JIPMAT 2022 (QA) - If () and () are the roots of the equation a x^2+b x+c=0, then b^2 is | PYQs + Solutions

JIPMAT 2022

Algebra

Quadratic Equations

Hard

If $\sin (\alpha)$ and $\cos (\alpha)$ are the roots of the equation $a x^{2}+b x+c=0$ , then $b^{2}$ is

c^{2}+2 a c

a^{2}+a c

a^{2}+2 a c

c^{2}+a c

Correct Option: 3

\sin(\alpha)

and

\cos(\alpha)

are roots:

a\sin^2(\alpha) + b\sin(\alpha) + c = 0

a\cos^2(\alpha) + b\cos(\alpha) + c = 0

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Adding equations:

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a[\sin^2(\alpha) + \cos^2(\alpha)] + b[\sin(\alpha) + \cos(\alpha)] + 2c = 0

a(1) + b[\sin(\alpha) + \cos(\alpha)] + 2c = 0

...(1)

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Product of roots =

\frac{c}{a}

So,

\sin(\alpha)\cos(\alpha) = \frac{c}{a}

...(2)

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Sum of roots =

-\frac{b}{a}

So,

\sin(\alpha) + \cos(\alpha) = -\frac{b}{a}

...(3)

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Therefore:

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[\sin(\alpha) + \cos(\alpha)]^2 = [-\frac{b}{a}]^2

But

[\sin(\alpha) + \cos(\alpha)]^2 = 1 + 2\sin(\alpha)\cos(\alpha)

1 + 2(\frac{c}{a}) = \frac{b^2}{a^2}

b^2 = a^2 + 2ac

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Therefore,

b^2 = a^2 + 2ac

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y = f(x)

JIPMAT 2023

Given below are two statements:

Statement (I): (x² + 3x + 1) = (x - 2)² is not a quadratic equation.

Statement (II): The nature of roots of quadratic equations x² + 2x√3 + 3 = 0 are real and equal.

In light of the above statements, choose the most appropriate answer from the options given below.