The number of solutions \((x, y, z)\) to the equation \(x\ –\ y \ –\ z = 25\), where x, y, and z are positive integers such that \(x ≤ 40, y ≤ 12\), and \(z ≤ 12\) is
- 101
- 99
- 87
- 105
The Correct Option is B
Solution and Explanation
The equation \(x−y−z=25\) can be expressed as \(x=25+y+z. \)
Given that y and z are positive integers with \(y≤12\) and \(z≤12\), the range for \(y+z\) is \(2≤(x+y)≤15\) when \(27≤x≤40. \)
The minimum value for x is 27.
For \(y=1, z\) can take 12 values.
Similarly, for \(y=2, z\) can take 12 values, and so on, until \(y=12\) where z can take 10 values.
Therefore, the total number of solutions is \(3+4+5+6+7+8+9+10+11+12+12+12=99. \)
Hence, the required result is \(99.\)
Top Questions on Number Systems
- Let \[ S = \{(m,n) : m,n \in \{1,2,3,\ldots,50\}\}. \] If the number of elements \((m,n)\) in \(S\) such that \(6^m + 9^n\) is a multiple of \(5\) is \(p\), and the number of elements \((m,n)\) in \(S\) such that \(m+n\) is a square of a prime number is \(q\), then \(p + q\) is equal to
- JEE Main - 2026
- Mathematics
- Number Systems
- Let $S$ be a set of $5$ elements and $P(S)$ denote the power set of $S$. Let $E$ be the event of choosing an ordered pair $(A,B)$ from $P(S)\times P(S)$ such that $A\cap B=\varnothing$. If the probability of the event $E$ is $\dfrac{3^p}{2^q}$, where $p,q\in\mathbb{N}$, then $p+q$ is equal to
- JEE Main - 2026
- Mathematics
- Number Systems
- The largest \( n \in \mathbb{N} \), for which \( 7^n \) divides \( 101! \), is :
- JEE Main - 2026
- Mathematics
- Number Systems
- The largest value of $n$, for which $40^n$ divides $60!$, is
- JEE Main - 2026
- Mathematics
- Number Systems
- \[ x - ny + z = 6 \\ x - (n - 2)y + (n + 1)z = 8 \\ (n - 1)y + z = 1 \] Let \( n \) be the number on the die when rolled randomly. Then \( P \) (that system equation has a unique solution) = \( \frac{k}{6} \). Then sum of value of \( k \) and all possible values of \( n \) is:
- JEE Main - 2026
- Mathematics
- Number Systems
Questions Asked in CAT exam
- The passage given below is followed by four summaries. Choose the option that best captures the essence of the passage.
In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
Cultural appropriation occurs when elements from a particular culture are borrowed without proper understanding, respect, or acknowledgment. This leads to the commodification of sacred symbols, the reinforcement of stereotypes, and the erasure of the cultural context from which these elements originated. It is essential to recognize that the impact of cultural appropriation extends beyond the realm of artistic expression, influencing societal perceptions and perpetuating power imbalances.- CAT - 2025
- Para Summary
- The number of distinct integers $n$ for which $\log_{\left(\frac14\right)}(n^2 - 7n + 11)>0$ is:
- CAT - 2025
- Linear Inequalities
For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which \[ (a_1)^1 \times (a_2)^2 \times \dots \times (a_{20})^{20} \;<\; a_{21} \times a_{22} \times \dots \times a_{20+m} \] is:
- CAT - 2025
- Linear Inequalities
- If $a - 6b + 6c = 4$ and $6a + 3b - 3c = 50$, where $a, b$ and $c$ are real numbers, the value of $2a + 3b - 3c$ is:
- CAT - 2025
- Linear & Quadratic Equations
The given sentence is missing in the paragraph below. Decide where it best fits among the options 1, 2, 3, or 4 indicated in the paragraph.
Sentence: While taste is related to judgment, with thinkers at the time often writing, for example, about “judgments of taste” or using the two terms interchangeably, taste retains a vital link to pleasure, embodiment, and personal specificity that is too often elided in post-Kantian ideas about judgment—a link that Arendt herself was working to restore.
Paragraph: \(\underline{(1)}\) Denneny focused on taste rather than judgment in order to highlight what he believed was a crucial but neglected historical change. \(\underline{(2)}\) Over the course of the seventeenth century and early eighteenth century, across Western Europe, the word taste took on a new extension of meaning, no longer referring specifically to gustatory sensation and the delights of the palate but becoming, for a time, one of the central categories for aesthetic—and ethical—thinking. \(\underline{(3)}\) Tracing the history of taste in Spanish, French, and British aesthetic theory, as Denneny did, also provides a means to recover the compelling and relevant writing of a set of thinkers who have been largely neglected by professional philosophy. \(\underline{(4)}\)
- CAT - 2025
- Para Completion