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Let a, b, c be any real numbers. Supp...

Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that `x""=""c y""+""b z ,""y""=""a z""+""c x""a n d""z""=""b x""+""a y` . Then `a^2+""b^2+""c^2+""2a b c` is equal to (1) 2 (2) `""1` (3) 0 (4) 1

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Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x""=""c y""+""b z , ""y""=""a z""+""c x"", ""z""=""b x""+""a y . Then a^2+""b^2+""c^2+""2a b c is equal to (1) 2 (2) ""1 (3) 0 (4) 1

Let a,b,c, be any real number. Suppose that there are real numbers x,y,z not all zero such that x=cy+bz,y=az+cx and z=bx+ay. Then a^(2)+b^(2)+c^(2) +2abc is equal to

Let z_1, z_2, z_3 be three complex numbers and a ,b ,c be real numbers not all zero, such that a+b+c=0a n da z_1+b z_2+c z_3=0. Show that z_1, z_2,z_3 are collinear.

Let z_1, z_2, z_3 be three complex numbers and a ,b ,c be real numbers not all zero, such that a+b+c=0 and a z_1+b z_2+c z_3=0. Show that z_1, z_2,z_3 are collinear.

If a^x=b ,\ b^y=c\ a n d\ c^z=a , prove that x y z=1

Show that |a b c a+2x b+2y c+2z x y z|=0

If the system of equations x=c y+b z y=a z+c x z=b x+a y has a non-trivial solution, show that a^2+b^2+c^2+2a b c=1

If the system of equations x=c y+b z ,\ \ y=a z+c x ,\ \ z=b x+a y has a non-trivial solution show that a^2+b^2+c^2+2a b c=1

If x , y , z are not all zero such that a x+y+z=0 , x+b y+z=0 , x+y+c z=0 then prove that 1/(1-a)+1/(1-b)+1/(1-c)=1

If a , b , c are non-zero real numbers and if the system of equations (a-1)x=y=z (b-1)y=z+x (c-1)z=x+y has a non-trivial solution, then prove that a b+b c+c a=a b c

ALLEN-DETERMINANTS-All Questions
  1. The value of |alpha| for which the system of equation alphax+y+z=alpha...

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  2. If D=|(1, 1, 1), (1, 1+x, 1), (1, 1, 1+y)| for x!=0, y!=0 then D is (...

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

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

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

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

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  8. The number of values of k for which the system of equations (k+1)x+8y=...

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  9. If a ,b ,c are sides of a scalene triangle, then value of |[a, b, c] ,...

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

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  11. Assertion : The system of linear equations x+(sinalpha)y+(cosalpha)z=0...

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

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  13. if alpha, beta , ne 0 " and " f(n) =alpha^(n)+beta^(n) " and " |{:(...

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

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  15. The system of linear equations x+lambday -z=0 lambdax -y-z=0 ...

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  16. The number of distinct real roots of the equation |{:(cosx, sinx, sinx...

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  17. Test the consistency and solve them when consistent the following s...

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  18. Let a ,b ,c be real numbers with a^2+b^2+c^2=1. Show that the eq...

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  19. The number of values of k for which the system of the equations (k+1)x...

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  20. Consider the system of equations x-2y+3z=1;-x+y-2z=; x-3y+4z=1. State...

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