Multiple choice

lf the equations $k(6x^{2}+3)+rx+(2x^{2}-1)=0$ and $6k(2x^{2}+1)+px+(4x^{2}-2)=0$ have both roots common, then the value $\cfrac{p}{r}$ is

  1. $\dfrac{1}{2}$
  2. $2$
  3. $1$
  4. $4$
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B Correct answer
Explanation

For the equations to have common roots, the coefficients must be proportional. Comparing the coefficients of the two equations leads to the ratio p/r = 2.

AI explanation

Rearranging the first equation gives (6k+2)x^2 + rx + (3k-1) = 0. The second equation simplifies to (12k+4)x^2 + px + (6k-2) = 0, which is exactly double the first equation's coefficients except for the x term, so p must equal 2r for the condition of common roots to hold. Dividing both sides by r yields p/r = 2. The result is 2.

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