Question

If $p,q$and $r$real and$p\ne q$, then the roots of the equation are $(p-q)x^2+5(p+q)x-2(p-q)=r$

• A. Real and equal
• B. Unequal and rational
• C. Unequal and irrational
• D. Nothing can be said

Answer 1

The correct option is D

Nothing can be said

Explanation for the correct option

If $p,q$and $r$real and $p\ne q$ then

$$\begin{gathered} \Rightarrow (p–q)x^2+5(p+q)x–2(p–q)=r \\ \Rightarrow (p–q)x^2+5(p+q)x–2(p–q)–r=0 \end{gathered}$$

$\begin{array}{rcl}\mathrm{Compair}\mathrm{with}{\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}& =& 0\end{array}$

$\because \mathrm{Discriminant}D=b^2-4ac$

$\begin{array}{rcl}\therefore D& =& [5{(p+q)}^2–4(p–q\left)\right(-2p+2q–r\left)\right]\\ & =& 25{[p+q]}^2+4(p–q)(2p+2q–r)\\ & =& 25{(p+q)}^2+4(2p^2–2pq–2pq+2q^2+pr–qr)\\ & =& 25{(p+q)}^2+8{(p–q)}^2+4r(p–q)\end{array}$

We can't determine the sign of $(p-q)$

Thus, nothing can be said about the nature of roots.

Hence option ‘D’ is correct .