Q96 / 155 · WBJEE 2020 · Mathematics

If $\mathrm{P}(\mathrm{x})=a \mathrm{x}^{2}+\mathrm{bx}+\mathrm{c}$ and $\mathrm{Q}(\mathrm{x})=-\mathrm{ax}^{2}+\mathrm{dx}+\mathrm{c}$, where $\mathrm{ac} \neq 0 \quad[\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$ are all real], then $\mathrm{P}(\mathrm{x}) \cdot \mathrm{Q}(\mathrm{x})=0$ has

  • Correct answer
    A
    at least two real roots
  • B
    two real roots
  • C
    four real roots
  • D
    no real root

Solution

Hint: If $P(x)=a x^{2}+b x+c, Q(x)=-a x^{2}+d x+c$ $\mathrm{D}_{1}=\mathrm{b}^{2}-4 \mathrm{ac}$ $\mathrm{D}_{2}=\mathrm{d}^{2}+4 \mathrm{ac}$ $\Rightarrow \mathrm{D}_{1}+\mathrm{D}_{2}>0$ Atleast two real roots.

Hence, the correct answer is option: 'A'.

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