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  • Let be the set of all <img alt="(\lambda, \mu)" src="https://entrancecorner.oncodecogs.com/gif.latex?

Let \mathrm{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, y) \mathrm{S}} 80\left(\lambda^2+\mu^2\right) is equal to

Option: 1

2130


Option: 2

2210


Option: 3

2290


Option: 4

2370


Answers (1)

best_answer

\begin{aligned} & \left|\begin{array}{ccc} \lambda & -1 & 1 \\ 1 & 2 & \mu \\ 3 & -4 & 5 \end{array}\right|=0 \quad \& \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)=5} 80\left(\lambda^2+\mu^2\right) \\ & =80\left(\frac{250}{16}+13\right) \\ & =1250+1040 \\ & =2290 \end{aligned}

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