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If the Equations X 2 + 2 X + 3 λ = 0 and 2 X 2 + 3 X + 5 λ = 0 Have a Non-zero Common Roots, Then λ = - Mathematics

MCQ

If the equations \[x^2 + 2x + 3\lambda = 0 \text { and } 2 x^2 + 3x + 5\lambda = 0\]  have a non-zero common roots, then λ =

Options

  • 1

  • -1

  • 3

  • none of these.

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Solution

-1

Let \[\alpha\] be the common roots of the equations, \[x^2 + 2x + 3\lambda = 0\] and \[2 x^2 + 3x + 5\lambda = 0\]

Therefore,

\[\alpha^2 + 2\alpha + 3\lambda = 0\]      ... (1)

\[2 \alpha^2 + 3\alpha + 5\lambda = 0\]       ... (2)

Solving (1) and (2) by cross multiplication, we get

\[\frac{\alpha^2}{10\lambda - 9\lambda} = \frac{\alpha}{6\lambda - 5\lambda} = \frac{1}{3 - 4}\]

\[ \Rightarrow \alpha^2 = - \lambda, \alpha = - \lambda\]

\[ \Rightarrow - \lambda = \lambda^2 \]

\[ \Rightarrow \lambda = - 1\]


 

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 14 Quadratic Equations
Q 15 | Page 17
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