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Let Deltar=|[r , x , (n(n+1))/2] , [2r-1...

Let `Delta_r=|[r , x , (n(n+1))/2] , [2r-1 , y , n^2] , [3r-2 , z , (n(3n-1))/2]|dot` Show that `sum_(r=1)^n Delta_r=0`

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To solve the problem, we need to show that \[ \sum_{r=1}^n \Delta_r = 0 \] where ...
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RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. If m is a positive integer and Dr=|2r-1\ ^m Cr1m^2-1 2^m m+1s in^2(...

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  2. Without expanding, show that "Delta"=|(a-x)^2(a-y)^2(a-z)^2(b-x)^2(b-y...

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  3. Let Deltar=|[r , x , (n(n+1))/2] , [2r-1 , y , n^2] , [3r-2 , z , (n(3...

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  4. If Deltar=|[2^(r-1)2 , 3^(r-1) , 4. 5^(r-1)] , [x , y , z] , [2^n-1 , ...

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  5. Find a quadratic polynomial phi(x) whose zeros are the maximum and mi...

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  6. Let f(x)=|secxcosxsec^2x+cotxcos e cxcos^2xcos^2x cos e c^2x1cos^2xcos...

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  7. Show that: |(a, b-c,c+b), (a+c, b, c-a), (a-b,b+a, c)|=(a+b+c)(a^2+b^2...

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  8. If a , b , c are real numbers, prove that |a b c b c a c a b|=-(a+b+c)...

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  9. Solve: |(a+x, a-x, a-x),(a-x, a+x, a-x),(a-x, a-x, a+x)|=0

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  10. Show that: |(3a,-a+b,-a+c),(-b+a,3b,-b+c),(-c+a,-c+b,3c)|=3(a+b+c)(a b...

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  11. If a,b,c are all distinct and |[a,a^3,a^4-1],[b,b^3,b^4-1],[c,c^3,c^4-...

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  12. Solve : |(x-2, 2x-3, 3x-4), (x-4, 2x-9, 3x-16) ,(x-8 ,2x-27,3x-64)|=0

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  13. If x+y+z=0 , prove that |[xa, yb,zc],[yc, za, xb], [zb, xc, ya]|=x y ...

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  14. If a , b , c are all positive and are p t h ,q t ha n dr t h terms of ...

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  15. Prove that: |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-...

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  16. |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-a^2,ca-b^2]|...

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  17. Prove that : |[(b+c)^2,a^2,a^2],[b^2,(c+a)^2,b^2],[c^2,c^2,(a+b)^2]|...

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  18. 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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  19. Prove that: |[a, b, ax+by],[ b, c, bx+cy], [ax+by, bx+cy,0]|=(b^2-a c)...

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

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