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NDA 2015
Exam
NDA
Year
2015
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2015 · Mathematics · Quadratic Equation

Consider the following statements in respect of the given equation...

Consider the following statements in respect of the given equation : \left(x^2+2\right)^2+8 x^2=6 x \left(x²+2\right) 1. All the roots of the equation are complex. 2. The sum of all the roots of the equation is 6. Which of the above statements is/are correct?

Options

  1. A1 only
  2. B2 only
  3. CBoth 1 and 2
  4. DNeither 1 nor 2

Correct answer

B. 2 only

Step-by-step solution

(x2+2)2+8x2=6x(x2+2)Let x2+2=yy2+8x2=6xyy2−6xy+8x2=0y=6x±36x2−32x22\begin{aligned} & \left(x^2+2\right)^2+8 x^2=6 x\left(x^2+2\right) \\ & \text{Let } x^2+2=y \\ & y^2+8 x^2=6 x y \\ & y^2-6 x y+8 x^2=0 \\ & y=\frac{6 x \pm \sqrt{36 x^2-32 x^2}}{2}\end{aligned}y=6x±2x2=3x±xy=4x,2x At y=4xx2+2=4xx2−4x+2=0\begin{aligned} & y=\frac{6 x \pm 2 x}{2}=3 x \pm x \\ & y=4 x, 2 x \\ & \text{ At } y=4 x \\ & x^2+2=4 x \\ & x^2-4 x+2=0 \end{aligned}Discriminant,Discriminant,\mathrm{D}=16-8=8 > 0Rootsarereal.SumofrootsRoots are real. Sum of roots=-(-4)=4 At x=2xx2+2=2xx2−2x+2=0D=4−8=−4<0\begin{aligned} & \text{ At } x=2 x \\ & x^2+2=2 x \\ & x^2-2 x+2=0 \\ & D=4-8=-4 < 0 \end{aligned}Rootsarecomplex.SumofrootsRoots are complex. Sum of roots=2SumofallrootsSum of all roots=4+2=6onlystatement2iscorrect.only statement 2 is correct.\therefore \quad$ Correct option is (B)
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