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

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

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To prove that the determinant \[ \begin{vmatrix} 1 & 1 & 1 & 1 \\ 1 & 1+x & 1 & 1 \\ 1 & 1 & 1+y & 1 \\ 1 & 1 & 1 & 1 ...
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RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. 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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  2. 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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  3. Prove that: |1 1 1 1 1+x1 1 1 1+y|=x y .

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

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

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

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

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  9. 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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  10. 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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  11. 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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  12. Using properties of determinants, show that |1a a^2-b c1bb^2-c a1cc...

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

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

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

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

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

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  18. Evaluate: =|10 ! 11 ! 12 ! 11 ! 12 ! 13 ! 12 ! 13 ! 14 !| .

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  19. Prove that: |x+y xx5x+4y4x2x 10 x+8y8x3x|=x^3 .

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  20. Show that: |1 1+p1+p+q2 3+2p1+3p+2q3 6+3p1+6p+3q|=1 .

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