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If roots of the equation $2{x^{^2}} - 4x + 2\sin \theta - 1 = 0$ are of opposite sign$(where \theta \in (0,\pi )),$ then $\theta $ belongs to
(A)$(\dfrac{\pi }{6},\dfrac{{5\pi }}{6})$
(B)$(0,\dfrac{\pi }{6}) \cup (\dfrac{{5\pi }}{6},\pi )$
(C)$(0,\dfrac{{5\pi }}{6})$
(D)$(0,\pi )$

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Last updated date: 26th Feb 2024
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IVSAT 2024
Answer
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Hint: Determinant of this quadratic equation should be greater than 0,roots of opposite sign mean if one is positive then second should be negative, so the product of both is negative.

Complete step-by-step answer:
$2{x^2} - 4x + 2\sin \theta - 1 = 0$
We are solving this question by using general equation, that is $a{x^2} + bx + c = 0$ if we suppose it`s root is $\alpha ,\beta $ and it`s of opposite sign that`s mean $\alpha .\beta < 0$. In these type of question the value of $\alpha ,\beta $ low root get value of $\dfrac{c}{a} < 0$
comparing $2{x^2} - 4x + 2\sin \theta - 1 = 0$ with general $a{x^2} + bx + c = 0$
a = 2 b = -4 c = $2\sin \theta $

In this equation both roots are of opposite sing so we apply $\dfrac{c}{a} < 0$ (same equation) $\dfrac{{2\sin \theta - 1}}{2} < 0$
$2\sin \theta - 1 < 0$
$\operatorname{Sin} \theta < \dfrac{1}{2}$
$D > 0$ [ where D = ${b^2} - 4ac$ ]
Value of D will be ${4^2} - 2.(2\sin \theta - 1) > 0$
= $16 - 4.2.(2\sin \theta - 1) > 0$
= $16 - 8(2\sin \theta - 1) > 0$
Now open the brackets and multiply by 8
=\[16 - 16\sin \theta + 8 > 0\]
Add the numbers which are 16+8
=$24 - 16\sin \theta > 0$
compare the equation
= $24 > 16\sin \theta $
= $\operatorname{Sin} \theta < \dfrac{{24}}{{16}}$
This quantity is greater than 1. This is true for all values of $\theta $ because the value of $\sin \theta $ is less than 1.
$\operatorname{Sin} \theta $ should be less than $\dfrac{3}{2} = 1.5$
$\operatorname{Sin} \theta > 0.5$
$\theta $=$(0,\dfrac{\pi }{6}),(\dfrac{{5\pi }}{6},\pi )$
So, the correct answer is “Option B”.

Note: Root of opposite sign means if one is positive then second should be negative, so the product of both must be negative. We can solve this question by solving other options. For roots of a given quadratic to be the opposite sign, the product of roots is negative. Sign of roots of a quadratic equation.
*Both roots are positive (If a and b are opposite in sign and a and c are same in sign)
*Both roots are negative (If a,b,c are all of same sign)
*Roots are of opposite sign (If a and c are of opposite sign)
*roots equal but opposite in sing (If b=0)