Question:
Let a, b ∈ R be such that the equation ax2 – 2bx + 15 = 0 has a repeated root α. If α and β are the roots of the equation x2 – 2bx + 21 = 0, then α2 + β2 is equal to
Let a, b ∈ R be such that the equation ax2 – 2bx + 15 = 0 has a repeated root α. If α and β are the roots of the equation x2 – 2bx + 21 = 0, then α2 + β2 is equal to
Updated On: May 18, 2025
- 37
- 58
- 68
- 92
Hide Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
The correct option is(B): 58.
ax2 – 2bx + 15 = 0 has repeated root so b2 = 15a
and
\(α=\frac{15}{b}\)
∵α is a root of x2 – 2bx + 21 = 0
So
\(\frac{225}{b^2}=9⇒b^2=25\)
Now α2 + β2 = (α + β)2 – 2αβ = 4b2 – 42 = 100 – 42
= 58
Was this answer helpful?
7
9
Top Questions on Differential equations
- Solve the differential equation: \[ x \cos\left(\frac{y}{x}\right) \frac{dy}{dx} = y \cos\left(\frac{y}{x}\right) + x \]
- CBSE CLASS XII - 2025
- Mathematics
- Differential equations
- Find the particular solution of the differential equation \( \frac{dy}{dx} - \frac{y}{x} + \csc\left(\frac{y}{x}\right) = 0 \); given that \( y = 0 \), when \( x = 1 \).
- CBSE CLASS XII - 2025
- Mathematics
- Differential equations
- The sum of the order and degree of the differential equation \[ \left( 1 + \left( \frac{dy}{dx} \right)^2 \right) \frac{d^2y}{dx^2} = \left( \frac{dy}{dx} \right)^3 \] is:
- CBSE CLASS XII - 2025
- Mathematics
- Differential equations
- The integrating factor of the differential equation \( \frac{dx}{dy} = \frac{x \log x}{2 \log x - y} \) is:
- CBSE CLASS XII - 2025
- Mathematics
- Differential equations
- The sum of the order and degree of the differential equation \[ \left( 1 + \left( \frac{dy}{dx} \right)^2 \right) \frac{d^2y}{dx^2} = \left( \frac{dy}{dx} \right)^3 \] is:
- CBSE CLASS XII - 2025
- Mathematics
- Differential equations
View More Questions
Questions Asked in JEE Main exam
- CrCl\(_3\).xNH\(_3\) can exist as a complex. 0.1 molal aqueous solution of this complex shows a depression in freezing point of 0.558°C. Assuming 100\% ionization of this complex and coordination number of Cr is 6, the complex will be:
- JEE Main - 2025
- Colligative Properties
- Suppose that the number of terms in an A.P. is \( 2k, k \in \mathbb{N} \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55, and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to:
- JEE Main - 2025
- Arithmetic Progression
- If \( \alpha + i\beta \) and \( \gamma + i\delta \) are the roots of the equation \( x^2 - (3-2i)x - (2i-2) = 0 \), \( i = \sqrt{-1} \), then \( \alpha\gamma + \beta\delta \) is equal to:
- JEE Main - 2025
- Complex numbers
- Let \( [x] \) denote the greatest integer function, and let \( m \) and \( n \) respectively be the numbers of the points, where the function \( f(x) = [x] + |x - 2| \), \( -2<x<3 \), is not continuous and not differentiable. Then \( m + n \) is equal to:
- JEE Main - 2025
- Differential Equations
- Let \( f(x) = \frac{x - 5}{x^2 - 3x + 2} \). If the range of \( f(x) \) is \( (-\infty, \alpha) \cup (\beta, \infty) \), then \( \alpha^2 + \beta^2 \) equals to.
- JEE Main - 2025
- Functions
View More Questions
Concepts Used:
Root
A Root refers to the vascular part of any plant which is pinpointed or discovered at the lowest level and is entirely responsible for connecting the plant with the soil. It absorbs water (H20), minerals, nutrients, etc., and further feeds the plant for its development. Roots are also responsible for acting as the repository for food and nutrients of the plant.
Read More: Plant Roots
Characteristics of Roots:
- Roots are homogeneous in nature due to their potential to work towards producing similar organs like secondary or tertiary roots from the pericycle.
- Roots can be examined as positively Geotropic and Hydrotropic but negatively Phototropic.
Functions of Roots:
- The roots are in charge of providing a systematic connection to the different parts of the plant.
- It is also accountable for the storage of the reserved food material and the synthesis for the growth of the plant.