Q167 / 270 · NDA 2023 (Phase 1) · Mathematics

For how many integral values of $k$, the equation $x^2-4 x+k$ $=0$, where $k$ is an integer, has real roots and both of them lie in the interval $(0,5)$ ?

  • A
    $3$
  • Correct answer
    B
    $4$
  • C
    $5$
  • D
    $6$

Solution

$x^2-4 x+K=0$ For real roots, $D \geq 0$ $\Rightarrow \quad(-4)^2-4(1)(K) \geq 0$ $16-4 K \geq 0$ $4-K \geq 0$ $K \leq 4$ but Both roots lie in $(0,5)$. $K=1,2,3,4$ (integers) Number of integral values $=4$.

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

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