If - - are roots of 4x2 -16x-c - 0- c 0 such that 1 - 2 - 3- then the number of integral value of c is

As α lies in (1, 2) f(1) 0 ⇒ (c 12) 0 ⇒ c 12 f(2) 0 ⇒ (c 16) 0 ⇒ c 16 ∴ c ∈ (12, 16) ⋯(1) As β lies in (2, 3) f(2) 0 ⇒ (c 16) 0 ⇒ c 16 f(3) 0 ⇒ c 1 ...

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Standard XIII

Mathematics

Location of Roots

If α, β are r...

Question

If $α, β$ are roots of $4 x^{2} - 16 x + c = 0, c > 0$ such that $1 < α < 2 < β < 3$ , then the number of integral value of $c$ is

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Solution

As $α$ lies in $(1, 2)$
$f (1) > 0 \Rightarrow (c - 12) > 0 \Rightarrow c > 12 f (2) < 0 \Rightarrow (c - 16) < 0 \Rightarrow c < 16 ∴ c \in (12, 16) \dots (1)$

As $β$ lies in $(2, 3)$
$f (2) < 0 \Rightarrow (c - 16) < 0 \Rightarrow c < 16 f (3) > 0 \Rightarrow c - 12 > 0 \Rightarrow c > 12 ∴ c \in (12, 16) \dots (2)$

Therefore, from equation $(1)$ and $(2)$ ,
$c \in (12, 16)$
Hence, the number of integral value of $c$ is $3$ .

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If α,β are roots of 4x2−16x+c=0, c>0 such that 1<α<2<β<3, then the number of integral value of c is

As α lies in (1,2) f(1)>0⇒(c−12)>0⇒c>12f(2)<0⇒(c−16)<0⇒c<16∴c∈(12,16)⋯(1) As β lies in (2,3) f(2)<0⇒(c−16)<0⇒c<16f(3)>0⇒c−12>0⇒c>12∴c∈(12,16)⋯(2) Therefore, from equation (1) and (2), c∈(12,16) Hence, the number of integral value of c is 3.

If $α, β$ are roots of $4 x^{2} - 16 x + c = 0, c > 0$ such that $1 < α < 2 < β < 3$ , then the number of integral value of $c$ is