JIPMAT 2023 (QA) - 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. | PYQs + Solutions

JIPMAT 2023

Algebra

Quadratic Equations

Easy

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.

Both Statement (I) and Statement (II) are true.

Both Statement (I) and Statement (II) are false.

Statement (I) is true but Statement (II) is false.

Statement (I) is false but Statement (II) is true.

Correct Option: 1

Statement (I):

(x^2 + 3x + 1) = (x - 2)^2

is not a quadratic equation

1) Let's expand

(x - 2)^2

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x^2 - 4x + 4

2) So the equation becomes:

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x^2 + 3x + 1 = x^2 - 4x + 4

3) Rearranging:

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x^2 + 3x + 1 - (x^2 - 4x + 4) = 0

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7x - 3 = 0

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x = \frac{3}{7}

4) This is a linear equation, not quadratic

Therefore Statement (I) is TRUE

Statement (II): The roots of

x^2 + 2x\sqrt{3} + 3 = 0

are real and equal

1) For quadratic equation

ax^2 + bx + c = 0

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Here,

a=1

b=2\sqrt{3}

c=3

2) For real and equal roots:

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Discriminant =

b^2 - 4ac = 0

3) Let's calculate:

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(2\sqrt{3})^2 - 4(1)(3)

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12 - 12

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0

4) Since discriminant = 0, roots are real and equal

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Therefore Statement (II) is TRUE

Hence both statements are true.