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If `a , b , c` are real numbers, prove that `\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{vmatrix}=`-(a+b+c)(c+b w+c w^2)(a+ bw^2+cw), where \ \ `w` is a complex cube root of unity.

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RD SHARMA ENGLISH-DETERMINANTS-All Questions
  1. Find dy/dx if y=cos^2x

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  2. If a+b+c!=0 and , \begin{vmatrix} a & b & c \\ b & c & a \\ ...

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  3. If a , b , c are real numbers, prove that \begin{vmatrix} a & b & ...

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  4. Show that: |a b-cc+b a+c b c-a a-bb+a c|=(a+b+c)(a^2+b^2+c^2)dot

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  5. Using properties of determinants. Prove that |[3a,-a+b,-a+c], [-b+a,3b...

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  6. Using properties of determinants, solve for x:|[a+x, a-x, a-x], [a-x...

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  7. Using properties of determinants, solve the following for x: |[x-2,...

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  8. 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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  9. If a , b , c are all positive and are p t h ,q th and r t h terms of a...

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  10. If x+y+z=0 prove that | xa yb zc yc za xb zb xc ya y|=x y z|a b c...

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  11. Prove that: |[b, c-a^2,c] ,[a-b^2,a b-c^2,c ],[a-b^2,a ,b-c^2b c-a^2a ...

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  12. 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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  13. Find a quadratic polynomial varphi(x) whose zeros are the maximum ...

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  14. f(x)=|(secx,cos x,sec^2x + cot x cosecx),(cos^2x,cos^2x,cosec^2 x),(1,...

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

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

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  17. If m is a positive integer and Dr=|(2r-1,\ ^m Cr,1),(m^2-1, 2^m ,m+1...

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  18. Without expanding evaluate the determinant "Delta"=|(1, 1, 1),(a, b,...

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

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  20. Prove that: |(-2a , a+b, a+c),( b+a,-2b, b+c),( c+a , c+b,-2c)|=4(a+b...

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