Question

If the values of $\lambda$ for which $\hat{i}+2\lambda \hat{j}-3\hat{k},2\hat{i}-2\lambda \hat{j}+\hat{k}$ and $-2\hat{i}-\hat{j}+2\lambda \hat{k}$ are linearly dependent are $\alpha$ and $\beta ,$ then which of the following is/are true

• A. $\alpha \beta =\frac{2}{3}$
• B. $\alpha \beta =-\frac{7}{12}$
• C. $\alpha +\beta =1$
• D. $\alpha +\beta =\frac{2}{3}$

Answer 1

Answer: (B), (D)

$\alpha +\beta =\frac{2}{3}$

For $\hat{i}+2\lambda \hat{j}-3\hat{k},2\hat{i}-2\lambda \hat{j}+\hat{k}$ and $-2\hat{i}-\hat{j}+2\lambda \hat{k}$ to be linearly dependent
$\left|\begin{array}{ccc}1& 2\lambda & -3\\ 2& -2\lambda & 1\\ -2& -1& 2\lambda \end{array}\right|=0\Rightarrow 1(-4{\lambda }^2+1)-2\lambda (4\lambda +2)-3(-2-4\lambda )=0\Rightarrow 12{\lambda }^2-8\lambda -7=0$
Let $\alpha$ and $\beta$ are roots of quadratic equation
$\Rightarrow \alpha +\beta =\frac{8}{12}=\frac{2}{3}$
$\Rightarrow \alpha \beta =-\frac{7}{12}$