If a-0 and the equation a x2-b x-c-0 has two real roots - and - such that - - 1- - 1- thenA- a-b-c - 0B- a-b-c - 0C- a-b-c - 0D- a-c - 0

The correct options are A a+b+c ≥ 0 B a b+c ≥ 0 C a+ | b |+c ≥ 0 D a c ≥ 0 Given y=ax^2+bx+c, a gt;0 As both the roots lie in the interval [ 1,1], ...

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Standard XI

Mathematics

Inequalities Involving Modulus Function

If a>0 and th...

Question

If a>0 and the equation $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ such that $| α | \leq 1, | β | \leq 1,$ then

$a + b + c \geq 0$

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$a - b + c \geq 0$

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$a + | b | + c \geq 0$

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$a - c \geq 0$

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Solution

The correct options are
A
$a + b + c \geq 0$

B
$a - b + c \geq 0$

C
$a + | b | + c \geq 0$

D
$a - c \geq 0$

Given $y = a x^{2} + b x + c, a > 0$
As both the roots lie in the interval [-1,1], we have
$y (- 1) \geq 0, y (1) \geq 0 \Rightarrow a - b + c \geq 0, a + b + c \geq 0 \Rightarrow a + | b | + c \geq 0 A l s o, α β \leq | α β | = | α | | β | \leq 1 \Rightarrow \frac{c}{a} \leq 1 o r c \leq a o r a - c \geq 0$

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If a>0 and the equation ax2+bx+c=0 has two real roots α and β such that |α|≤1, |β|≤1, then

The correct options are A a+b+c≥0 B a−b+c≥0 C a+|b|+c≥0 D a−c≥0 Given y=ax2+bx+c, a>0 As both the roots lie in the interval [-1,1], we have y(−1)≥0,y(1)≥0⇒a−b+c≥0,a+b+c≥0⇒a+|b|+c≥0Also, αβ≤|αβ|=|α||β|≤1⇒ca≤1 or c≤ a or a−c≥0

If a>0 and the equation $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ such that $| α | \leq 1, | β | \leq 1,$ then