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If the sum of two roots of the equation $x^3-2 p x^2+3 q x-4 r=0$ is zero, then the value of $r$ is
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Verified Answer
The correct answer is:
$\frac{3 p q}{2}$
Let the two roots of equation is $m,-m$.
Now, sum of three roots $=2 p$
Hence, third root will be $2 p$.
Now, $m \times(-m)+m \times(2 p)+(-m) \times 2 p=3 q$
$$
\Rightarrow \quad-m^2=3 q
$$
Now, $\quad m \times(-n) \times 2 p=4 r$
$$
\begin{array}{rlrl}
\Rightarrow & 3 q \times 2 p & =4 r \\
& \Rightarrow & r & =\frac{3 p q \times 2}{4} \Rightarrow r=\frac{3 p q}{2}
\end{array}
$$
Now, sum of three roots $=2 p$
Hence, third root will be $2 p$.
Now, $m \times(-m)+m \times(2 p)+(-m) \times 2 p=3 q$
$$
\Rightarrow \quad-m^2=3 q
$$
Now, $\quad m \times(-n) \times 2 p=4 r$
$$
\begin{array}{rlrl}
\Rightarrow & 3 q \times 2 p & =4 r \\
& \Rightarrow & r & =\frac{3 p q \times 2}{4} \Rightarrow r=\frac{3 p q}{2}
\end{array}
$$
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