+1=0,3x+y-2=0, and a x+2y-b=0 are consistent, t...

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  7. If w is a complex cube root of unity, then value of =|[a1+b1w, a1w^2+b...

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  8. Let f(x)=|[cos(x+x^2),sin(x+x^2),-cos(x+x^2)],[sin(x-x^2),cos(x-x^2), ...

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  9. Which of the following has/have value equal to zero? |[8, 2, 7],[ 12, ...

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  10. If f(x),g(x) and h(x) are three polynomials of degree 2, then prove th...

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  11. If a ,b ,c are in G.P. with common ratio r1a n dalpha,beta,gamma are i...

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  12. Show 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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  13. In triangle A B C , if |[1, 1, 1],[cot(A/2),cot(B/2),cot(C/2)],[tan(B/...

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  14. If a ,b ,c ,d ,e, and fare in G.P. then the value of |((a^2),(d^2),x)...

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  15. Prove that: |[(b+c)^2,a^2,a^2],[b^2,(c+a)^2,b^2],[c^2,c^2,(a+b)^2]|=2a...

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  16. If xi=ai bi ci,i=1,2,3 are three-digit positive integer such that ea...

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  17. The system of linear equations x+lambday-z=0, lambdax-y-z=0, x+y-lam...

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  18. If a , b ,c are all different and |[a,a^3,a^4-1],[b,b^3,b^4-1],[c,c^...

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  19. If p q r!=0 and the system of equation (p+a)x+b y+c z=0, a x+(q+b)y+c...

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  20. If c<1 and the system of equations x+y-1=0,2x-y-c=0,a n d -b x+3b y-c=...

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