The equation${e^x} - x - 1 = 0$ has

Below are four equations in $x$. Assume that $0 < r < 4$. Which of the following equations has the largest solution for $x$ ?

If two roots of the equation ${x^3} - 3x + 2 = 0$ are same, then the roots will be

If $x+\frac{1}{x}=a, x^2+\frac{1}{x^3}=b$, then $x^3+\frac{1}{x^2}$ is

Let $\alpha$ and $\beta$ be two real numbers such that $\alpha+\beta=1$ and $\alpha \beta=-1 .$ Let $p _{ n }=(\alpha)^{ n }+(\beta)^{ n },p _{ n -1}=11$ and $p _{ n +1}=29$ for some integer $n \geq 1 .$ Then, the value of $p _{ n }^{2}$ is .... .

If ${\log _2}x + {\log _x}2 = \frac{{10}}{3} = {\log _2}y + {\log _y}2$ and $x \ne y,$ then $x + y = $