Friday, February 3, 2023

### The probability that a ticketless traveler is caught during a trip...

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Question:The probability that a ticketless traveler is caught during a trip is 0.1. If the traveler makes 4 trips , the probability that...

### Linear Transformation Part II – Inverse Linear Transformation and Isomorphism

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Definition: A function $T$ from $V$ into $W$ is called invertible if there exists a function $S$ from $W$ to $V$ such that $TS$ is...

### Cayley Hamilton Theorem

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Statement:   Let T be a linear operator on a finite dimensional vector space V. If  f is the characteristic polynomial for T, Then $f(T)=0$, i.e....

### Out of 7 consonants and 4 vowels, how many words of 3 consonants and...

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Question: Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed? A. 25200               B....

### Two circles $k_1$ and $k_2$ intersect in such a way that outside their intersection...

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Question:Two circles $k_1$ and $k_2$ intersect in such a way that outside their intersection is 10% of the area of the circle $k_1$ and...

### Linear Functionals, Annihilator and Double Dual

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Definition:If $V$ is a vector space over the field $F$ and $S$ be a subset of $V$, the annihilator of $S$ is $S^0$...

### A metric space (X, d) is connected iff every continuous function is constant

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Let (X, d) be a connected metric space and suppose that there exists a continuous function $f:X\to {\lbrace0, 1\rbrace}$. Claim: f is constant.

### Three boys go to the same school. Every boy needs two cubes of the...

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Question:Three boys go to the same school. Every boy needs two cubes of the same color to play a game, but the cubes...

### Every continuous function f defined on ([a, b]) is Riemann integrable

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Since, every function defined on closed and bounded interval is uniformly continuous function on [a, b], $\therefore$ By definition of uniformly continuous function,

### A bounded function ​( f:[a, b]to mathbb{R} )​ is Riemann integrable iff for every...

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Suppose f is Riemann integrable on [a, b]. Let $\epsilon>0$ be arbitrary. $\int\limits_\underline{a}^bf(x)dx=\int\limits_a^\underline{b}f(x)dx$     ..... (1)

### Three boys go to the same school. Every boy needs two cubes of the same color to play a game, but the cubes that...

Question:Three boys go to the same school. Every boy needs two cubes of the same color to play a game, but the cubes...

### The product of 2 numbers is 1575 and their quotient is $frac{9}{7}$. Then the sum of the numbers is

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### Any two closed subset of metric space are connected iff they are disjoint

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