Let S be the set of all ( , ) for which the vectors i - j + k , i… - Vidyadip

MCQ

Let $S$ be the set of all $(\lambda, \mu)$ for which the vectors $\lambda \hat{ i }-\hat{ j }+\hat{ k }, \hat{ i }+2 \hat{ j }+\mu \hat{ k }$ and $3 \hat{ i }-4 \hat{ j }+5 \hat{ k }$, where $\lambda-\mu=5$, are coplanar, then $\sum_{(\lambda, \mu) \in S} 80\left(\lambda^2+\mu^2\right)$ is equal to :

A
$2370$
B
$2130$
✓
$2290$
D
$2210$

✓

Answer

Correct option: C.

$2290$

c
$\left|\begin{array}{ccc}\lambda & -1 & 1 \\ 1 & 2 & \mu \\ 3 & -4 & 5\end{array}\right|=0 \quad and\, \lambda-\mu=5$

$\lambda(10+4 \mu)+(5-3 \mu)+(-10)=0$

$(\mu+5)(4 \mu+10)+5-3 \mu-10=0$

$\mu=-15 ; \lambda=5 / 4$

$\mu=-3 ; \lambda=2$

$\text { Hence } \sum_{(\lambda, \mu) \in S} 80\left(\lambda^2+\mu^2\right)$

$=80\left(\frac{250}{16}+13\right)$

$=1250+1040$

$=2290$

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