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if |{:(1,,1,,1),(a,,b,,c),(a^(3),,b^(3),...

if `|{:(1,,1,,1),(a,,b,,c),(a^(3),,b^(3),,c^(3)):}|= (a-b)(b-c)(c-a)(a+b+c)`
where a,b,c are all different then the determinant
`|{:(1,,1,,1),((x-a)^(2),,(x-b)^(2),,(x-c)^(2)),((x-b)(x-c),,(x-c)(x-a),,(x-a)(x-b)):}|` vanishes when

A

`a + b + c = 0`

B

`x = (1)/(3) (a + b +c)`

C

`x = (1)/(2) (a + b +c)`

D

`x = a + b + c`

Text Solution

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The correct Answer is:
B
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OBJECTIVE RD SHARMA ENGLISH-DETERMINANTS-Exercise
  1. If the determinant |(b -c,c -a,a -b),(b' -c',c' -a',a' -b'),(b'' -c'',...

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  2. If a !=b, then the system of equation ax + by + bz = 0 bx + ay + bz ...

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  3. if |{:(1,,1,,1),(a,,b,,c),(a^(3),,b^(3),,c^(3)):}|= (a-b)(b-c)(c-a)(a+...

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  4. Show that a x+b y+r=0,b y+c z+p=0a n dc z+a x+q=0 are perpendicular to...

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  5. Consider the function f(x) = |{:(a^(2)+x,,ab,,ac),(ab,,b^(2)+x,,bc),(a...

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  6. The equation |(x-a,x-b,x-c),(x-b,x-a,x-c),(x-c,x-b,x-a)|=0 (a,b,c are ...

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  7. Let A = [(1,sin theta,1),(- sin theta,1,sin theta),(-1,-sin theta,1)],...

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  8. If a,b,c are non-zero real number such that |(bc,ca,ab),(ca,ab,bc),(ab...

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  9. The value of the determinant |{:(ka,,k^(2)+a^(2),,1),(kb,,k^(2)+b^(2)...

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  10. The system of simulataneous equations kx + 2y -z = 1 (k -1) y -2z ...

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  11. The value of the determinant |(1,omega^(3),omega^(5)),(omega^(3),1,ome...

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  12. If a,b,c are non-zero real number such that |(bc,ca,ab),(ca,ab,bc),(ab...

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  13. If the system of equations x + ay + az = 0 bx + y + bz = 0 cx + ...

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  14. The determinant D=|{:(cos(alpha+beta),-sin(alpha+beta),cos2beta),(sina...

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  15. If omega is a cube root of unity, then Root of polynomial is |(x + 1...

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  16. " If " Delta(1) =|{:(x,,b,,b),(a,,x,,b),(a,,a,,x):}|" and " Delta(2)...

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  17. If y= sin px and y(n) is the nth derivative of y, then |{:(y,y(1),y(...

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  18. If |p b c a q c a b r|=0 , find the value of p/(p-a)+q/(q-b)+r/(r-c),\...

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  19. Using properties of determinants, show that |{:(x, p, q), ( p, x, q)...

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  20. The factors of |(x,a,b),(a,x,b),(a,b,x)|, are

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