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Q.

If a>b>c and the system of equations ax+by+cz=0, bx+cy+az=0 ,cx+ay+bz=0 has a non trivial solution, then both the roots of the quadratic equation at2+bt+c=0 are 

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a

complex

b

real

c

of opposite sign

d

positive

answer is A, B.

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Detailed Solution

a>b>c and given equations are
ax+by+cz=0    bx+cy+az=0cx+ay+bz=0    
For non trivial solution
Δ=abcbcacab=03abca3+b3+c3=0 a+b+c=0
If α,β be the roots of at2+bt+c=0
α+β=ba and αβ=cab24ac=(ac)24ac=(ac)2>0
For opposite sign  ||αβ∣>0
 (α+β)24αβ>0b2a24ca>0 b24ac>0(ac)24ac>0 (ac)2>0 true 
Hence, roots are real and have opposite sign.

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