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Prove: |(a^2,b c, a c+c^2),(a^2+a b,b^2,...

Prove: `|(a^2,b c, a c+c^2),(a^2+a b,b^2,a c ),(a b,b^2+b c,c^2)|=4a^2b^2c^2`

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`Delta=|(a^2,b c, a c+c^2),(a^2+a b,b^2,a c ),(a b,b^2+b c,c^2)|`

`Delta= |(a^2+2ab,2b^2+2bc, 2c^2+2ac),(a^2+2a b,2b^2+bc,c^2+a c ),(a b,b^2+b c,c^2)|`
`underset(1)(R)->underset(1)(R)+underset(3)(R)+underset(2)(R)`
`underset(2)(R)->underset(2)(R)+underset(3)(R)`
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RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. Prove: |(a, a+b, a+2b),( a+2b, a ,a+b ),(a+b, a+2b, a)|=9(a+b)b^2

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  2. Prove: |(1,a, b c),(1,b ,c a),(1,c ,a b)|=|(1,a ,a^2),( 1,b,b^2),( 1,c...

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  3. Prove: |(a^2,b c, a c+c^2),(a^2+a b,b^2,a c ),(a b,b^2+b c,c^2)|=4a^2b...

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  4. Prove: |(x+4,x,x),(x,x+4,x),(x,x,x+4)|=16(3x+4)

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  5. Show that |[1,1+p,1+p+q],[2,3+2p,1+3p+2q],[3,6+3p,1+6p+3q]|=

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  6. Prove: |(a, b-c,c-b),( a-c, b, c-a),( a-b,b-a, c)|=(a+b-c)(b+c-a)(c+a-...

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  7. Prove: |a^2+1a b a c a bb^2+1b cc a c b c^2+1|=1+a^2+b^2+c^2

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  8. Prove: |1a a^2a^2 1a a a^2 1|=(a^3-1)^2

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  9. Prove: |b+c a a b c+a b cc a+b|=4a b c

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  10. Show that |[b^2+c^2,ab,ac],[ba,c^2+a^2,bc],[ca,cb,a^2+b^2]|=4a^2b^2c^2

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  11. Prove: |(0,b^2a, c^2a),( a^2b,0,c^2b),( a^2c, b^2c,0)|=2a^3b^3c^3

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  12. Prove: |((a^2+b^2)/c,c,c),( a,(b^2+c^2)/a ,a),( b,b,(c^2+a^2)/b)|=4a b...

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  13. Prove: |(-bc, b^2+b c,c^2+bc), (a^2+a c,-a c,c^2+a c),( a^2+a b,b^2+a ...

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  14. |[x+lambda, 2x, 2x], [2x, x+lambda, 2x], [2x, 2x, x+lambda]| =(5x+ lam...

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

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

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

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

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

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

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