Question

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

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

Answer 1

Answer: (B)

$(\frac{-1}{4},2)$

$b\left(\frac{5}{4}\right)=\left(\frac{5}{4}\right)\hat{i}-(\frac{5}{4}-\left[\frac{5}{4}\right])\hat{j}+[\frac{5}{4}+1]\hat{k}$
$=\hat{i}-\frac{1}{4}\hat{j}+2\hat{k}$
$b\left(\frac{5}{4}\right)=k(\hat{i}+\lambda \hat{j}+\mu \hat{k})$
$\hat{i}-\frac{1}{4}\hat{j}+2\hat{k}=k\hat{i}+k\lambda \hat{j}+\mu k\hat{k}$
$k=1$ $\lambda =\frac{-1}{4}$ $\mu =2$