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Let `a, b, c, d` be real numbers in `G.P.` If `u, v, w` satisfy the system of equations `u + 2y +3w = 6,4u + 5v + 6w =12 and 6u + 9v = 4` then show that the roots of the equation `(1/u+1/v+/w)x^2+[(b-c)^2+(c-a)^2+(d-b)^2]x+u+v+w=0` and 20x^2+10(a-d)^2 x-9=0` are reciprocals of each other.

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ALLEN-DETERMINANTS-All Questions
  1. . For what values of lambda and mu the system of equations x+y+z=6, x+...

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  2. Find those values of c for which the equations: 2x+3y=3,(c+ 2)x + (c +...

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  3. Let a, b, c, d be real numbers in G.P. If u, v, w satisfy the system...

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  4. If a^2+b^2+c^2=1 then show that value of the determinant |a^2+(b^2+c^2...

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  5. If Sr = alpha^r + beta^r + gamma^r then show that |[S0,S1,S2],[S1,S2,S...

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  6. Solve the system of equations : z+ay+a^2x+a^3=0 z+by+b^2x+b^3=0 z+cy+c...

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  7. If x ,\ y , z are not all zero and if a x+b y+c z=0; b x+c y+a z=0; c ...

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  8. If a^2+b^2+c^2=-2\ a b n df(x)=|1+a^xx(1+b^2)x(1+c^2)x(1+a^2)x1+b^2x(1...

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  9. The value of |alpha| for which the system of equation alphax+y+z=alpha...

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  10. If D=|1 1 1 1 1+x1 1 1 1+y|"f o r"x!=0, y!=0 then D is (1) divisibl...

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  11. Let a, b, c be any real numbers. Suppose that there are real number...

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  12. Let a, b, c be such that b(a""+""c)!=0 . If |a a+1a-1-bb+1b-1cc-1c...

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  13. Consider the system of linear equations; x-1\ +2x2+x3=3 2x1+3x2+3=3 3x...

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  14. The number of value of k for which the linear equations 4x+k y+2z=0k x...

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  15. If the trivial solution is the only solution of the system of equation...

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  16. The number of values of k, for which the system of equations (k""+"...

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  17. If a ,b ,c are sides of a scalene triangle, then value of |a b c b c a...

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  18. If the system of linear equations x1+2x2+3x3=6,\ x1+3x2+5x3=9,\ 2x2+5x...

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  19. Statement 1: The system of linear equations x+(sinalpha)y+(cosalpha)z=...

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  20. Consider the system of equations x+a y=0,\ y+a z=0.\ Then the set of ...

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