JEE Main Quadratic Equation and Inequalities Practice Questions With Solutions

Let $\alpha ,\beta$\alpha , \beta be the roots of the equation $x^2-\sqrt{2}x+2=0$x^{2} - \sqrt{2} x + 2 = 0. Then ${\alpha }^{14}+{\beta }^{14}$\alpha^{14} + \beta^{14} is equal to

The set of all $a\in \mathbb{R}$a \in \mathbb{R} for which the equation $x|x-1|+|x+2|+a=0$x \left|\right. x - 1 \left|\right. + \left|\right. x + 2 \left|\right. + a = 0 has exactly one real root, is :

Let $\alpha ,\beta$\alpha , \beta be the roots of the quadratic equation $x^2+\sqrt{6}x+3=0$x^{2} + \sqrt{6} x + 3 = 0. Then $\frac{{\alpha }^{23}+{\beta }^{23}+{\alpha }^{14}+{\beta }^{14}}{{\alpha }^{15}+{\beta }^{15}+{\alpha }^{10}+{\beta }^{10}}$\frac{\alpha^{23} + \beta^{23} + \alpha^{14} + \beta^{14}}{\alpha^{15} + \beta^{15} + \alpha^{10} + \beta^{10}} is equal to :

Let $\alpha ,\beta ,\gamma$\alpha , \beta , \gamma be the three roots of the equation $x^3+bx+c=0$x^{3} + b x + c = 0. If $\beta \gamma =1=-\alpha$\beta \gamma = 1 = - \alpha, then $b^3+2c^3-3{\alpha }^3-6{\beta }^3-8{\gamma }^3$b^{3} + 2 c^{3} - 3 \alpha^{3} - 6 \beta^{3} - 8 \gamma^{3} is equal to :

Let $A=\{x\in R:[x+3]+[x+4]\le 3\},$A = \left{\right. x \in R : \left[\right. x + 3 \left]\right. + \left[\right. x + 4 \left]\right. \leq 3 \left.\right} ,

$B=\left\{x\in R:3^x{\left(\sum _{r=1}^{\mathrm{\infty }}\frac{3}{{10}^r}\right)}^{x-3}<3^{-3x}\right\},$B = \left{\right. x \in R : 3^{x} \left(\right. \sum_{r = 1}^{\infty} \frac{3}{10^{r}} \left.\right)^{x - 3} < 3^{- 3 x} \left.\right} , where [t] denotes greatest integer function. Then,

The sum of all the roots of the equation $|x^2-8x+15|-2x+7=0$\left|\right. x^{2} - 8 x + 15 \left|\right. - 2 x + 7 = 0 is :

The number of integral values of k, for which one root of the equation $2x^2-8x+k=0$2 x^{2} - 8 x + k = 0 lies in the interval (1, 2) and its other root lies in the interval (2, 3), is :

Let $S=\left\{x:x\in \mathbb{R}\mathrm{and}{(\sqrt{3}+\sqrt{2})}^{x^2-4}+{(\sqrt{3}-\sqrt{2})}^{x^2-4}=10\right\}$S = \left{\right. x : x \in \mathbb{R} and \left(\left(\right. \sqrt{3} + \sqrt{2} \left.\right)\right)^{x^{2} - 4} + \left(\left(\right. \sqrt{3} - \sqrt{2} \left.\right)\right)^{x^{2} - 4} = 10 \left.\right}. Then $n(S)$n \left(\right. S \left.\right) is equal to

The number of real roots of the equation $\sqrt{x^2-4x+3}+\sqrt{x^2-9}=\sqrt{4x^2-14x+6}$\sqrt{x^{2} - 4 x + 3} + \sqrt{x^{2} - 9} = \sqrt{4 x^{2} - 14 x + 6}, is :