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Show that |[a ,b ,c],[ a^2,b^2,c^2],[bc,...

Show that `|[a ,b ,c],[ a^2,b^2,c^2],[bc, ca, ab]|=|[1, 1, 1],[a^2,b^2,c^2],[a^3,b^3,c^3]|=(a-b)(b-c)(c-a)(a b+b c+c a)` .

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Part a) $$ \Delta=\left|\begin{array}{ccc} a & a^{2} & b c \\ b & b^{2} & c a \\ c & c^{2} & a b \end{array}\right| $$ ...
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RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. In a triangleABC, if |[1,1,1][1+sinA,1+sinB,1+sinC],[sinA+sin^2A,sinB+...

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

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  3. 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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  4. 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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  5. 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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  6. Using properties of determinants, show that |1a a^2-b c1bb^2-c a1cc...

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

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

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

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

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

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

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

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

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  15. Show that: |a a+b a+b+c2a3a+2b4a+3b+2c3a6a+3b 10 a+6b+3c|=a^3 .

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  16. Show that: |b+cc+a a+b q+r r+p p+q y+z z+xx+y|=2|a b c p q r x y z| .

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  17. Prove that: |1+a1 1 1 1+b1 1 1 1+c|=a b c(1+1/a+1/b+1/c)=a b c+b c+c a...

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  18. If a ,\ b ,\ c are the roots of the equation x^3+p x+q=0 , then fin...

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  19. Prove that: |(b+c)^2a^2a^2b^2(c+a)^2b^2c^2c^2(a+b)^2|=2a b c(a+b+c)^3

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  20. Prove that: |(a,b, ax+by),(b,c,bx+cy), (ax+by, bx+cy,0)|=(b^2-a c)(a x...

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