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MHTCET 23 Apr 2025 Shift 1 Paper

Question : 146 of 150

Marks: +1, -0

x=-2+\sqrt{-3}

2 x^{4}+5 x^{3}+7 x^{2}-x+38

is equal to

Solution:

Concept:

We form a quadratic equation from the given value of

x

, then use polynomial division to simplify the expression.

Explanation:

Given

x = -2 + \sqrt{-3} = -2 + i\sqrt{3}

x + 2 = i\sqrt{3}

Squaring both sides:

(x+2)^2 = (i\sqrt{3})^2

x^2 + 4x + 4 = -3

Therefore:

x^2 + 4x + 7 = 0

Now divide the given expression by

x^2 + 4x + 7

2x^4 + 5x^3 + 7x^2 - x + 38 = (x^2+4x+7)(2x^2 - 3x + 5) + 3

Since

x^2 + 4x + 7 = 0

, the product is

0

Hence the value is

0 + 3 = 3

Answer:

The required value is

3

, which is option C.

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