SATHEE: Theory of Equations 1 Question 6

Theory of Equations 1 Question 6

7. If $λ$ be the ratio of the roots of the quadratic equation in $x, 3 m^{2} x^{2} + m (m - 4) x + 2 = 0$ , then the least value of $m$ for which $λ + \frac{1}{λ} = 1$ , is

(2019 Main, 12 Jan I)

(a) $- 2 + \sqrt{2}$

(b) $4 - 2 \sqrt{3}$

(d) $2 - \sqrt{3}$

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Answer:

Correct Answer: 7. (c)

Solution:

Let the given quadratic equation in $x$ ,

$3 m^{2} x^{2} + m (m - 4) x + 2 = 0, m \neq 0$ have roots $α$ and $β$ , then

$α + β = - \frac{m (m - 4)}{3 m^{2}} and α β = \frac{2}{3 m^{2}}$

Also, let $\frac{α}{β} = λ$

Then, $λ + \frac{1}{λ} = 1 \Rightarrow \frac{α}{β} + \frac{β}{α} = 1$

(given)

$\Rightarrow α^{2} + β^{2} = α β$

$\Rightarrow (α + β)^{2} = 3 α β$

$\Rightarrow \frac{m^{2} (m - 4)^{2}}{9 m^{4}} = 3 \frac{2}{3 m^{2}}$

$\Rightarrow (m - 4)^{2} = 18$

$[∵ m \neq 0]$

$\Rightarrow m - 4 = \pm 3 \sqrt{2}$

$\Rightarrow m = 4 \pm 3 \sqrt{2}$

The least value of $m = 4 - 3 \sqrt{2}$

Theory of Equations 1 Question 6

7. If $λ$ be the ratio of the roots of the quadratic equation in $x, 3 m^{2} x^{2} + m (m - 4) x + 2 = 0$ , then the least value of $m$ for which $λ + \frac{1}{λ} = 1$ , is

Answer:

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Theory of Equations 1 Question 6

7. If λ be the ratio of the roots of the quadratic equation in x,3m2x2+m(m−4)x+2=0, then the least value of m for which λ+1λ=1, is

Answer:

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Medical Preparation

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Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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7. If $λ$ be the ratio of the roots of the quadratic equation in $x, 3 m^{2} x^{2} + m (m - 4) x + 2 = 0$ , then the least value of $m$ for which $λ + \frac{1}{λ} = 1$ , is