QJEEM19Ma03cb: JEE Main 2019 Mathematics - Class 11 JEEM19Ma03cb Solution: If a, b, c be three distinct real numbers in G.P. and a + b + c …

a, b, c are in G.P.

So, b = ar

and c = ar²

given   a + b + c = xb

$\Rightarrow$   a + br + ar² = x(ar)

$\Rightarrow$   1 + r + r² = xr

$\Rightarrow$   x = 1 + r + ${1 \over r}$

let sum of r + ${1 \over r}$ = M

$\therefore$   r² + 1 = Mr

$\Rightarrow$   r² $-$ Mr + 1 = 0

this quadratic equation will have

real solution when discriminant is $\ge$ 0

$\therefore$   b² $-$ 4ac $\ge$ 0

M² $-$ 4.1.1 $\ge$ 0

$\Rightarrow$   M² $\ge$ 4

M $\ge$ 2 or M $\le$ $-$ 2

$\therefore$   M $\in$ ( $-$ $\propto$ , $-$ 2] $\cup$ [2, $\propto$ )

As   x = 1 + r + ${1 \over r}$

= 1 + M

$\therefore$   x $\in$ ( $-$ $\propto$ , $-$ 1] $\cup$ [3, $\propto$ )

$\therefore$   x can't be 0, 1, 2.