Let $M$ be a measurable space (a sigma algebra of measurable sets) then a function $f:E\longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ on $\varnothing \neq E\in M$ given as $\left\{ x\in E:f(x)>a\right\} $ is measurable if $\forall a,$----------under $f$ is measurable set.

• A) image of $\left( x,\infty \right) $
• B) inverse image of $\left( a,\infty \right) $
• C) image of $\left( a,\infty \right) $
• D) inverse image of $\left( a,\infty \right) $

{\text{Which of the following sets is a Borel set?}}

• A) {F_\sigma }{\text{ set}}
• B) {\text{All of these}}
• C) {\text{The set of rational numbers}}
• D) {G_\delta }{\text{ set}}

\[{\text{Which of the following sets is countable and also a Borel set with measure zero?}}\]

• A) \[{\text{The interval [0,1]}}\]
• B) \[{\text{The set of natural numbers}}\]
• C) \[{\text{The set of irrational numbers}}\]
• D) \[{\text{The set of real numbers}}\]

Let $x,y\in \left[ 0,1\right) ,$then the sum modulo $1;x\widehat{+}y$ of $\ x $ and $\ y$ is defined by_________.

• A) $x+y$ if $x+y\geq 1$ and $x+y-1$ if $x+y<1$
• B) $x+y$ if $x+y\leq 1$ and $x+y-1$ if $x+y>1$
• C) $x+y$ if $x+y<1$ and $x+y-1$ if $x+y\geq 1$
• D) $x+y-1$

\[{\text{Which of the following best describes a measurable function?}}\]

• A) \[{\text{A function that maps Borel sets to Lebesgue measurable sets}}{\text{.}}\]
• B) \[{\text{A function that can be represented as the union of countably many open sets}}{\text{.}}\]
• C) \[{\text{A function that maps measurable sets to measurable sets}}{\text{.}}\]
• D) \[{\text{A function that preserves the order of elements in its domain}}{\text{.}}\]

\[{\text{A singleton set }}\{ x\} {\text{ can be written as:}}\]

• A) \[\bigcup\limits_{n = 1}^\infty {\left( {\frac{1}{n},{\text{ }}\frac{1}{n}} \right)} \]
• B) \[\bigcup\limits_{n = 1}^\infty {\left( {x - \frac{1}{n},{\text{ }}x + \frac{1}{n}} \right)} \]
• C) \[\bigcap\limits_{n = 1}^\infty {\left( { - \frac{1}{n},{\text{ }}\frac{1}{n}} \right)} \]
• D) \[\bigcap\limits_{n = 1}^\infty {\left( {x - \frac{1}{n},{\text{ }}x + \frac{1}{n}} \right)} \]

\[{\text{Let }}f:\mathbb{R} \to \mathbb{R}{\text{ defined by }}f(x) = 9.{\text{ The set }}\{ x \in \mathbb{R}:f(x) > 7\} {\text{ is equal to which of the following sets?}}\]

• A) \[\emptyset \]
• B) \[\mathbb{R}\]
• C) \[\left( {7,\infty } \right)\]
• D) \[\left( { - \infty ,7} \right)\]

The common ratio of the sequence -7, -14, -28, … is ____________.

• A) -2
• B) -7
• C) 7
• D) 2

The $n^{th}$-term of a geometric sequence with common ratio $r$ is given by:

• A) $a_n = a_1 r^{\,n-1}$
• B) $a_n = a_1 r^{\,n}$
• C) $a_n = n a_1 r$
• D) $a_n = n a_1 r^{\,n}$

A geometric sequence is characterized by the presence of which of the following properties?

• A) None of the given choices
• B) A common ratio
• C) Sum of the terms
• D) A common difference

The general notation $a_n$ for a sequence represents the __________.

• A) sum of all the terms
• B) $n^{th}$ term of the sequence
• C) number of terms in the sequence
• D) none of them

An infinite series is best defined as:

• A) A sequence with infinitely many terms
• B) A function defined on the real numbers
• C) The limit of a sequence
• D) The sum of the terms of a sequence

0, 2, 4, 8, ... is a/an ____________ sequence.

• A) harmonic
• B) alternative
• C) geometric
• D) arithmetic

If a function $f(x)$ is continuous over the interval [a,b), then $\int_a^b f(x)\,dx =$_____________.

• A) $\lim_{t \to a^+} \int_t^b f(x)\,dx$
• B) $\lim_{t \to a^-} \int_t^b f(x)\,dx$
• C) $\lim_{t \to b^+} \int_a^t f(x)\,dx$
• D) $\lim_{t \to b^-} \int_a^t f(x)\,dx$

In a geometric sequence, the constant value obtained by dividing any term by its preceding term is called:

• A) Average rate
• B) Common ratio
• C) Fixed difference
• D) Growth factor

If $\lim_{t \to b^-} \int_a^t f(x)\, dx$ results in a finite real number, the integral is said to be _________.

• A) undefined
• B) proper integral
• C) divergent
• D) convergent

The fourth term of the sequence 1, 10, 19, ... is _________.

• A) 30
• B) 29
• C) 27
• D) 28

What happens to the sum of an absolutely convergent series if its terms are rearranged?

• A) The sum becomes zero.
• B) The sum changes to a different finite value.
• C) The sum remains the same.
• D) The series diverges.

Which of the following is a necessary condition for a series to be conditionally convergent?

• A) The series must diverge, but the series of absolute values must converge.
• B) The series must converge, but the series of absolute values must diverge.
• C) The series must have only positive terms.
• D) The terms must approach a non-zero constant.

All of the following are true for trade secrets except:

• A) It is protecting the secrecy itself
• B) The firm must take all possible measures to keep it secret
• C) It should not be known to others or general public
• D) It is the legal right to use the work of someone