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Without expanding or evaluating show that `|0b-a c-a a-b0c-b a-c b-c0|=0` .

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$$ \Delta=\left|\begin{array}{ccc} 0 & b-a & c-a \\ a-b & 0 & c-b \\ a-c & b-c & 0 \end{array}\right| $$ $$ ...
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RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. If A is a skew-symmetric matrix of odd order n , then |A|=O .

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  2. Prove that: |0a-b-a0-c b c0|=0 .

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  3. Without expanding or evaluating show that |0b-a c-a a-b0c-b a-c b-c...

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  4. Without expanding, prove that |a+b x c+dx p+q x a x+b c x+d p x+q u...

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  5. Prove that: |-a^2a b a c b a-b^2b c a c b c-c^2|=4a^2b^2c^2 .

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  6. Prove that: |1 1 1 1 1+x1 1 1 1+y|=x y .

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  7. Evaluate: |1a a^2 1bb^2 1cc^2| .

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  8. Show that : |x y z x^2y^2z^2x^3y^3z^3|=x y z(x-y)(y-z)(z-x)dot

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  9. Prove that: |alphabetagammaalpha^2beta^2gamma^2beta+gammagamma+alphaal...

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  10. In a triangleABC, if |[1,1,1][1+sinA,1+sinB,1+sinC],[sinA+sin^2A,sinB+...

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  11. In a A B C ,if|1 1 1 1+cosA1+cosB1+cosCcos^2A+cosBcos^2A+cosBcos^2+co...

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  12. Show that |[a ,b ,c],[ a^2,b^2,c^2],[bc, ca, ab]|=|[1, 1, 1],[a^2,b^2,...

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  13. If x!=y!=z and |xx^2 1+x^3y y^2 1+y^3z z^2 1+z^3|=0 , then prove that ...

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  14. For any scalar p prove that =|xx^2 1+p x^3y y^2 1+p y^3z z^2 1+p z^3|=...

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  15. Using properties of determinants, show that |1a a^2-b c1bb^2-c a1cc...

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  16. Prove that: |a^2+2a2a+1 1 2a+1a+2 1 3 3 1|=(a-1)^3 .

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  17. Let a ,\ b and c denote the sides B C ,\ C A and A B respectively of ...

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  18. If f(x)=|a-1 0a x a-1a x^2a x a| , using properties of determinants...

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  19. Show that: |x p q p x q q q x|=(x-p)(x^2+p x-2q^2) .

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  20. If m in N and mgeq2 prove that: |1 1 1\ ^m C1\ ^(m+1)C1\ ^(m+2)C1\ ^m...

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