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|[y+z ,x, y],[ z+y, z, x],[x+y, y ,z]|=(...

`|[y+z ,x, y],[ z+y, z, x],[x+y, y ,z]|=(x+y+z)(x-z)^2`

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`LHS=|[y+z ,x, y],[ z+y, z, x],[x+y, y ,z]|`
`underset(1)(R)->underset(1)(R)+underset(2)(R)+underset(3)(R)`
`implies Delta=|[2(x+y+z) ,x+y+z, x+y+z],[ z+y, z, x],[x+y, y ,z]|`
`implies (x+y+z)|[2 ,1, 1],[ z+y, z, x],[x+y, y ,z]|`
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Prove that : |{:(y+z,x,y),(z+x,z,x),(x+y,y,z):}|=(x+y+z)(x-z)^(2)

Prove the identities: |[z, x, y],[ z^2,x^2,y^2],[z^4,x^4,y^4]|=|[x, y, z],[ x^2,y^2,z^2],[x^4,y^4,z^4]|=|[x^2,y^2,z^2],[x^4,y^4,z^4],[x, y, z]| =x y z (x-y)(y-z)(z-x)(x+y+z)

Prove that Det [[x + y + 2z, x, y], [z, y + z + 2x, y], [z, x, z + x + 2y]] = 2 (x + y + z) ^ 3

|[z+y,x,x] , [y,z+x,y] , [z,z,x+y]|=

Value of [[x+y, z,z ],[x, y+z, x],[y, y, z+x]], where x ,y ,z are nonzero real number, is equal to x y z b. 2x y z c. 3x y z d. 4x y z

Prove that |[x+y, y+z, z+x] , [z+x, x+y, y+z] , [y+z, z+x, x+y]|=2|[x,y,z] , [z,x,y] , [y,z,x]|

Prove: |(y+z, z, y),( z, z+x,x),( y, x,x+y)|=4\ x y z

RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. Prove: |(1+a,1, 1),( 1, 1+a, 1),(1, 1, 1+a)|=a^3+3a^2

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  2. Prove: |(2y, y-z-x,2y),(2z,2z, z-x-y),( x-y-z,2x,2x)|=(x+y+z)^3

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  3. |[y+z ,x, y],[ z+y, z, x],[x+y, y ,z]|=(x+y+z)(x-z)^2

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  4. Using properties of determinants, prove that |(a+x, y, z),( x, a+y, z...

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

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  6. Without expanding, prove that |(a, b, c),( x, y, z),( p, q ,r)|=|(x, y...

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  7. Show that |x+1x+2x+a x+2x+3x+b x+3x+4x+c|=0 where a ,\ b ,\ c are in A...

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  8. Show that [[x-3,x-4,x-alpha],[x-2,x-3,x-beta],[x-1,x-2,x-gamma]]=0 whe...

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  9. If a ,\ b ,\ c are real numbers such that |b+cc+a a+b c+a a+bb+c a+...

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  10. a!=p , b!=q,c!=r and |(p,b,c),(a,q,c),(a,b,r)|=0 the value of p/(p-...

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  11. Show that x=2 is a root of the equation |x-6-1 2-3xx-3-3 2xx+2|=0 a...

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  12. Solve the following determinant equations: |(x+a, b, c),( a, x+b, c)...

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  13. Solve the following: |[1,x,x^2], [1,a, b^2], [1,b,c^2]|=0 , a!=b

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  14. If a ,b and c are all non-zero and |1+a1 1 11+b1 1 11+c|=0, then pro...

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  15. Find the area of the triangle with vertices A\ (5,\ 4),\ \ B(-2,\ 4...

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  16. Show that pointsA (a , b + c), B (b , c + a), C (c , a + b)are collin...

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  17. If the points (a1, b1),\ \ (a2, b2) and (a1+a2,\ b1+b2) are collinear,...

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  18. If the points (2,\ -3),\ (lambda,\ -1) and (0,\ 4) are collinear, find...

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  19. Using determinants, find the area of the triangle whose vertices are...

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  20. If A(x1, y1),B(x2, y2)a n dC(x3,y3) are vertices of an equilateral tri...

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