Question

If $\alpha ,\beta$ are roots of the equation $x+2x+5=0$ and $\overline{a}=(\alpha +\beta )\overline{i}+\alpha \beta \overline{j},\overline{b}=\alpha \beta \overline{i}+(\alpha +\beta )\overline{j}+({\alpha }^2+{\beta }^2)\overline{k}$ then $\overline{a}\times \overline{b}=$

• A. $\overline{i}+12\overline{j}+12\overline{k}$
• B. $-30\overline{i}+12\overline{j}-5\overline{k}$
• C. $-30\overline{i}+12\overline{j}-21\overline{k}$
• D. $\overline{i}-12\overline{j}+29\overline{k}$

Answer 1

The correct option is C $-30\overline{i}+12\overline{j}-21\overline{k}$

$\alpha +\beta =-2$ $\alpha \beta =5$

$\alpha +{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta$

$=4-10$

$=-6$

$\overline{a}=-2\hat{i}+5\hat{j}$

$\overline{b}=5\hat{i}-2\hat{j}-6\hat{k}$

$\overline{a}\times \overline{b}\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ -2& 5& 0\\ 5& -2& -6\end{array}\right|$

$\hat{i}(-30)-\hat{j}(-12)+\hat{b}(4-25)$

$=-30\hat{i}+12\hat{j}-21\hat{k}$