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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2021 · Mathematics · Quadratic Equation

If the roots of the equation 4 x^2-(5 k+1) x+5 k=0 differ by unity,...

If the roots of the equation 4x2−(5k+1)x+5k=04 x^2-(5 k+1) x+5 k=0 differ by unity, then which one of the following is a possible value of kk ?

Options

  1. A-3
  2. B-1
  3. C−15-\frac{1}{5}
  4. D−35-\frac{3}{5}

Correct answer

C. −15-\frac{1}{5}

Step-by-step solution

Let roots of 4x2−(5k+1)x+5k=04 x^2-(5 k+1) x+5 k=0 are α\alpha and β\beta. α−β=1( Given ),α+β=5k+14,αβ=5k4\alpha-\beta=1(\text{ Given }), \alpha+\beta=\frac{5 k+1}{4}, \alpha \beta=\frac{5 k}{4} Now, (α−β)2=(α+β)2−4αβ(\alpha-\beta)^2=(\alpha+\beta)^2-4 \alpha \beta. ⇒1=(5k+14)2−4⋅5k4⇒25k2−70k−15=0⇒(k−3)(25k+5)=0⇒k=3,k=−15\begin{aligned} & \Rightarrow 1=\left(\frac{5 k+1}{4}\right)^2-4 \cdot \frac{5 k}{4} \\ & \Rightarrow 25 k^2-70 k-15=0 \\ & \Rightarrow(k-3)(25 k+5)=0 \\ & \Rightarrow k=3, k=-\frac{1}{5} \end{aligned}
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