Consider the information given below and answer the two items (02) that follow:
What is the number of students who are good in either Hindi or Mathematics but not in English?
125
Understanding the Student Data and the Problem
The problem provides data about students in a class based on their proficiency in three subjects: Hindi, Mathematics, and English. We are given the number of students good in only one subject, in two subjects, and in all three subjects.
The specific question asks for the number of students who are good in either Hindi or Mathematics but are explicitly not good in English. This type of problem can be effectively solved using concepts from set theory, often visualized with a Venn diagram.
Breaking Down the Given Information
Let's list the information provided:
- Students good in Hindi only: 54
- Students good in Mathematics only: 63
- Students good in English only: 41
- Students good in both Hindi and Mathematics: 18
- Students good in all three subjects (Hindi, Mathematics, and English): 10
Identifying the Target Group
We need to find the number of students who belong to the set (Hindi or Mathematics) AND (not English). In terms of set operations, this is equivalent to $(H \cup M) \setminus E$, where $H$ is the set of students good in Hindi, $M$ is the set of students good in Mathematics, and $E$ is the set of students good in English. The symbol $\cup$ denotes union (either/or) and $\setminus$ denotes set difference (excluding elements of another set).
The students who are good in either Hindi or Mathematics but not English include:
- Students good in Hindi only.
- Students good in Mathematics only.
- Students good in both Hindi and Mathematics, but not English.
Calculating Students Good in Hindi and Mathematics Only (Not English)
We are given that 18 students are good in both Hindi and Mathematics. This group of 18 includes students who are good in Hindi and Mathematics only, AND students who are good in Hindi, Mathematics, and English (all three). Since 10 students are good in all three subjects, the number of students good in Hindi and Mathematics only (meaning not in English) is the total number good in Hindi and Mathematics minus those good in all three.
Number of students good in Hindi and Mathematics only (not English) = (Total good in Hindi and Mathematics) - (Good in all three subjects)
Number of students good in Hindi and Mathematics only (not English) = $18 - 10 = 8$
Calculating the Total Number of Students in the Target Group
Now we can sum the numbers for the groups identified as being in (Hindi or Mathematics) and (not English):
- Students good in Hindi only: 54
- Students good in Mathematics only: 63
- Students good in Hindi and Mathematics only (not English): 8
Total number of students good in either Hindi or Mathematics but not in English = (Hindi only) + (Mathematics only) + (Hindi and Mathematics only)
Total number of students = $54 + 63 + 8$
Total number of students = $125$
Summary of Calculation
| Group Description | Number of Students |
|---|---|
| Students good in Hindi only | 54 |
| Students good in Mathematics only | 63 |
| Students good in Hindi and Mathematics only (not English) | 18 (Total Hindi & Math) - 10 (All three) = 8 |
| Total (Hindi or Mathematics but not English) | 54 + 63 + 8 = 125 |
Therefore, 125 students are good in either Hindi or Mathematics but not in English.
Revision Table: Key Student Groups
| Group | Description | Count | Included in Target (Hindi or Math, not English)? |
|---|---|---|---|
| Hindi Only | Good in Hindi, not Math, not English | 54 | Yes |
| Mathematics Only | Good in Math, not Hindi, not English | 63 | Yes |
| English Only | Good in English, not Hindi, not Math | 41 | No |
| Hindi & Mathematics Only | Good in Hindi & Math, not English | $18 - 10 = 8$ | Yes |
| Hindi & English Only | Good in Hindi & English, not Math (Data not provided) | - | No (good in English) |
| Mathematics & English Only | Good in Math & English, not Hindi (Data not provided) | - | No (good in English) |
| All Three Subjects | Good in Hindi, Math, & English | 10 | No (good in English) |
| Target Group Sum | Hindi or Math, but not English | $54 + 63 + 8 = 125$ | - |
Additional Information: Set Theory and Venn Diagrams
Problems involving overlapping groups of students based on subjects are classic examples of set theory applications. A Venn diagram is a visual tool that helps represent these sets and their intersections (students good in multiple subjects) and unions (students good in at least one subject). Each circle in the Venn diagram represents a subject (Hindi, Math, English), and the overlapping regions represent students good in combinations of these subjects.
The region representing "Hindi only" is the part of the Hindi circle that does not overlap with the Math or English circles. Similarly for "Mathematics only" and "English only". The region for "Hindi and Mathematics only" is the overlap between the Hindi and Math circles, excluding the part that also overlaps with the English circle (which is the "all three" region). The region for "all three" is the central overlap of all three circles.
In this problem, "either Hindi or Mathematics but not in English" corresponds to the union of the "Hindi only", "Mathematics only", and "Hindi and Mathematics only" regions in the Venn diagram.
Important Questions from Relations and Functions
Consider the following statements:
1. The relation f defined by \(f(x)= \begin{cases}x^3, & 0 \leq x \leq 2 \\ 4 x, & 2 \leq x \leq 8\end{cases}\) is a function.
2. The relation g defined by \(g(x)= \begin{cases}x^2, & 0 \leq x \leq 4 \\ 3 x, & 4 \leq x \leq 8\end{cases}\) is a function.
Which of the statements given above is/are correct?
A function satisfies \(f(x-y)=\frac{f(x)}{f(y)}\), where f(y) ≠ 0. If f(1) = 0.5, then what is f(2) + f(3) + f(4) + f(5) + f(6) equal to ?
A mapping f : A → B defined as \(f(x)=\frac{2 x+3}{3 x+5}, x \in A\) If f is to be onto, then what are A and B equal to ?
If f(x) = x(4x2 - 3), then what is f(sinθ) equal to ?
Let R be a relation on the set N of natural numbers defined by ‘nRm ⟺ n is a factor of m’. Then which one of the following is correct?
