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If `a ,\ b ,\ c` are the roots of the equation `x^3+p x+q=0` , then find the value of the determinant `|1+a1 1 1 1+b1 1 1 1+c|` .

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RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. 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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  2. 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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  3. If a ,\ b ,\ c are the roots of the equation x^3+p x+q=0 , then fin...

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  4. 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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  5. 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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  6. Without expanding the determinant, show that (a+b+c) is a factor of...

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  7. If a ,\ b ,\ c are roots of the equation x^3+p x^{2}+q=0 , prove tha...

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  8. If a, b, c are positive and unequal, show that value of the determinan...

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  9. If a+b+c!=0 and |a b c b c a c a b|=0 , then prove that a=b=c dot

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  10. 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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  11. 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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  12. Using properties of determinants. Prove that |3a-a+b-a+c-b+a3b-b+c-c+a...

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  13. Using properties of determinants, solve for x:|a+x a-x a-x a-x a+x a...

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

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  15. 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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  16. 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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  17. If x+y+z=0 prove that |a x b y c z c y a z b x b z c x a y|=x y z|a b ...

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  18. Prove that: |b c-a^2c a-b^2a b-c^2c a-b^2a b-c^2b c-a^2a b-c^2b c-a^2c...

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

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  20. Let "Delta"r=|r x(n(n+1))/2 2r-1y n^2 3r-2z(n(3n-1))/2| . Show that su...

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