Let S be the set of all (λ, μ) for which the vectors λ ^ i - ^ j + ^ k , ^ j + 2 ^ j + μ ^ k and 3 ^ i - 4 ^ j + 5 ^ k , where λ - μ = 5 , are coplanar, then ∑ _ (λ, μ) ∈ S 80 ( λ ^ 2 + μ ^ 2 ) is equal to

Correct answer: C. 2290 The given equations are coplanar. ⇒ | cases λ & - 1 & 1 1 & 2 & μ 3 & - 4 & 5 cases | = 0 ⇒ λ[10 + 4μ] + 1[5 - 3μ] + 1[- 10] = 0 ⇒ 4λμ + 10λ - 3μ = 5 . . . . . . . (i) And λ - μ = 5 So, by (1) ⇒ 4(5 + μ)μ + 10(5 + μ) - 3μ = 5 ⇒ 4 μ ^ 2 + 27μ + 45 = 0 ⇒ 4 μ ^ 2 + 15μ + 12μ + 45 = 0 ∴ μ = - 3, (- 15)/(4) ⇒ λ = 2, (5)/(4) ∴ ∑ _ λμ ∈ S 80( λ ^ 2 + μ ^ 2 ) = 80[(9 + 4) + (225)/(16) + (25)/(16)] = 80[13 + (125)/(8)] = 10 × 229 = 2290 Hence this is the correct option.

JEE Main 202315 AprilMorning Shift

Exam: JEE Main
Year: 2023
Date: 15 April
Shift: Morning Shift
Subject: Mathematics
Chapter: Vector Algebra

JEE Main 2023 · 15 April · Morning Shift · Mathematics · Vector Algebra

Let S be the set of all (λ, μ) for which the vectors λ ^ i - ^ j +...

Let

S

be the set of all \left(\lambda, \mu\right)

for which the vectors

\lambda\overset{^}{i} - \overset{^}{j} + \overset{^}{k}, \overset{^}{j} + 2\overset{^}{j} + \mu\overset{^}{k} and 3\overset{^}{i} - 4\overset{^}{j} + 5\overset{^}{k}

, where

\lambda - \mu = 5

, are coplanar, then

\mathop{∑}_{\left(\lambda, \mu\right)$ \in S}80 \left({\lambda}^{2} + {\mu}^{2}\right) is equal to

Options

A $2210$
B $2130$
C $2290$
D $2370$

Correct answer

C. $2290$

Step-by-step solution

The given equations are coplanar.
$⇒ \left|$ \begin{cases}\lambda & - 1 & 1 \\ 1 & 2 & \mu \\ 3 & - 4 & 5\end{cases} $\right| = 0$
$⇒ \lambda\left[10 + 4\mu\right] + 1\left[5 - 3\mu\right] + 1\left[- 10\right] = 0$
$⇒ 4\lambda\mu + 10\lambda - 3\mu = 5 . . . . . . . \left(i\right)$
And $\lambda - \mu = 5$
So, by \left(1\right) $$ ⇒ 4\left(5 + \mu\right)\mu + 10\left(5 + \mu\right) - 3\mu = 5 $$ ⇒ 4{\mu}^{2} + 27\mu + 45 = 0 $$ ⇒ 4{\mu}^{2} + 15\mu + 12\mu + 45 = 0 $$ ∴ \mu = - 3, \dfrac{- 15}{4} $$ ⇒ \lambda = 2, \dfrac{5}{4} $$ ∴ \mathop{∑}_{\lambda\mu \in S}80\left({\lambda}^{2} + {\mu}^{2}\right) = 80\left[\left(9 + 4\right) + \dfrac{225}{16} + \dfrac{25}{16}\right] $$ = 80\left[13 + \dfrac{125}{8}\right] = 10 \times 229 $$ = 2290
Hence this is the correct option.

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