List-1 List-II Types of Interval for variable x Mathematical Representation
(Given a, b $\in$ R with b > a)(A) Closed Interval (I) [a,b) = {x$\in$R:a $\le$ x < b} (B) Open Interval (II) [a, b] = {x$\in$R: a $\le$ x $\le$ b} (C) Unbounded Interval (III) [$\alpha$,$\infty$) = {x$\in$R:a $\le$ x} (D) Half Open Interval (IV) (a, b) = {x$\in$R: a < x < b}
Understanding Real Number Interval Types
This question involves matching different categories of real number intervals with their specific mathematical representations. Intervals are crucial for defining ranges of values on the number line, commonly used in algebra and calculus.
Defining Key Interval Types
Let's break down each type of interval presented in List-I:
- (A) Closed Interval: This type of interval includes all real numbers between two specified endpoints, including the endpoints themselves.
- (B) Open Interval: An open interval consists of all real numbers between two specified endpoints, but it excludes both endpoints.
- (C) Unbounded Interval: An unbounded interval represents a set of real numbers that extends infinitely in one or both directions. It does not have finite upper or lower bounds.
- (D) Half Open Interval: Also known as a half-closed interval, this type includes one endpoint but excludes the other.
Mathematical Representations of Intervals
Here are the mathematical representations given in List-II:
- (I) [a,b) = {x$\in$R:a $\le$ x < b}: This notation defines a set of real numbers 'x' such that 'x' is greater than or equal to 'a' AND strictly less than 'b'. This describes a half-open interval.
- (II) [a, b] = {x$\in$R: a $\le$ x $\le$ b}: This notation represents all real numbers 'x' where 'x' is greater than or equal to 'a' AND less than or equal to 'b'. This is the definition of a closed interval.
- (III) [$\alpha$,$\infty$) = {x$\in$R:a $\le$ x}: This representation shows an interval starting at 'a' (or '$\\alpha$' as noted in the interval notation, representing the starting point) and extending towards positive infinity. The condition $a \le x$ means 'x' is greater than or equal to 'a'. This clearly defines an unbounded interval.
- (IV) (a, b) = {x$\in$R: a < x < b}: This notation signifies all real numbers 'x' that are strictly greater than 'a' AND strictly less than 'b'. This is characteristic of an open interval.
Matching Interval Types with Representations
We can now match each type from List-I to its corresponding mathematical representation from List-II:
| List-I (Type of Interval) | List-II (Mathematical Representation) | Reasoning |
|---|---|---|
| (A) Closed Interval | (II) [a, b] = {x$\in$R: a $\le$ x $\le$ b} | The square brackets [a, b] and the condition $a \le x \le b$ explicitly include both endpoints, defining a closed interval. |
| (B) Open Interval | (IV) (a, b) = {x$\in$R: a < x < b} | The parentheses (a, b) and the condition $a < x < b$ indicate that neither endpoint is included, which is the definition of an open interval. |
| (C) Unbounded Interval | (III) [$\alpha$,$\infty$) = {x$\in$R:a $\le$ x} | The notation involving infinity ($\\infty$) signifies an interval that does not have an upper bound. The starting point $a \le x$ confirms it's an unbounded interval. |
| (D) Half Open Interval | (I) [a,b) = {x$\in$R:a $\le$ x < b} | The mix of square bracket [a and parenthesis b) along with the condition $a \le x < b$ shows that 'a' is included while 'b' is excluded, characterising a half-open interval. |
Identifying the Correct Option
Based on the matching process:
- (A) Closed Interval matches with (II)
- (B) Open Interval matches with (IV)
- (C) Unbounded Interval matches with (III)
- (D) Half Open Interval matches with (I)
Therefore, the correct combination is (A)-(II), (B)-(IV), (C)-(III), (D)-(I).
Important Questions from Algebra (Notes)
- What is the remainder when 2023²⁰²⁴ + 2025²⁰²⁴ is divided by 2024?
In an examination, a student scores 4 marks for every correct answer and loses 1 mark for every wrong answer. If she/he attempts all 60 questions and secures 130 marks, the number of questions she/he attempts wrongly, are?
Match List-I with List-II
List-1 List-II (A) If $\begin{bmatrix}\lambda-1 & 0 \\ 0 & \lambda-1 \end{bmatrix} $, then $\lambda$ is (I) 0 (B) If A=$ \begin{bmatrix}1 & 2 \\2 & 4 \end{bmatrix} $, then $\Delta$ is (II) 1 (C) If A = $ \begin{bmatrix}1 & 0 \\0 & \frac{1}{2} \end{bmatrix} $, then $|A^{-1}|$ is (III) -2 (D) If $ \begin{bmatrix}a+1 & 1 \\1 & 2 \end{bmatrix} = \begin{bmatrix}-1 & 1 \\1 & 2 \end{bmatrix} $, then a is (IV) 2 Choose the correct answer from the options given below:
- If (x - 1) is a factor of $2x^2 - 5x + k = 0$, then the value of k is:
- If $x = (2+\sqrt{3})^{\frac{1}{3}} + (2+\sqrt{3})^{-\frac{1}{3}}$ and $x^3-3x + k = 0$, then the value of k is:
