<p><strong>Let <math><mi>λ</mi><mo>≠</mo><mn>0</mn></math> be in R. If <math><mi>α</mi></math> and <math><mi>β</mi></math> are the roots of the equation, <math><msup><mrow><mrow><mi>x</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mo>-</mo><mi>x</mi><mo>+</mo><mn>2</mn><mi>λ</mi><mo>=</mo><mn>0</mn></math> and <math><mi>α</mi></math> and <math><mi>γ</mi></math> are the roots of the equation, <math><mn>3</mn><msup><mrow><mrow><mi>x</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mo>-</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>27</mn><mi>λ</mi><mo>=</mo><mn>0</mn></math> , then <math><mfrac><mrow><mrow><mi>β</mi><mi>γ</mi></mrow></mrow><mrow><mrow><mi>λ</mi></mrow></mrow></mfrac></math> is equal to:</strong></p>

f(x)=a?⋅(b?×c?)=x-23-2x-17-2x=x3-27x+26f'(x)=3x2-27=0⇒x=±3 and f''(-3)<0⇒ local maxima at x=x0=-3Thus, a?=-3iˆ-2jˆ+3kˆ,b?=2iˆ-3jˆ-kˆ , and c?=7iˆ-2jˆ-3kˆ⇒a?⋅b?+b?⋅c?+c?⋅a?=9-5-26=-22

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Maths NCERT Exemplar Solutions Class 12th Chapter Twelve Maths Class 12th Complex Numbers and Quadratic Equations Maths NCERT Exemplar Solutions Class 12th Chapter Five

Let $λ \neq 0$ be in R. If $α$ and $β$ are the roots of the equation, $x^{2} - x + 2 λ = 0$ and $α$ and $γ$ are the roots of the equation, $3 x^{2} - 10 x + 27 λ = 0$ , then $\frac{β γ}{λ}$ is equal to:

Option 1 -

Option 2 -

Option 3 -

Option 4 -

27(Quadratic)

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1 Answer

Answered by

alok kumar singh | Contributor-Level 10

2 months ago

Correct Option - 2

Detailed Solution:

$f (x) = \overset{?}{a} \cdot (\overset{?}{b} \times \overset{?}{c}) = |\begin{matrix} x & - 2 & 3 \\ - 2 & x & - 1 \\ 7 & - 2 & x \end{matrix}|$

$= x^{3} - 27 x + 26$

$f^{'} (x) = 3 x^{2} - 27 = 0 \Rightarrow x = \pm 3$ and $f^{''} (- 3) < 0$

$\Rightarrow$ local maxima at $x = x_{0} = - 3$

Thus, $\overset{?}{a} = - 3 \hat{i} - 2 \hat{j} + 3 \hat{k}, \overset{?}{b} = 2 \hat{i} - 3 \hat{j} - \hat{k}$ , and $\overset{?}{c} = 7 \hat{i} - 2 \hat{j} - 3 \hat{k}$

$\Rightarrow \overset{?}{a} \cdot \overset{?}{b} + \overset{?}{b} \cdot \overset{?}{c} + \overset{?}{c} \cdot \overset{?}{a} = 9 - 5 - 26 = - 22$

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Kindly consider the following figure

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Where $α = e^{i (2 π / 6)}$

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Let $λ \neq 0$ be in R. If $α$ and $β$ are the roots of the equation, $x^{2} - x + 2 λ = 0$ and $α$ and $γ$ are the roots of the equation, $3 x^{2} - 10 x + 27 λ = 0$ , then $\frac{β γ}{λ}$ is equal to:

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Let λ≠0 be in R. If α and β are the roots of the equation, x2-x+2λ=0 and α and γ are the roots of the equation, 3x2-10x+27λ=0 , then βγλ is equal to:

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Let $λ \neq 0$ be in R. If $α$ and $β$ are the roots of the equation, $x^{2} - x + 2 λ = 0$ and $α$ and $γ$ are the roots of the equation, $3 x^{2} - 10 x + 27 λ = 0$ , then $\frac{β γ}{λ}$ is equal to: