Consider the data sets, A and $B,$ of two dimensional features shown below, where each sample point is represented as a small symbol $\text{'}x\text{'}$ or $\text{'}0\text{'}.$

$($a$)$ For data set A$,$ make a quick copy of the axes and rough distributions of marks$]$ the data, and sketch the principal components that you would obtain with principal component analysis $($PCA$).$ Label the first principal component as a and the second as b$.$ Briefly justify the principal components you drew.

$($b$)$ In data set B$,$ there are two classes $-$ samples of Class $1$ are shown with marks$]$ a symbol $\text{'}0\text{'}$ and those of Class $2$ with a symbol $\text{'}x\text{'}.$ Explain what preprocessing you would apply to the data set so that a logistic regression model classifies the samples into the two classes reasonably well.