Let \(S={z∈C:|z−2|≤1,and(1+i)+\overline{z}(1−i)≤2}. \)Let \(|z−4i|\) attains minimum and maximum values, respectively, at z1∈s and z2∈s. If \(5(|z_1|^2+|z_2|^2=α+β√5\) where α and β are integers, then the value of α+β is equal to _________.
Correct Answer: 26
Solution and Explanation
The correct answer is: 26
S represents the shaded region shown in the diagram.
Clearly, z1 will be the point of intersection of PA and given circle.
PA : 2x + y = 4 and given circle has equation
(x – 2)2 + y2 = 1.
On solving, we get
\(z_1=()+\frac{2}{5}i=|z_1|^2=5-\frac{4}{\sqrt5}\)
z2 will be either B or C.
So
\(5(|z_1|^2+|z_2|^2=30-4√5\)
Clearly α = 30 and β = –4 ⇒α + β = 26
Top Questions on Relations and functions
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is:
- JEE Main - 2025
- Mathematics
- Relations and functions
- Let $f: A \to B$ be defined by $f(x) = \frac{x - 2}{x - 3}$, where $A = \mathbb{R} - \{3\}$ and $B = \mathbb{R} - \{1\}$. Discuss the bijectivity of the function.
- CBSE CLASS XII - 2025
- Mathematics
- Relations and functions
- Let \( R \) be a relation defined on a set \( \mathbb{N} \) of natural numbers such that \( R = \{(x, y) : xy \text{ is a square of a natural number}, x, y \in \mathbb{N} \} \). Determine if the relation \( R \) is an equivalence relation.
- CBSE CLASS XII - 2025
- Mathematics
- Relations and functions
- Show that the function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = 4x^3 - 5 \), \( \forall x \in \mathbb{R} \), is one-one and onto.
- CBSE CLASS XII - 2025
- Mathematics
- Relations and functions
- If $f(x) = -2x^2$, then the correct statement is:
- CBSE CLASS XII - 2025
- Mathematics
- Relations and functions
Questions Asked in JEE Main exam
The velocity-time graph of an object moving along a straight line is shown in the figure. What is the distance covered by the object between \( t = 0 \) to \( t = 4s \)?
- JEE Main - 2025
- Kinetic Energy
- If the system of equations \[ (\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \] \[ \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \] \[ (\lambda + 1)x + (\lambda + 2)y - (\lambda + 2)z = 9 \] has infinitely many solutions, then \( \lambda^2 + \lambda \) is equal to:
- JEE Main - 2025
- Matrices and Determinants
A bob of mass \(m\) is suspended at a point \(O\) by a light string of length \(l\) and left to perform vertical motion (circular) as shown in the figure. Initially, by applying horizontal velocity \(v_0\) at the point ‘A’, the string becomes slack when the bob reaches at the point ‘D’. The ratio of the kinetic energy of the bob at the points B and C is:
- JEE Main - 2025
- Kinetic Energy
- The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is:
- JEE Main - 2025
- permutations and combinations
- A bar magnet has total length \( 2l = 20 \) units and the field point \( P \) is at a distance \( d = 10 \) units from the centre of the magnet. If the relative uncertainty of length measurement is 1%, then the uncertainty of the magnetic field at point P is:
- JEE Main - 2025
- Magnetic Field
JEE Main Notification
GMC Azamgarh Admission 2025: Dates, Courses, Fees, Eligibility, SelectJuly 22, 2025Concepts Used:
Relations
A relation in mathematics defines the relationship between two different sets of information. If two sets are considered, the relation between them will be established if there is a connection between the elements of two or more non-empty sets. Therefore, we can say, ‘A set of ordered pairs is defined as a relation.’
Read Also: Relation and Function
Types of Relations:
There are 8 main types of relations which are:
- Empty Relation - An empty relation is one in which there is no relation between any elements of a set.
- Universal Relation - A universal is a type of relation in which every element of a set is related to each other. Now one of the universal relations will be R = {x, y} where, |x – y| ≥ 0. For universal relation, R = A × A
- Identity Relation - In an identity relation, every element of a set is related to itself only. For example, in a set A = {a, b, c}, the identity relation will be I = {a, a}, {b, b}, {c, c}.
- Inverse Relation - It is seen when a set has elements which are inverse pairs of another set. For example if set A = {(a, b), (c, d)}, then inverse relation will be R-1 = {(b, a), (d, c)}.
- Reflexive Relation - If every element of set A maps for itself, then set A is known as a reflexive relation.It is represented as a∈ A, (a,a) ∈ R.
- Symmetric Relation - A relation R on a set A is known as asymmetric relation if (a, b) ∈R then (b, a) ∈R , such that for all a and b ∈A.
- Transitive Relation - For transitive relation, if (x, y) ∈ R, (y, z) ∈ R, then (x, z) ∈ R. For a transitive relation, aRb and bRc ⇒ aRc ∀ a, b, c ∈ A
- Equivalence Relation - If a relation is reflexive, symmetric and transitive at the same time it is known as an equivalence relation.
Representation of Relations:
There are two ways by which a relation can be represented-
- Roster method
- Set-builder method
The roster form and set-builder for for a set integers lying between -2 and 3 will be-
Roster form
I= {-1,0,1,2}
Set-builder form
I= {x:x∈I,-2<x<3}