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Prove: |(y+z, z, y),( z, z+x,x),( y, x,x...

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

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`LHS=|(y+z, z, y),( z, z+x,x),( y, x,x+y)|`
`underset(1)(R)->underset(1)(R)-underset(2)(R)-underset(3)(R)`
`implies |(y+z-z-y, z-z-x-x, y-x-x-y),( z, z+x,x),( y, x,x+y)|`
`implies |(0, -2x, -2x),( z, z+x,x),( y, x,x+y)|`
...
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RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. |[x+lambda, 2x, 2x], [2x, x+lambda, 2x], [2x, 2x, x+lambda]| =(5x+ lam...

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  2. सारणिकों के गुणधर्मो का प्रयोग करके प्रश्न 6 से 14 तक को सिद्ध कीजिए: ...

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

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  4. Show that | (-a(b^2 + c^2 - a^2), 2b^3, 2c^3), (2a^3, -b(c^2 + a...

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  5. Prove: |(1+a,1, 1),( 1, 1+a, 1),(1, 1, 1+a)|=a^3+3a^2

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

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

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

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

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

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  11. 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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  12. 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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  13. If a ,\ b ,\ c are real numbers such that |b+cc+a a+b c+a a+bb+c a+...

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  14. 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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  15. 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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  16. Solve the following determinant equations: |(x+a, b, c),( a, x+b, c)...

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

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

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

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