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If the roots of the equation b x^2+""c x...

If the roots of the equation `b x^2+""c x""+""a""=""0` be imaginary, then for all real values of x, the expression `3b^2x^2+""6b c x""+""2c^2` is (1) greater than 4ab (2) less than 4ab (3) greater than -`4a b` (4) less than -`4a b`

A

greater than 4ab.

B

less then 4ab

C

grreater than - 4ab.

D

less than - 4ab.

Text Solution

Verified by Experts

The correct Answer is:
3
` bx^(2) + cx + a = 0 `
Since the roots are imaginary,
` c^(2) - 4ab lt 0 rArr c^(2) lt 4ab rArr - c^(2) gt - 4ab`
Now consider ` 3b^(2) x^(2) + 6bcx + 2c^(2)`
Since ` 3b^(2) gt 0 ` , the given expression has minimum value
So, minimum value = `(4(3b^(2)) (2c^(2)) - 36 b^(2) c^(2))/(4(3b^(2)))`
` = - (12 b^(2) c^(2))/(12b^(2))`
` - c^(2) gt - 4 ab`
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