JAMIA 2022 — Mathematics' PYQ

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Question

JAMIA 2022 — Mathematics' PYQ

JAMIA | Mathematics' | 2022

The value of $b$ for which the equations $x^{2} + b x - 1 = 0$ and $x^{2} + x + b = 0$
have no root in common is

Choose the correct answer:

Explanation

x^2 + bx - 1 = 0 \quad \dots (1)

x^2 + x + b = 0 \quad \dots (2)

For common root $x$ :

(x^2 + bx - 1) - (x^2 + x + b) = 0

(b - 1)x - (1 + b) = 0

x = \frac{b + 1}{b - 1}

Substitute $x$ in (2):

\left( \frac{b + 1}{b - 1} \right)^2 + \left( \frac{b + 1}{b - 1} \right) + b = 0

(b + 1)^2 + (b + 1)(b - 1) + b(b - 1)^2 = 0

b^2 + 2b + 1 + b^2 - 1 + b(b^2 - 2b + 1) = 0

b^3 + 3b = 0

b(b^2 + 3) = 0

Common root values:

b = 0, \pm i\sqrt{3}

Since $-i\sqrt{5}$ is not in $\{0, i\sqrt{3}, -i\sqrt{3}\}$ , it satisfies the condition for no common root.

Correct Option:

-i\sqrt{5}