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# If α, β Are the Roots of the Equation X 2 + P X + Q = 0 Then − 1 α + 1 β Are the Roots of the Equation - Mathematics

MCQ

If α, β are the roots of the equation $x^2 + px + q = 0 \text { then } - \frac{1}{\alpha} + \frac{1}{\beta}$ are the roots of the equation

#### Options

• $x^2 - px + q = 0$

• $x^2 + px + q = 0$

• $q x^2 + px + 1 = 0$

• $q x^2 - px + 1 = 0$

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#### Solution

$q x^2 - px + 1 = 0$

Given equation:

$x^2 + px + q = 0$

Also,

$\alpha$ and $\beta$ are the roots of the given equation.
Then, sum of the roots = $\alpha + \beta = - p$

Product of the roots = $\alpha\beta = q$

Now, for roots

$- \frac{1}{\alpha} , - \frac{1}{\beta}$ , we have:

Sum of the roots = $- \frac{1}{\alpha} - \frac{1}{\beta} = - \frac{\alpha + \beta}{\alpha\beta} = - \left( \frac{- p}{q} \right) = \frac{p}{q}$

Product of the roots = $\frac{1}{\alpha\beta} = \frac{1}{q}$

Hence, the equation involving the roots $- \frac{1}{\alpha}, - \frac{1}{\beta}$ is as follows:

$x^2 - \left( \alpha + \beta \right)x + \alpha\beta = 0$

$\Rightarrow x^2 - \frac{p}{q}x + \frac{1}{q} = 0$

$\Rightarrow q x^2 - px + 1 = 0$

Is there an error in this question or solution?
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 14 Quadratic Equations
Q 21 | Page 17
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