Question

Let $\overline{f}\left(t\right)=\left[t\right]\overline{i}-\left(t-\left[t\right]\right)\overline{j}+\left[t+1\right]\overline{k}$ be vector. Where $[.]$ denotes greatest integer function. If $\overline{f}\left(\frac{5}{4}\right)$ and $\overline{i}+\lambda \overline{j}+\mu \overline{k}$ are parallel vectors then $(\alpha ,\beta )$

• A. $(1,1)$
• B. $(-\frac{1}{4},2)$
• C. $(\frac{1}{2},2)$
• D. $(\frac{1}{4},4)$

Answer 1

The correct option is B $(-\frac{1}{4},2)$

$\overline{f}\left(\frac{5}{4}\right)=\left[\frac{5}{4}\right]\hat{i}-(\frac{5}{4}-\left[\frac{5}{4}\right])\hat{j}+[\frac{5}{4}+1]\hat{k}$

$=\left(1\right)\hat{i}-(\frac{5}{4}-1)\hat{j}+(1+1)\hat{k}$

$=\hat{i}-\frac{i}{4}+2\hat{k}$

We know, if $\overline{a}=a_1,\hat{i}+a_2\hat{j}+a_3\hat{k}$

and $\overline{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$

$\frac{a_1}{b_1}=\frac{a_2}{b_2}=\frac{a_3}{b_3}$

$\therefore \frac{1}{1}=\frac{\lambda }{(-1/4)}=\frac{\mu }{\left(2\right)}$

$\Rightarrow \lambda =\frac{-1}{4}$ and $\mu =2$

There is a MISPRINT is the question it asks the value of

$(\lambda ,\mu )$ and not $(\alpha ,\beta )$

$\therefore$ the ans is $\left(B\right)$