Paragraph: Let α and β be the roots of the equation x^2-(1-2 a^2) x+(1-2 a^2)=0 Question: Under what condition does the above equation have real roots?

Correct answer: D. a^2 ≥ (1)/(2) Using a x^2+b x+c=0a=1, b=- (1-2 a²) \& c= (1-2 a²) For roots to be real, b^2-4 ac ≥0 aligned & [-(1-2 a^2) ]^2-4(1)(1-2 a^2) ≥0 & 4 a^4+4 a^2-3 ≥ 0 & (2 a^2-1)(2 a^2+3) ≥ 0 & a² ≥ (1)/(2) or a^2 ≤-(3)/(2) aligned

NDA 2016

Exam: NDA
Year: 2016
Subject: Mathematics
Chapter: Quadratic Equation

NDA 2016 · Mathematics · Quadratic Equation

Paragraph: Let α and β be the roots of the equation x^2-(1-2 a^2)...

Paragraph: Let

\alpha

and

\beta

be the roots of the equation

x^2-\left(1-2 a^2\right) x+\left(1-2 a^2\right)=0

Question: Under what condition does the above equation have real roots?

Options

A $\mathrm{a}^2 < \frac{1}{2}$
B $\mathrm{a}^2 > \frac{1}{2}$
C $\mathrm{a}^2 \leq \frac{1}{2}$
D $\mathrm{a}^2 \geq \frac{1}{2}$

Correct answer

D. $\mathrm{a}^2 \geq \frac{1}{2}$

Step-by-step solution

Using

a x^2+b x+c=0a=1, b=-

\left(1-2 a²\right) \& c=

\left(1-2 a²\right)

For roots to be real,

\mathrm{b}^2-4 \mathrm{ac} \geq0

\begin{aligned} & \Rightarrow \quad\left[-\left(1-2 \mathrm{a}^2\right)\rright]^2-4(1)\left(1-2 \mathrm{a}^2\right) \geq0 \\ & \Rightarrow \quad 4 \mathrm{a}^4+4 \mathrm{a}^2-3 \geq 0 \\ & \Rightarrow \quad\left(2 \mathrm{a}^2-1\right)\left(2 \mathrm{a}^2+3\right) \geq 0 \\ & \Rightarrow \quad a² \geq \frac{1}{2} \text{ or }a^2 \leq-\frac{3}{2}\end{aligned}

Practice Quadratic Equation on Quantrex Academy →

More from Quadratic Equation

For a given k , what is the minimum value of x^2+kx+k^2 ? NDA 2026 If the highest degree coefficient is equal to 1 , then what is the total number of… NDA 2026 Let α and β be the roots of the quadratic equation x^2-2bx+c^2=0 where b , c are positive… NDA 2026 Consider the following statements : I. x + x+1=0 has two irrational roots. II. 5 x -x-4=0… NDA 2026 For the following two (02) items: Let f(x) = ax^2 + bx + c be a quadratic polynomial such… NDA 2025 For the following two (02) items: Let α and β be the roots of the quadratic equation x^2… NDA 2025 For the following two (02) items: Let f(x) = ax^2 + bx + c be a quadratic polynomial such… NDA 2025 If one root of the equation x^2 - kx + k = 0 exceeds the other by 2 3 , then which one of… NDA 2025 For the following two (02) items: Let α and β be the roots of the quadratic equation x^2… NDA 2025 If x = (1111)_2 , y = (1001)_2 and z = (110)_2 , then what is x^3 - y^3 - z^3 - 3xyz… NDA 2025 If x^2 - x + 1 = 0 , then what is (x - (1)/(x))^2 + (x - (1)/(x))^4 + (x - (1)/(x))^8… NDA 2025 If x + (5)/(y) = 4 and y + (5)/(x) = -4 , then what is (x + y) equal to? NDA 2025 For the following two (02) items: Consider the equation abx^2 + bcx + ca = cax^2 + abx +… NDA 2025 For the following two (02) items: Consider the equation abx^2 + bcx + ca = cax^2 + abx +… NDA 2025 If n is a root of the equation x^2+p x+m=0 and m is a root of the equation x^2+p x+n=0… NDA 2024 The roots of the equation 7 x^2-6 x+1=0 are α and β , where 2 α and 2 β are the angles of… NDA 2024 Full Quadratic Equation list Quadratic Equation JEE Maths PYQs JEE Mathematics PYQs All NDA PYQs