Q.

Let S be the set of all column matrices  b1b2b3 such that  b1,b2,b3  and the system of equations (in real variables)   x + 2y + 5z = b1,  2x  4y + 3z  =  b2,x2y+2z=b3 has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each  b1b2b3S

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a

x+2y+3z = b1,4y+5z = b2 and  x+2y+6z = b3

b

x + 2y + 5z = b1,  2x + 3z = b2  and x + 4y - 5z = b3

c

x+2y-5z = b1, 2x4y+10z = b2 and x2y+5z = b3

d

x+y+3z = b1, 5x + 2y + 6z = b2 and 2 xy3z = b3

answer is A, D.

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Detailed Solution

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Δ=|125243122|=0 given equation represent family of lines
2x4y+3zb2+λ(x+2y+5zb1)=0  is same as  x2y+2zb3=0 for same λ
 2λ1=2λ42=5λ+32=(b1λ+b2)b3        λ=17and13b3=b1+7b2
 A|123045126|0
B|113526213|0=0 as planes are non parallel is must represent family of planes for solution to exit
 (5μ+1)x+(2μ+1)y+(6μ+3)z=b1+μb2
Must be  2x+b+3z+b3=0 which is not true always
Cx2y+5z=b1,x2y+5z=b22,x2y+5z=b3 as these are parallel for specific case 
when b1=b22=b3
D|125203145|0

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