Two distinct polynomials f(x) and g(x) are defined as follows: f(x)=x^2+a x+2 ; g(x)=x^2+2 x+a .If e

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4-2.Quadratic Equations and Inequations

normal

Two distinct polynomials $f(x)$ and $g(x)$ are defined as follows:

$f(x)=x^2+a x+2 ; g(x)=x^2+2 x+a$.If the equations $f(x)=0$ and $g(x)=0$ have a common root, then the sum of the roots of the equation $f(x)+g(x)=0$ is

$-\frac{1}{2}$

$0$

$\frac{1}{2}$

$1$

(KVPY-2015)

Solution

(c)

We have,

$f(x)=x^2+a x+2$ and $g(x)=x^2+2 x+a$

Let $\alpha$ be the common root of $f(x)=0$ and $g(x)=0$.

$\frac{\alpha^2}{a^2-4}=\frac{\alpha}{2-a}$

$\Rightarrow \alpha=\begin{array}{c}a^2-4 \\ 2-a\end{array}=\frac{(a+2)(a-2)}{-(a-2)}=-(a+2)$

and $\quad \frac{\alpha}{2-a}=\frac{1}{2-a} \Rightarrow \alpha=1$

$\therefore \quad-(a+2)=1$

$a+2=-1 \Rightarrow a=-3$

Now $\quad f(x)+g(x)=0$

$\therefore x^2-3 x+2+x^2+2 x-3=0$

$2 x^2-x-1=0$

Sum of roots $=\frac{1}{2} \quad\left[\because \alpha+\beta=\frac{-b}{a}\right]$

Standard 11

Mathematics

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hard

(JEE MAIN-2023)

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normal

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hard

(JEE MAIN-2024)

The population of cattle in a farm increases so that the difference between the population in year $n+2$ and that in year $n$ is proportional to the population in year $n+1$. If the populations in years $2010, 2011$ and $2013$ were $39,60$ and $123$,respectively, then the population in $2012$ was

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(KVPY-2014)

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normal

(KVPY-2017)

Two distinct polynomials $f(x)$ and $g(x)$ are defined as follows: $f(x)=x^2+a x+2 ; g(x)=x^2+2 x+a$.If the equations $f(x)=0$ and $g(x)=0$ have a common root, then the sum of the roots of the equation $f(x)+g(x)=0$ is

Solution

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Two distinct polynomials $f(x)$ and $g(x)$ are defined as follows:

$f(x)=x^2+a x+2 ; g(x)=x^2+2 x+a$.If the equations $f(x)=0$ and $g(x)=0$ have a common root, then the sum of the roots of the equation $f(x)+g(x)=0$ is