Algebra

If $ \alpha \quad and \beta $ are the roots of the equation $ {ax}^{2}+bx+c=0 $, then $ \underset { x\rightarrow \cfrac { \pi }{ 2 } }{ lim } \cfrac { tan\left[ \left( \alpha +\beta \right) x \right] }{ sin\left[ \left( \alpha \beta \right) x \right] } $ is equal to :

Consider the expression $g(x) = \sin^{2}x - (b + 1)\sin x + 3(b - 2)$ where $b$ is a real parameter, then the equation $g(x) = 0$ have two distinct roots in $(0, \pi)$ then $b$ lie in the interval,

If $\alpha$ be the bigger root of the equation $f(x)=ax^{2}-(4-2a)x-8=0$ where $a\in \left(\dfrac{4}{3}, 2\right)$ and if the value of $\cot^{-1}(2\tan\alpha)-\tan^{-1}\left(\dfrac{\cot\alpha}{2}\right)=2L$ then the value of $\dfrac{6L}{\pi}$ is

If $abc=p$ and $A=\begin{bmatrix} a & b & c \ c & a & b \ b & c & a \end{bmatrix}$ such that $AA'+I,$ then $a,b,c$ are the roots of the equation

Let $\alpha, \beta$ be the roots of the equation $ax^2 + bx + c = 0$. Let $S_n = \alpha^n + \beta^n$ for $n \geq 1$ Evaluate the determinant $\begin{vmatrix}3 & 1 + S_1 & 1 + S_2\ 1 + S_1 & 1 + S_2 & 1 + S_3\ 1 + S_2 & 1 + S_3 & 1 + S_4\end{vmatrix}$

If $\alpha ,\beta $ are roots of the equation $25{x^2} - 5x - 2 = 0$ and ${A_n} = {\alpha ^n} + {\beta ^n}$, then $\mathop {\lim }\limits_{x \to \infty } \sum\limits_{r = 1}^n {{A_r} = } $

Let $\alpha$ & $\beta$ be the roots of the equation, $ax^{2}+bx+c=0$ where $1<\alpha<\beta$, then $\displaystyle\lim_{x\to m}\frac{|ax^{2}+bx+c|}{ax^{2}+bx+c}=1$, then which of the following is correct?

Equation $x^n-1=0, n > 1, n\epsilon N$, has roots $1, a_2, ....a_n$. The value of $\displaystyle \sum_{r=2}^{n}\frac {1}{2-a_r}$, is

Consider the quadratic equation $f(x)=ax^{2}+bx+c$. Where $a, b, c\in R$ and $a\neq 0$ such that $f(x)=f(2-x)$ for all real number $x$. The sum of the roots of $f(x)$ is :

The value of $k$ so that the sum of the cubes of the roots of the equation ${ x }^{ 2 }-kx+\left( 2k-3 \right) =0$ assumes the minimum value, is