(B) Let $S = \sum_{k=0}^{99} \left[ \frac{1}{2} + \frac{k}{100} \right]$.For $0 \le k \le 49$,we have $0 \le \frac{k}{100} \le 0.49$,so $\frac{1}{2} +

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(B) Let $S = \sum_{k=0}^{99} \left[ \frac{1}{2} + \frac{k}{100} \right]$.For $0 \le k \le 49$,we have $0 \le \frac{k}{100} \le 0.49$,so $\frac{1}{2} + \frac{k}{100} = 0.5 + \frac{k}{100} For $50 \le k \le 99$,we have $0.50 \le \frac{k}{100} \le 0.99$,so $1 \le \frac{1}{2} + \frac{k}{100} The number of terms from $k = 50$ to $k = 99$ is $99 - 50 + 1 = 50$.Therefore,the sum is $0 \times 50 + 1 \times 50 = 50$.

Class 11 Mathematics Set Theory Mix Examples of Set Theory

Medium

For a real number $x$,$[x]$ denotes the greatest integer less than or equal to $x$. The value of $\left[ \frac{1}{2} \right] + \left[ \frac{1}{2} + \frac{1}{100} \right] + \left[ \frac{1}{2} + \frac{2}{100} \right] + \dots + \left[ \frac{1}{2} + \frac{99}{100} \right]$ is

IIT 1994 All IIT

A
$49$
B
$50$
C
$48$
D
$51$

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Set Theory

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Mix Examples of Set Theory

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IIT 1994 Paper All IIT years →

Let $A = \{x \in (0, \pi) - \{\frac{\pi}{2}\} : \log_{(2/\pi)}|\sin x| + \log_{(2/\pi)}|\cos x| = 2\}$ and $B = \{x \geq 0 : \sqrt{x}(\sqrt{x} - 4) - 3|\sqrt{x} - 2| + 6 = 0\}$. Then $n(A \cup B)$ is equal to:

JEE MAIN 2025

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Find the simplest form of the following switching circuit.

DifficultMHT CET 2026

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If $U$ is the universal set and $A \cup B \cup C = U$,then ${(A - B) \cup (B - C) \cup (C - A)}'$ is equal to:

Medium

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$A$ two-digit number $\overline{ab}$ is called almost prime if one obtains a two-digit prime number by changing at most one of its digits $a$ or $b$. (For example,$18$ is an almost prime number because $13$ is a prime number). The number of almost prime two-digit numbers is:

KVPY 2019

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Let $S = \{ x \in \mathbb{R} : x \ge 0 \text{ and } 2|\sqrt{x} - 3| + \sqrt{x}(\sqrt{x} - 6) + 6 = 0 \}$. Then $S$:

DifficultJEE MAIN 2018

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For a real number $x$,$[x]$ denotes the greatest integer less than or equal to $x$. The value of $\left[ \frac{1}{2} \right] + \left[ \frac{1}{2} + \frac{1}{100} \right] + \left[ \frac{1}{2} + \frac{2}{100} \right] + \dots + \left[ \frac{1}{2} + \frac{99}{100} \right]$ is

Similar Questions

Let $A = \{x \in (0, \pi) - \{\frac{\pi}{2}\} : \log_{(2/\pi)}|\sin x| + \log_{(2/\pi)}|\cos x| = 2\}$ and $B = \{x \geq 0 : \sqrt{x}(\sqrt{x} - 4) - 3|\sqrt{x} - 2| + 6 = 0\}$. Then $n(A \cup B)$ is equal to:

Find the simplest form of the following switching circuit.

If $U$ is the universal set and $A \cup B \cup C = U$,then ${(A - B) \cup (B - C) \cup (C - A)}'$ is equal to:

$A$ two-digit number $\overline{ab}$ is called almost prime if one obtains a two-digit prime number by changing at most one of its digits $a$ or $b$. (For example,$18$ is an almost prime number because $13$ is a prime number). The number of almost prime two-digit numbers is:

Let $S = \{ x \in \mathbb{R} : x \ge 0 \text{ and } 2|\sqrt{x} - 3| + \sqrt{x}(\sqrt{x} - 6) + 6 = 0 \}$. Then $S$:

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