If α and β are the roots of 4x2-16x+t=0, ∀ t0 such that 1

Question

If α and β are the roots of 4x2-16x+t=0, ∀ t>0 such that 1 < α < 2 < β < 3, then the number of integral values of t are (A) 2 (B) 3 (C) 4 (D) 5

Tardigrade Admin · Accepted Answer

f(x)=4x2-16x+t Opening upward parabola Solution

f(1)>0, f(2) < 0, f(3)>0 f(1)>0⇒ 4-16+t>0⇒ t>12 f(2) < 0⇒ 16-32+t < 0⇒ t < 16 f(3)>0⇒ 36-48+t>0⇒ t>12 ⇒ 12 < t < 16⇒ t=13, 14, 15 Hence, the number of integral values (t = 13,14,15) is 3.

Q. If $α$ and $β$ are the roots of $4 x^{2} - 16 x + t = 0, \forall t > 0$ such that $1 < α < 2 < β < 3,$ then the number of integral values of $t$ are

$2$

$3$

$4$

$5$

Q. If α and β are the roots of 4x2−16x+t=0,∀t>0 such that 1<α<2<β<3, then the number of integral values of t are

2

3

4

5

f(x)=4x2−16x+t Opening upward parabola f(1)>0,f(2)<0,f(3)>0 f(1)>0⇒4−16+t>0⇒t>12 f(2)<0⇒16−32+t<0⇒t<16 f(3)>0⇒36−48+t>0⇒t>12 ⇒12<t<16⇒t=13,14,15 Hence, the number of integral values (t=13,14,15) is 3.

Q. If $α$ and $β$ are the roots of $4 x^{2} - 16 x + t = 0, \forall t > 0$ such that $1 < α < 2 < β < 3,$ then the number of integral values of $t$ are

$2$

$3$

$4$

$5$