SATHEE: Theory of Equations 1 Question 9

Theory of Equations 1 Question 9

10. The value of $λ$ such that sum of the squares of the roots of the quadratic equation, $x^{2} + (3 - λ) x + 2 = λ$ has the least value is

(2019 Main, 10 Jan II)

(a) $\frac{4}{9}$

(b) 1

(d) 2

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Answer:

Correct Answer: 10. (d)

Solution:

Given quadratic equation is

$x^{2} + (3 - λ) x + 2 = λ$

$x^{2} + (3 - λ) x + (2 - λ) = 0$

Let Eq. (i) has roots $α$ and $β$ , then $α + β = λ - 3$ and $α β = 2 - λ$

$\begin{array}{r} [∵ For a x^{2} + b x + c = 0, sum of roots = - \frac{b}{a} \\ and product of roots = \frac{c}{a}] \end{array}$

Now, $α^{2} + β^{2} = (α + β)^{2} - 2 α β$

$\begin{aligned} = (λ - 3)^{2} - 2 (2 - λ) \\ = λ^{2} - 6 λ + 9 - 4 + 2 λ \\ = λ^{2} - 4 λ + 5 = (λ^{2} - 4 λ + 4) + 1 \\ = (λ - 2)^{2} + 1 \end{aligned}$

Clearly, $a^{2} + β^{2}$ will be least when $λ = 2$ .

Theory of Equations 1 Question 9

10. The value of $λ$ such that sum of the squares of the roots of the quadratic equation, $x^{2} + (3 - λ) x + 2 = λ$ has the least value is

Answer:

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Theory of Equations 1 Question 9

10. The value of λ such that sum of the squares of the roots of the quadratic equation, x2+(3−λ)x+2=λ has the least value is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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10. The value of $λ$ such that sum of the squares of the roots of the quadratic equation, $x^{2} + (3 - λ) x + 2 = λ$ has the least value is