If the sum of the squares of the roots of the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let the roots of the equation $x^2 - (\sin \alpha - 2)x - (1 + \sin \alpha) = 0$ be $x_1$ and $x_2$.From the properties of quadratic equations,we

Vedclass · Accepted Answer

$\frac{\pi}{2}$

Vedclass · Answer

(D) Let the roots of the equation $x^2 - (\sin \alpha - 2)x - (1 + \sin \alpha) = 0$ be $x_1$ and $x_2$.From the properties of quadratic equations,we have:$x_1 + x_2 = \sin \alpha - 2$$x_1 x_2 = -(1 + \sin \alpha)$We want to minimize $S = x_1^2 + x_2^2$.$S = (x_1 + x_2)^2 - 2x_1 x_2$$S = (\sin \alpha - 2)^2 - 2(-(1 + \sin \alpha))$$S = \sin^2 \alpha - 4 \sin \alpha + 4 + 2 + 2 \sin \alpha$$S = \sin^2 \alpha - 2 \sin \alpha + 6$To minimize $S$,let $u = \sin \alpha$,where $u \in [-1, 1]$.$S(u) = u^2 - 2u + 6 = (u - 1)^2 + 5$.The minimum value occurs at $u = 1$.Since $\sin \alpha = 1$,we have $\alpha = \frac{\pi}{2}$.

If the sum of the squares of the roots of the equation $x^2 - (\sin \alpha - 2)x - (1 + \sin \alpha) = 0$ is least,then $\alpha$ is

Similar Questions

If $\alpha$ and $\beta$ are the roots of the equation $2x^2-4x+3=0$,then $\frac{2(\alpha^4+\beta^4)+3(\alpha^2+\beta^2)}{\alpha+\beta} = $

If the roots of the given equation $(2k + 1)x^2 - (7k + 3)x + k + 2 = 0$ are reciprocal to each other,then the value of $k$ will be

If one root of the quadratic equation $x^2 + px + q = 0$ is the square of the other,then:

If $1, 2, 3$ and $4$ are the roots of the equation $x^4+ax^3+bx^2+cx+d=0$,then $a+2b+c$ is equal to

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 2(m^2 + 1)x + m^4 + m^2 + 1 = 0$,then find the value of $\alpha^2 + \beta^2$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module