Question

If the roots of the quadratic equation $(4p-p^2-5)x^2-(2p-1)x+3p=0$ lie on either side of unity, then the number of integral values of $p$ is

• A. $1$
• B. $4$
• C. $3$
• D. $2$

Answer 1

Answer: (D)

$2$

$(4{p-p}^2-5)x^2-(2p-1)x+3p=0=f\left(x\right)$

Roots lie on either side of unity.

$\therefore f\left(1\right)>0$ as $(4{p-p}^2-5<0)⟹(4{p-p}^2-5)1^2-(2p-1)1+3p>0⟹{-p}^2+5p-4>0⟹p^2-5p+4<0⟹(p-4)(p-1)<0⟹p\in (1,4)\therefore p=2,3$

So two integral values of $p$.