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

Assume that $\alpha, \beta, \gamma$ are the roots of $2x^3+5x^2+5x+2=0$. For $h \in R$,if $\alpha+h, \beta+h, \gamma+h$ are roots of $a(h)x^3+b(h)x^2+c(h)x+d(h)=0$,then:

If $\alpha$ and $\beta$ are the roots of the equation ${x^2} - (1 + {n^2})x + \frac{1}{2}(1 + {n^2} + {n^4}) = 0$,then the value of ${\alpha ^2} + {\beta ^2}$ is

If the sum of the roots of the equation $\lambda x^2 + 2x + 3\lambda = 0$ is equal to their product,then $\lambda = $

If $x_1$ and $x_2$ are the real roots of the equation $x^2-kx+c=0$,then the distance between the points $A(x_1, 0)$ and $B(x_2, 0)$ is

If $\alpha, \beta, \gamma$ are the roots of $x^3+ax^2+bx+c=0$,then find the value of $\sum \frac{1}{\alpha}$,given that $\alpha, \beta, \gamma$ are non-zero.

Vedclass Test Series

Exam Paper Generator

Online Exam Module