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If the system of linear equations x+a...

If the system of linear equations `x+a y+a z=0,\ x+b y+b z=0,\ x+c y+c z=0` has a non zero solution then System is always non trivial solutions System is consistent only when `a=b=c` If `a!=b!=c` then `x=0,\ y=t ,\ z=-t\ AAt in R` If `a=b=c\ t h e n\ y=t_1, z=t_2,\ x=-a(t_1+t_2)AAt_1, t_2 in R`

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ALLEN -DETERMINANTS-All Questions
  1. Prove that: {:|(1,a, a^2-bc), (1,b,b^2-ca),(1,c,c^2-ab)|:}=0

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  2. If u=a x^2+2b x y+c y^2,\ y^(prime)=a^('2)+2b^(prime)x y+c^(prime)y^2,...

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  3. If the system of linear equations x+a y+a z=0,\ x+b y+b z=0,\ x+c y...

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  4. If x,y and y are not all zero and if ax+by+cz=0,bx+cy+az=0 and cx...

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  5. Prove that the system of equations in x\ a n d\ y ; a x(h y)+g=0,\ h x...

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

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  7. The system of equations ax+y+z=a-1,x+ay+z=a-1andx+y+az=a-1 does not h...

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  8. If D=|[1, 1, 1], [1, 1+x, 1], [1, 1, 1+y]| for x!=0, y!=0 then D is (1...

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  9. Let a,b,c, be any real number. Suppose that there are real numbers x,y...

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  10. Let a,b,c be such that b(a+c) ne 0. If |{:(a,a+1,a-1),(-b,b+1,b-1)...

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  11. Consider the system of linear equations; x1 +2x2+x3=3 2x1+3x2+x3=3 3x1...

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

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

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

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  15. If a ,b ,c are sides of a scalene triangle, then value of {:|(a, b, c)...

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

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

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

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  19. If f(n)=alpha^n+beta^n then show that |{:(3,1+f(1),1+f(2)),(1+f(1),...

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  20. The set of all values of lamda for which the system of linear equation...

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