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If f(x)=|[a-1, 0,a] , [x,a-1,a], [x^2a, ...

If `f(x)=|[a-1, 0,a] , [x,a-1,a], [x^2a, x, a]|` , using properties of determinants, find the value of `f(2x)-f(x)` .

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RD SHARMA ENGLISH-DETERMINANTS-All Questions
  1. Prove that: |a^2+2a2a+1 1 2a+1a+2 1 3 3 1|=(a-1)^3 .

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

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  3. If f(x)=|[a-1, 0,a] , [x,a-1,a], [x^2a, x, a]| , using properties of ...

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  4. Show that: x p q p x q q q xx-p)(x^2+p x-2q^2) .

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

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

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  7. Prove that: |x+yx x5x+4y4x2x 10 x+8y8x3x|=x^3 .

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

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  9. Show that: |[a, a+b ,a+b+c],[2a,3a+2b,4a+3b+2c],[3a,6a+3b, 10 a+6b+3c]...

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  10. Show that: |[b+c,c+a ,a+b],[ q+r, r+p, p+q],[ y+z ,z+x,x+y]|=2|[a, b, ...

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  11. Prove that: determinant \begin{vmatrix} 1+ a & 1 & 1 \\ 1 & 1+b...

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

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  13. 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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  14. Show that: |(b+c)^2b a c a a b(c+a)^2c b a c b c(a+b)^2|=2a b c(a+b+c)...

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  15. Show that: |1+a^2-b^2 2a b-2b2a b1-a^2+b^2 2a2b-2a1-a^2-b^2|=(1+a^2+b^...

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  16. Show that: |b^2+c^2a b a c b a c^2+a^2b cc a c b a^2+b^2|=4a^2\ b^2\ c...

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  17. Prove that: |a b a x+b y b c b x+c y a x+b y b x+c y0|=(b^2-a c)(a x^2...

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  18. Without expanding the determinant, show that (a+b+c) is a factor of ...

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  19. If a ,\ b ,\ c are roots of the equation x^3+p x+q=0 , prove that d...

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  20. Find dy/dx if y=cos^2x

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