JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Quadratic Equations and Inequalities

The sum of all the roots of the equation $(\mathrm{x}-1{)}^2-5|\mathrm{x}-1|+6=0$, is:

Let $\alpha$ and $\beta$ be the roots of equation $x^2+2ax+(3a+10)=0$ such that $\alpha <1<\beta$. Then the set of all possible values of $a$ is :

Let $\alpha ,\beta$ be the roots of the quadratic equation $12{\mathrm{x}}^2-20\mathrm{x}+3\lambda =0,\lambda \in \mathbb{Z}$. If $\frac{1}{2}\le |\beta -\alpha |\le \frac{3}{2}$, then the sum of all possible values of $\lambda$ is :

If $\alpha$ and $\beta$ ($\alpha <\beta$) are the roots of the equation $(-2+\sqrt{3})(|\sqrt{x}-3|)+(x-6\sqrt{x})+(9-2\sqrt{3})=0$, $x\ge 0$, then $\sqrt{\frac{\beta }{\alpha }}+\sqrt{\alpha \beta }$ is equal to :

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in:

The smallest positive integral value of $a$, for which all the roots of ${\mathrm{x}}^4-{\mathrm{ax}}^2+9=0$ are real and distinct, is equal to