(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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Mix Examples of Set Theory

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IIT 1994 Paper All IIT years →

If a set $A$ has $5$ elements,then the number of ways of selecting two subsets $P$ and $Q$ from $A$ such that $P$ and $Q$ are mutually disjoint,is

EasyAP EAMCET 2016

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Let $A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\}$ and $B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\}$. If $C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\}$,then $\sum_{(x, y) \in C} |x + y|$ is :

DifficultJEE MAIN 2025

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Let $A = \{ x \in R : [x + 3] + [x + 4] \leq 3 \}$ and $B = \{ x \in R : 3^x \left( \sum_{n=1}^{\infty} \frac{3}{10^n} \right)^{x-3} < 3^{-3x} \}$,where $[t]$ denotes the greatest integer function. Then,

DifficultJEE MAIN 2023

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If $P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$ is a polynomial such that $P(0) = 1, P(1) = 2, P(2) = 5, P(3) = 10$ and $P(4) = 17$,then $P(5) =$

MediumAP EAMCET 2024

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Consider the two sets: $A = \{m \in R : \text{both the roots of } x^{2} - (m+1)x + m+4 = 0 \text{ are real}\}$ and $B = [-3, 5)$. Which of the following is not true?

DifficultJEE MAIN 2020

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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

If a set $A$ has $5$ elements,then the number of ways of selecting two subsets $P$ and $Q$ from $A$ such that $P$ and $Q$ are mutually disjoint,is

Let $A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\}$ and $B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\}$. If $C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\}$,then $\sum_{(x, y) \in C} |x + y|$ is :

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If $P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$ is a polynomial such that $P(0) = 1, P(1) = 2, P(2) = 5, P(3) = 10$ and $P(4) = 17$,then $P(5) =$

Consider the two sets: $A = \{m \in R : \text{both the roots of } x^{2} - (m+1)x + m+4 = 0 \text{ are real}\}$ and $B = [-3, 5)$. Which of the following is not true?

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