Question

All the values of m for which both roots of the equation \[{x^2} - 2mx + {m^2} - 1 = 0\] are greater than -2 but less than 4, lie in the interval
1) \[ - 2 < m < 0\]
2) \[m > 3\]
3) \[ - 1 < m < 3\]
4) \[1 < m < 4\]

Answer 1

Hint: This question is based on the basic concepts of quadratic equations and involves elementary concepts like finding the roots of the equation. Once you find out the roots of the equation just restrict them to the given condition and find the common set of the solution and you will land up with the correct answer.

Complete step-by-step answer:
 Let’s begin solving this question by finding out the roots of the equation,
We have,
\[ \Rightarrow {x^2} - 2mx + {m^2} - 1 = 0\]
Clearly, we have, as a perfect square,
\[ \Rightarrow {(x - m)^2} - 1 = 0\]
Now, adding one on both the sides of the equation,
\[ \Rightarrow {(x - m)^2} = 1\]
And raising the power by ½ on both the sides we get,
\[ \Rightarrow (x - m) = \pm 1\], (as 1 can be positive or negative after solving the root)
Therefore, the roots of the equation are
\[ \Rightarrow {x_1} = 1 + m,\]
\[ \Rightarrow {x_2} = - 1 + m\]
Now, the condition given is
\[ \Rightarrow - 2 < {x_0} < 4\]
Now, applying to the first root of the equation we get,
\[ \Rightarrow - 2 < {x_1} < 4\]
\[ \Rightarrow - 2 < 1 + m < 4\]
\[ \Rightarrow - 3 < m < 3 - - - - (i)\]
Similarly, applying the condition to the second root we get,
\[ \Rightarrow - 2 < {x_2} < 4\]
\[ \Rightarrow - 2 < - 1 + m < 4\]
\[ \Rightarrow - 1 < m < 3 - - - - (ii)\]
Now, let’s compare the equations \[(i)\]and \[(ii)\]
\[ \Rightarrow - 3 < m < 3 - - - - (i)\]
\[ \Rightarrow - 1 < m < 3 - - - - (ii)\]
Now, After taking a common set, we get the value of m as,
\[ \Rightarrow - 1 < m < 3\]
Thus, option(3) is the correct answer.
So, the correct answer is “Option 3”.

Note: This question involves the use of quadratic equation’s properties. One should be well versed with them before attempting this question. It also requires one to represent the range of a variable on a number line and find the common section. Do not commit calculation mistakes, and be sure of the final answer.