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NDA 2019
Exam
NDA
Year
2019
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2019 · Mathematics · Quadratic Equation

Consider the following statements in respect to the quadratic...

Consider the following statements in respect to the quadratic equation 4(x−p)(x−q)−r2=04(x-p)(x-q)-r^2=0 where p,qp, q and rr are real numbers : 1. The roots are real 2. The roots are equal if p=qp=q and r=0r=0 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

C. Both 1 and 2

Step-by-step solution

4(x−p)(x−q)−r2=0=(x−p)(x−q)−r24=0=x2−(p+q)x+pq−r24=0\begin{aligned} & 4(x-p)(x-q)-r^2=0 \\ & =(x-p)(x-q)-\frac{r^2}{4}=0 \\ & =x^2-(p+q) x+p q-\frac{r^2}{4}=0 \end{aligned} Discriminat, D=[−(p+q)]2−4(pq−r24)=0D=(p+q)2−4pq+r2=0D=(p−q)2+r2=0\begin{aligned} & \mathrm{D}=[-(p+q)]^2-4\left(p q-\frac{r^2}{4}\right)=0 \\ & \mathrm{D}=(p+q)^2-4 p q+r^2=0 \\ & \mathrm{D}=(p-q)^2+r^2=0 \end{aligned} We have D≥0⇒\mathrm{D} \geq 0 \Rightarrow real roots. and if p=q,r=0theD=0∴p=q, r=0 the \mathrm{D}=0\therefore equal roots ∴\therefore both statements are correct.
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