If - - are the roots of the quadratic equation 4x2 - 16x - k - 0- k- R such that 1 - - - 2 and 2 - - - 3- then the number of integral values of k is

The correct option is C 3 Given, f(x)=4x^2 16x+k=0, k∈ R and 1 lt;α lt;2 and nbsp; nbsp; 2 lt;β lt;3 As a=4 gt;0 ,(i) ...

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Standard XII

Mathematics

Sum of n Terms

If α, β are...

Question

If $α, β$ are the roots of the quadratic equation $4 x^{2} - 16 x + k = 0, k \in R$ such that $1 < α < 2$ and $2 < β < 3$ , then the number of integral values of $k$ is

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Solution

The correct option is C $3$
Given,
$f (x) = 4 x^{2} - 16 x + k = 0, k \in R$
and $1 < α < 2$ and $2 < β < 3$
As $a = 4 > 0$ ,
(i) $f (1) > 0$
$\Rightarrow 4 - 16 + k > 0$
$k > 12$
(ii) $f (3) > 0$
$\Rightarrow 36 - 48 + k > 0$
$k > 12$
(iii) Discriminant, $D > 0$
$\Rightarrow 256 - 16 k > 0$
$k < 16$
(iv) $f (2) < 0$
$16 - 32 + k < 0$
$\Rightarrow k < 16$
The integral value of $k = 13, 14, 15$ .
The no. of integral values are $3$ .
The correct answer is option C.

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If α,β are the roots of the quadratic equation 4x2−16x+k=0,k∈R such that 1<α<2 and 2<β<3, then the number of integral values of k is

If $α, β$ are the roots of the quadratic equation $4 x^{2} - 16 x + k = 0, k \in R$ such that $1 < α < 2$ and $2 < β < 3$ , then the number of integral values of $k$ is