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4.2 Quadratic equations and inequations — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS4.2 Quadratic equations and inequations1 Mark
MCQ
If the roots of ${x^2} + x + a = 0$exceed $a$, then
  • A
    $2 < a < 3$
  • B
    $a > 3$
  • C
    $ - 3 < a < 3$
  • ✓
    $a < - 2$
✓

Answer

Correct option: D.
$a < - 2$
d
(d) If the roots of the quadratic equation $a{x^2} + bx + c = 0$ exceed a number k, then $a{k^2} + bk + c > 0$ if $a > 0,$ ${b^2} - 4ac \ge 0$ and sum of the roots $ > 2k$

Therefore, if the roots of ${x^2} + x + a = 0$ exceed a number a, then

${a^2} + a + a > 0,1 - 4a \ge 0$ and $ - 1 > 2a$

==> $a(a + 2) > 0,$$a \le \frac{1}{4}$and $a < - \frac{1}{2}$

==> $a > 0\,{\rm{or}}\,a < - 2,a < \frac{1}{4}$and $a < - \frac{1}{2}$

Hence $a < - 2$.

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