Use of Monotonicity for Proving Inequalities

Q. Which of the following statements is/are correct in stating the number of solutions of the equation
$x\log x+x=\lambda ,$ where $\lambda \in R.$

1. If $\lambda =0$, then the equation has exactly two solutions in $x$.
2. If $\lambda \in (0,\frac{1}{e^2})$, then the equation has exactly one solution in $x$.
3. If $\lambda \in (\frac{-1}{e^2},0)$, then equation has at least one solution in $x$.
4. the equation has at least one solution in $[1,\lambda ]$

Q. Let $A=\{a,b,c,d,e\}$ and $B=\{a,c,g,f\}.$ Find $A-B$ and $B-A.$

1. $\{b,d,e\}$; $\{g,f\}$
2. $\{b,d\}$; $\left\{f\right\}$
3. $\left\{\varphi \right\}$; $\{g,f\}$
4. $\{a,b,c,d,e\}$; $\{g,f\}$

Q.

x $\ge$ sin(x) for $\forall$ x $\ge$ 0

1. True

2. False

Q. If $f\left(x\right)=x+\sin x;g\left(x\right)=e^{-x};u=\sqrt{c+1}-\sqrt{c};v=\sqrt{c}-\sqrt{c-1};(c>1),$ then

1. $f\circ g\left(u\right)<f\circ g\left(v\right)$
2. $g\circ f\left(u\right)<g\circ f\left(v\right)$
3. $g\circ f\left(u\right)>g\circ f\left(v\right)$
4. $f\circ g\left(u\right)>f\circ g\left(v\right)$