Q:

Suppose $P(|X| < 1) = 1$ and $P(|Y| = 2) = 1$.

Suppose $P(|X| < 1) = 1$ and $P(|Y| = 2) = 1$.
Then which of the following is true?

The standard deviation of $X$ is smaller than that of $Y$.
The mean of $X$ is smaller than that of $Y$.
The variance of $X$ is larger than that of $Y$.
The median of $X$ is equal to the median of $Y$.
None of the above is correct.

Since $X$ is between $-1$ and $1$, and $Y$ is neither $-2$ or $2$, so that the s.d. of $X$ is smaller than that of $Y$, isn't it?

A:

To construct counterexamples: 

Let $Y\equiv 2$ and $X \sim U(-1,1)$. Then $Var(X)>0=Var(Y)$.
Let $Y \equiv -2$ and $X\sim U(-1,1)$ or simply $X\equiv 0$. Then $E[X]=0>-2=E[Y]$.
Let $Y\in\{-2,2\}$ with $P(Y=-2)=p=1-P(Y=2)$ and $p>0$ and $X\equiv 0$. Then $Var(Y)>0=Var(X)$.
Let $Y\equiv2$ and $X\sim U(-1,1)$. Then median $Y=2$ and median $X=0$.

Go with 5.

