Find whether the following mathematical sentences are constants or variables:
(a) 'a' represents the number of districts in Nepal.
(b) 'y' represents the districts of province 2.
(a) '2x' where 'x' represents the counting numbers less than 5.
(b) 'm' represents the whole numbers between 5 and 7.
(a) The values of 'b' are 3, 6, 9 and 12.
(b) If x + y = 7, find the possible values of x and y. Are they variables? [They are natural numbers]
(a) 'N' denotes the natural numbers less than 2.
(b) 'W' denotes the whole numbers less than 2.
(a) If x - y = 4 and x = {8, 7, 6}, find y. Are they constants or variables?
(b) 3x + 2, where x represents the prime numbers less than 4.
(a) 'X' denotes the number of students in your class.
(b) 4x-3, where x represents the odd numbers less than 5.
(a) 'Y' represents the major rivers of Nepal.
(b) The values of m are 4, 5, 6 and 7.
(a) 'P' represents the prime numbers between 6 and 16.
(b) 'b' represents the games which start from the letter b.
[tex]m^2 - n^2[/tex] where m = 4 and n = -2
'm' represents the ministry of Nepal.
'L' represents the Lakes of Chitwan district.
'd' represents the multiples of 6 less than 30.
Answer (2)
In mathematics, constants and variables are key components. Let's examine each provided statement to determine whether they represent constants or variables.
'a' represents the number of districts in Nepal.
As of recent data, Nepal has a fixed number of districts, which is 77. Hence, 'a' is a constant because it represents a specific, unchanging value.
'y' represents the districts of province 2.
This refers to a specific set of districts, making 'y' a constant , assuming the number and identity of districts don't change.
'2x' where 'x' represents the counting numbers less than 5.
Here, 'x' can take any of the values 1, 2, 3, or 4 (counting numbers less than 5). Because 'x' can vary, it is a variable .
'm' represents the whole numbers between 5 and 7.
The whole numbers between 5 and 7 are 6. So 'm' is a constant , as it has a specific value.
The values of 'b' are 3, 6, 9, and 12.
'b' can be any of these values, thus 'b' is a variable .
If x + y = 7, find the possible values of x and y. Are they variables? [They are natural numbers]
Possible pairs for (x, y) as natural numbers could be (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1). Both x and y are variables as they can take multiple values.
'N' denotes the natural numbers less than 2.
The natural number less than 2 is 1. Therefore, 'N' is a constant .
'W' denotes the whole numbers less than 2.
Whole numbers less than 2 are 0 and 1. So 'W' can represent multiple values, making it a variable .
If x - y = 4 and x = {8, 7, 6}, find y. Are they constants or variables?
Solving for y: if x is 8, y = 4; if x is 7, y = 3; if x is 6, y = 2. Each solution for y corresponds to a fixed x, indicating that each result of y is a constant in the context of each x.
3x + 2, where x represents the prime numbers less than 4.
* Prime numbers less than 4 are 2 and 3. So 'x' can take values 2 or 3, making it a **variable**.
'X' denotes the number of students in your class.
* This depends on the specific context and census of the class, so 'X' is a **variable**.
4x-3, where x represents the odd numbers less than 5.
* Odd numbers less than 5 are 1 and 3. Thus, 'x' can vary, making it a **variable**.
'Y' represents the major rivers of Nepal.
* This refers to a specific set of entities, so 'Y' is a **constant**.
The values of m are 4, 5, 6 and 7.
* 'm' can be any of these values, thereby making 'm' a **variable**.
'P' represents the prime numbers between 6 and 16.
* Prime numbers in this range are 7, 11, 13. So 'P' is a **variable**.
'b' represents the games which start from the letter b.
* As 'b' can vary depending on which game is being referred to, 'b' is a **variable**.
m^2 - n^2 where m = 4 and n = -2
* Substituting m and n gives a specific value. Therefore, this expression results in a **constant**.
'm' represents the ministry of Nepal.
* This typically refers to specific entities, so 'm' can be considered a **constant**.
'L' represents the Lakes of Chitwan district.
* This also refers to specific entities and is a **constant**.
'd' represents the multiples of 6 less than 30.
* Multiple values like 6, 12, 18, and 24 suggest that 'd' is a **variable**.
In summary, whether a mathematical term is a constant or a variable depends on its ability to assume multiple values or remain fixed. Variables can take on a range of values, while constants refer to single, definite values.
In the provided statements, constants are numerical representations that remain unchanged, while variables can assume multiple values. For example, 'a' is a constant (77 districts in Nepal), while 'x' in '2x' is a variable as it can be 1, 2, 3, or 4. Understanding the difference helps with mathematical concepts involving algebra and equations.
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