Answer
Verified
394.8k+ views
Hint: We solve this question by going through the nature of roots of a quadratic equation using discriminant $\Delta ={{b}^{2}}-4ac$ and find the range when the discriminant is greater than zero.
Then we take the sum of roots using formula $\alpha +\beta =\dfrac{-b}{a}$ and find another range of m. Then we plot the rough graph from the given information and find the range of m when x=1 and x=5 by substituting those values in the equation and by inequality that their values are greater than zero.
Complete step-by-step solution:
First, let us start by going through the concept of the nature of roots before starting to solve the question.
For any quadratic equation $a{{x}^{2}}+bx+c=0$ having roots $\alpha $ and $\beta $,
Sum of the roots is $\alpha +\beta =\dfrac{-b}{a}$ and product of the roots is $\alpha \beta =\dfrac{c}{a}$.
The nature of the roots can be said by the discriminant of the quadratic equation $a{{x}^{2}}+bx+c=0$, that is $\Delta ={{b}^{2}}-4ac$.
If the roots are real and distinct, then the discriminant is greater than zero, which is ${{b}^{2}}-4ac>0$.
If the roots are equal, then the discriminant is equal to zero, which is ${{b}^{2}}-4ac=0$.
If the roots are imaginary, then the discriminant is less than zero, which is ${{b}^{2}}-4ac<0$.
We were given that roots of the equation ${{x}^{2}}-mx+4=0$ are distinct.
So, from above as the roots are real and distinct, its discriminant is greater than zero.
$\begin{align}
& \Rightarrow \Delta ={{\left( -m \right)}^{2}}-4\left( 1 \right)\left( 4 \right)>0 \\
& \Rightarrow {{m}^{2}}-16>0 \\
& \Rightarrow \left( m-4 \right)\left( m+4 \right)>0 \\
& \Rightarrow m\in \left( -\infty ,-4 \right)\cup \left( 4,\infty \right)............\left( 1 \right) \\
\end{align}$
So, we get that the range of m as above.
Let the roots of given equation be $\alpha $ and $\beta $.
So, by using the formula for sum of the roots of a quadratic equation, we have
$\begin{align}
& \Rightarrow \alpha +\beta =-\left( -m \right) \\
& \Rightarrow \alpha +\beta =m \\
\end{align}$.
We were also given that the roots of the given quadratic equation lie between 1 and 5, that is
$\begin{align}
& \Rightarrow 1<\alpha <5 \\
& \Rightarrow 1<\beta < 5 \\
\end{align}$
Adding the above two inequalities we get,
$\begin{align}
& \Rightarrow 1+1< \alpha +\beta < 5+5 \\
& \Rightarrow 2< \alpha +\beta < 10 \\
& \Rightarrow 2< m <10.............\left( 2 \right) \\
\end{align}$
Now let us plot the graph of the quadratic equation using the given information.
As we see in the graph, when x=1 and when x=5 the value of the equation is positive. So,
When x=1,
$\begin{align}
& \Rightarrow {{\left( 1 \right)}^{2}}-m\left( 1 \right)+4>0 \\
& \Rightarrow 1-m+4 > 0 \\
& \Rightarrow 5-m >0 \\
& \Rightarrow m < 5..............\left( 3 \right) \\
\end{align}$
When x=5,
$\begin{align}
& \Rightarrow {{\left( 5 \right)}^{2}}-m\left( 5 \right)+4>0 \\
& \Rightarrow 25-5m+4 >0 \\
& \Rightarrow 29-5m> 0 \\
& \Rightarrow 5m< 29 \\
& \Rightarrow m< \dfrac{29}{5}..............\left( 4 \right) \\
\end{align}$
From equations (1), (2), (3) and (4) we can find the region common to all of them. So the region common to $m\in \left( -\infty ,-4 \right)\cup \left( 4,\infty \right)$, $2< m< 10$, $m< 5$, $m< \dfrac{29}{5}$ is $m\in \left( 4,5 \right)$.
So, m lies in the interval $\left( 4,5 \right)$.
Hence, the answer is Option A.
ote: The major mistake that one does in this question is while finding the intervals they forget to check the condition that the equation is positive when x=1 and when x=5. In that case they have only two equations of intervals $m\in \left( -\infty ,-4 \right)\cup \left( 4,\infty \right)$ and $2< m< 10$, and the region common to them is $\left( 4,10 \right)$. But it is wrong. So, one should consider all the possible ways of finding the interval.
