Statement -1 If `x,y,z` be real variables satisfying `x+y+z=6` and `xy+yz+z=8`, the range of variables x,y and z are identical.
Statement -2 `x+y+z=6` and `xy+yz+zx=8` remains same if `x,y,z` interchange their positions.

Options

A. Statement -1 is true, Statement -2 is true, Statement -2 is a correct explanation for Statement-1
B. Statement -1 is true, Statement -2 is true, Statement -2 is not a correct explanation for Statement -1
C. Statement -1 is true, Statement -2 is false
D. Statement -1 is false, Statement -2 is true

Answer

A

Explanation

`:'x,y,z epsilonR`
`x+y+z=6`…….i
and `xy+yz+zx=8` ………ii
`impliesxy+(x+y){6-(x+y)}=8` [fromEq. (i)]
`impliesxy+6x+6y-(x^(2)-2xy+y^(2))=8`
or `y^(2)+(x-6)y+x^(2)-6x+8=0`
`:.(x-6)^(2)-4.1.(x^(2)-6x+8)ge0, AA y epsilonR`
`implies-3x^(2)+12x+4ge0` or `3x^(2)-12x-14le0`
or `2-4/(sqrt(3))lexle2+4/(sqrt(3))`
or `x epsilon [2-4/(sqrt(3)),2+4/(sqrt(3))]`
Similarly `y epsilon [2-4/(sqrt(3)),2+4/(sqrt(3))]`
and `z epsilon [2-4/(sqrt(3)),2+4/(sqrt(3))]`
Since Eqs i and ii remains same if `x,y,z` interchange their positions.
Hence both statements are true and Statement 2 is a correct explanation of Statement 1.