Question

If both the roots of the quadratic equation `x^(2) mx+4=0` are real and distinct and they lie in the interval `[1,5]` then m lies in the interval
A. `(4,5)`
B. `(-5,-4)`
C. `(5,6)`
D. `(3,4)`

Answer 1

Correct Answer - A
According to given information, we have the following graph

Now, the folowing conditions should satisfy
`(i) Dgt0impliesb^(2)-4ac gt0`
`impliesm^(2)-4xx1xx4gt0`
`implies m^(2)-16gt0`
`implies(m-4)(m+4)gt0`
`impliesm in (-oo,-4)uu(4,oo)`
(ii) The value of the parabola should lie between `x=1and x=5`
`therefore-(b)/(2a)in(1,5)impliesltm/2lt5impliesm in (2,10)`
(iii) `f(I)gt0implies1-m+4 gt 0`
`impliesmlt5impliesm in (-oo,5)`
(iv) `f(5)gt0implies25-5m+4gt0implies5mlt29impliesm in(-oo,(29)/(5))`
From the values of m obtained in (i),(ii),(iii),and (iv), we get `m in (4,5).`