Then we take the sum of roots using formula $\alpha +\beta =\dfrac{-b}{a}$ and find another range of m. Then we plot the rough graph from the given information and find the range of m when x=1 and x=5 by substituting those values in the equation and by inequality that their values are greater than zero.
Complete step-by-step solution:
First, let us start by going through the concept of the nature of roots before starting to solve the question.
For any quadratic equation $a{{x}^{2}}+bx+c=0$ having roots $\alpha $ and $\beta $,
Sum of the roots is $\alpha +\beta =\dfrac{-b}{a}$ and product of the roots is $\alpha \beta =\dfrac{c}{a}$.
The nature of the roots can be said by the discriminant of the quadratic equation $a{{x}^{2}}+bx+c=0$, that is $\Delta ={{b}^{2}}-4ac$.
If the roots are real and distinct, then the discriminant is greater than zero, which is ${{b}^{2}}-4ac>0$.
If the roots are equal, then the discriminant is equal to zero, which is ${{b}^{2}}-4ac=0$.
If the roots are imaginary, then the discriminant is less than zero, which is ${{b}^{2}}-4ac<0$.
We were given that roots of the equation ${{x}^{2}}-mx+4=0$ are distinct.
So, from above as the roots are real and distinct, its discriminant is greater than zero.
$\begin{align}
& \Rightarrow \Delta ={{\left( -m \right)}^{2}}-4\left( 1 \right)\left( 4 \right)>0 \\
& \Rightarrow {{m}^{2}}-16>0 \\
& \Rightarrow \left( m-4 \right)\left( m+4 \right)>0 \\
& \Rightarrow m\in \left( -\infty ,-4 \right)\cup \left( 4,\infty \right)............\left( 1 \right) \\
\end{align}$
So, we get that the range of m as above.
Let the roots of given equation be $\alpha $ and $\beta $.
So, by using the formula for sum of the roots of a quadratic equation, we have
$\begin{align}
& \Rightarrow \alpha +\beta =-\left( -m \right) \\
& \Rightarrow \alpha +\beta =m \\
\end{align}$.
We were also given that the roots of the given quadratic equation lie between 1 and 5, that is
$\begin{align}
& \Rightarrow 1<\alpha <5 \\
& \Rightarrow 1<\beta < 5 \\
\end{align}$
Adding the above two inequalities we get,
$\begin{align}
& \Rightarrow 1+1< \alpha +\beta < 5+5 \\
& \Rightarrow 2< \alpha +\beta < 10 \\
& \Rightarrow 2< m <10.............\left( 2 \right) \\
\end{align}$
Now let us plot the graph of the quadratic equation using the given information.
As we see in the graph, when x=1 and when x=5 the value of the equation is positive. So,
When x=1,
$\begin{align}
& \Rightarrow {{\left( 1 \right)}^{2}}-m\left( 1 \right)+4>0 \\
& \Rightarrow 1-m+4 > 0 \\
& \Rightarrow 5-m >0 \\
& \Rightarrow m < 5..............\left( 3 \right) \\
\end{align}$
When x=5,
$\begin{align}
& \Rightarrow {{\left( 5 \right)}^{2}}-m\left( 5 \right)+4>0 \\
& \Rightarrow 25-5m+4 >0 \\
& \Rightarrow 29-5m> 0 \\
& \Rightarrow 5m< 29 \\
& \Rightarrow m< \dfrac{29}{5}..............\left( 4 \right) \\
\end{align}$
From equations (1), (2), (3) and (4) we can find the region common to all of them. So the region common to $m\in \left( -\infty ,-4 \right)\cup \left( 4,\infty \right)$, $2< m< 10$, $m< 5$, $m< \dfrac{29}{5}$ is $m\in \left( 4,5 \right)$.
So, m lies in the interval $\left( 4,5 \right)$.
Hence, the answer is Option A.
ote: The major mistake that one does in this question is while finding the intervals they forget to check the condition that the equation is positive when x=1 and when x=5. In that case they have only two equations of intervals $m\in \left( -\infty ,-4 \right)\cup \left( 4,\infty \right)$ and $2< m< 10$, and the region common to them is $\left( 4,10 \right)$. But it is wrong. So, one should consider all the possible ways of finding the interval.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Two charges are placed at a certain distance apart class 12 physics CBSE
Difference Between Plant Cell and Animal Cell
What organs are located on the left side of your body class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
The planet nearest to earth is A Mercury B Venus C class 6 social science CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is BLO What is the full form of BLO class 8 social science CBSE