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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2024 · Mathematics · Quadratic Equation

If - 2 and 3 are roots of the equation a_0+a_1 x+a_2 x^2+a_3...

If −2-\sqrt{2} and 3arerootsoftheequation\sqrt{3} are roots of the equationa_0+a_1 x+a_2 x^2+a_3 x^3+x^4=0wherewherea_0, a_1, a_2, a_3areintegers,thenwhichoneofthefollowingiscorrect?are integers, then which one of the following is correct?

Options

  1. Aa2=a3=0a_2=a_3=0
  2. Ba2=0a_2=0 and a3=−5a_3=-5
  3. Ca0=6,a3=0a_0=6, a_3=0
  4. Da1=0a_1=0 and a2=5a_2=5

Correct answer

C. a0=6,a3=0a_0=6, a_3=0

Step-by-step solution

∵−2\because-\sqrt{2} and 3\sqrt{3} are roots of the given equation Therefore 2\sqrt{2} and −3-\sqrt{3} are also roots of the given  equation x4+a3x3+a2x2+a1x+a0=(x+2)(x−2)(x−3)(x+3)=(x2−2)(x2−3)=x4−3x2−2x2+6=x4−5x2+6∴a3=0,a2=−5,a1=0 and  a0=6.\begin{aligned} & \text{ equation } x^4+a_3 x^3+a_2 x^2+a_1 x+a_0 \\ & =(x+\sqrt{2})(x-\sqrt{2})(x-\sqrt{3})(x+\sqrt{3}) \\ & =\left(x^2-2\right)\left(x^2-3\right) \\ & =x^4-3 x^2-2 x^2+6 \\ & =x^4-5 x^2+6 \\ & \therefore \quad a_3=0, a_2=-5, a_1=0 \text{\text{ and } } a_0=6 . \end{aligned}
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