Let $\alpha$ and $\beta$ be the two disinct roots of the equation $x^3 + 3x^2 -1 = 0.$ The equation which has $(\alpha \beta )$ as its root is equal to
$x^3 -3x -1 =0$
$x^3 -3x^2 + 1 = 0$
$x^3 + x^2 -3x + 1 = 0$
$x^3 + x^2 + 3x -1 = 0$
If $72^x \cdot 48^y=6^{x y}$, where $x$ and $y$ are non-zero rational numbers, then $x+y$ equals
The sum of integral values of $a$ such that the equation $||x\ -2|\ -|3\ -x||\ =\ 2\ -a$ has a solution
Let $\alpha $ and $\beta $ be the roots of the quadratic equation ${x^2}\,\sin \,\theta - x\,\left( {\sin \,\theta \cos \,\,\theta + 1} \right) + \cos \,\theta = 0\,\left( {0 < \theta < {{45}^o}} \right)$ , and $\alpha < \beta $. Then $\sum\limits_{n = 0}^\infty {\left( {{\alpha ^n} + \frac{{{{\left( { - 1} \right)}^n}}}{{{\beta ^n}}}} \right)} $ is equal to
If $a,b,c$ are distinct real numbers and $a^3 + b^3 + c^3 = 3abc$ , then the equation $ax^2 + bx + c = 0$ has two roots, out of which one root is