Let S be the set of all for which the vectors i ^ - j ^ + k ^ , i ^ + 2 j ^ + k ^ and 3 i ^ - 4 j ^ + 5 k ^ , where = 5 , are coplanar. Then S 80 ( 2 + 2 ) is equal to [2023]

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Solution

Q.
Let S be the set of all $(λ, μ)$ for which the vectors $λ \hat{i} - \hat{j} + \hat{k},$ $\hat{i} + 2 \hat{j} + μ \hat{k}$ and $3 \hat{i} - 4 \hat{j} + 5 \hat{k}$ , where $λ - μ = 5$ , are coplanar. Then $\sum_{(λ, μ) \in S} 80 (λ^{2} + μ^{2})$ is equal to [2023]

1 2290

2 2210

3 2370

4 2130

Ans.
(1)

$Since, the given vectors are coplanar:$

$∴$ $| \begin{matrix} λ & - 1 & 1 \\ 1 & 2 & μ \\ 3 & - 4 & 5 \end{matrix} |$ $= 0$

$\Rightarrow λ (10 + 4 μ) + 1 (5 - 3 μ) + 1 (- 4 - 6) = 0$

$\Rightarrow 10 λ + 4 λ μ + 5 - 3 μ - 10 = 0$

$\Rightarrow 10 (5 + μ) + 4 μ (5 + μ) - 5 - 3 μ = 0$ $[∵ λ - μ = 5]$

$\Rightarrow 50 + 10 μ + 20 μ + 4 μ^{2} - 5 - 3 μ = 0$

$\Rightarrow 4 μ^{2} + 27 μ + 45 = 0$

$\Rightarrow 4 μ^{2} + 12 μ + 15 μ + 45 = 0$

$\Rightarrow (4 μ + 15) (μ + 3) = 0$

$\Rightarrow μ = - 3 and μ = \frac{- 15}{4}$ $\Rightarrow λ = 2 and λ = \frac{5}{4}$

$∴$ $\sum_{(λ, μ) \in S} 80 (λ^{2} + μ^{2}) = 80 [(4 + 9) + (\frac{25}{16} + \frac{225}{16})]$

$= \frac{458}{16} \times 80 = 2290$

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