Question
The values of a for which( a2−1)x2 + 2(a-1)x + 2 is positive for any x are
Answer: Option D
:
D
We know that the expression ax2 + bx + c > 0 for all x, if a > 0 and b2 < 4ac
∴ (a2−1)x2 + 2(a-1)x + 2 is positive for allx, ifa2−1 > 0 and 4(a−1)2−8(a2−1) < 0
⇒ a2−1 > 0 and -4(a - 1)(a + 3) < 0
⇒ a2−1 > 0 and (a - 1)(a + 3) >0
⇒ a2 > 1 and a < -3 or a > 1
⇒ a < -3 or a > 1
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:
D
We know that the expression ax2 + bx + c > 0 for all x, if a > 0 and b2 < 4ac
∴ (a2−1)x2 + 2(a-1)x + 2 is positive for allx, ifa2−1 > 0 and 4(a−1)2−8(a2−1) < 0
⇒ a2−1 > 0 and -4(a - 1)(a + 3) < 0
⇒ a2−1 > 0 and (a - 1)(a + 3) >0
⇒ a2 > 1 and a < -3 or a > 1
⇒ a < -3 or a > 1
Was this answer helpful ?
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