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\lhead{\textcolor{StatBlue}{\textbf{Statstrive ExamIQ}}}
\rhead{ISI MSQE PEA 2022}
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\begin{document}

\begin{center}
{\Huge \textbf{Statstrive ExamIQ}}\\[6pt]
{\Large ISI MSQE PEA 2022 Question Paper}\\[4pt]
{\small Regenerated from the local solution TeX source. Items missing full source detail are marked Needs Review.}
\end{center}

\vspace{0.5cm}
\begin{longtable}{c l c}
\toprule
Question & Topic & Website Status \\
\midrule
\endhead
1 & Consumer Theory & Ready \\
2 & Consumer Theory & Ready \\
3 & Consumer Theory & Ready \\
4 & Production and Cost & Ready \\
5 & Market Structure & Ready \\
6 & Consumer Theory & Ready \\
7 & Consumer Theory & Ready \\
8 & Consumer Theory & Ready \\
9 & Optimization & Ready \\
10 & Production Theory & Ready \\
11 & Consumer Theory & Ready \\
12 & Market Structure & Ready \\
13 & Market Structure & Ready \\
14 & Macroeconomics & Ready \\
15 & Combinatorics & Ready \\
16 & Algebra / Functions & Ready \\
17 & Functions & Ready \\
18 & Calculus & Ready \\
19 & Calculus & Ready \\
20 & Linear Algebra & Ready \\
21 & Calculus & Ready \\
22 & Calculus & Ready \\
23 & Combinatorics & Ready \\
24 & Functions & Ready \\
25 & Linear Algebra & Ready \\
26 & Linear Algebra & Needs Review \\
27 & Linear Algebra & Ready \\
28 & Probability & Ready \\
29 & Probability & Ready \\
30 & Functions & Ready \\
\bottomrule
\end{longtable}

\newpage
\tableofcontents
\newpage

\section*{Question 1}

\addcontentsline{toc}{section}{Question 1}

\noindent\textbf{Topic:} Consumer Theory \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



Consider an economy with two goods $X$ and $Y$. Let the utility function be given by
$u(x,y) = A\sqrt{xy}$ where $A > 0$, $x \geq 0$ and $y \geq 0$. The budget constraint is
$P_X x + P_Y y \leq M$, with $M > 0$. Let $P_X = P_Y > 1$ and let $(x^*, y^*)$ denote the
utility maximizing bundle. Then,
\begin{enumerate}[label=(\Alph*)]
\item it must always be that $x^* > y^*$
\item it must always be that $x^* = y^*$
\item it must always be that $x^* < y^*$
\item it must always be that $x^* + y^* = M$
\end{enumerate}

\newpage

\section*{Question 2}

\addcontentsline{toc}{section}{Question 2}

\noindent\textbf{Topic:} Consumer Theory \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



For the linear utility $u(x_1,x_2) = 3x_1 + 2x_2$ with budget $2x_1 + 3x_2 \leq M$,
$x_1, x_2 \geq 0$, the Lagrangian is
\[
L(x_1,x_2;\lambda) = 3x_1 + 2x_2 + \lambda[M - 2x_1 - 3x_2].
\]
Determine the equilibrium $(x_1^*, x_2^*, \lambda^*)$.
\begin{enumerate}[label=(\Alph*)]
\item $(x_1^* = M/2,\ x_2^* = 0,\ \lambda^* = 3/2)$
\item $(x_1^* = 0,\ x_2^* = M/3,\ \lambda^* = 2/3)$
\item $(x_1^* = M/2,\ x_2^* = 0,\ \lambda^* = 2/3)$
\item $(x_1^* = 0,\ x_2^* = M/3,\ \lambda^* = 3/2)$
\end{enumerate}

\newpage

\section*{Question 3}

\addcontentsline{toc}{section}{Question 3}

\noindent\textbf{Topic:} Consumer Theory \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



In month 1: $p_X = 2, p_Y = 3$. Consumer $A$ chose $(3,8)$, consumer $B$ chose $(6,6)$.
In month 2: $p_X = 3, p_Y = 2$. Consumer $A$ chose $(8,3)$, consumer $B$ chose $(4,9)$.
Which statement is correct?
\begin{enumerate}[label=(\Alph*)]
\item Both consumers satisfy WARP
\item Neither consumer satisfies WARP
\item Consumer $A$ satisfies WARP but not consumer $B$
\item Consumer $B$ satisfies WARP but not consumer $A$
\end{enumerate}

\newpage

\section*{Question 4}

\addcontentsline{toc}{section}{Question 4}

\noindent\textbf{Topic:} Production and Cost \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



Production function: $Y(L,K) = \min\{2L, K\}$; cost $C = wL + rK$, $w,r>0$.
Find $(L^*, K^*)$ minimizing cost subject to $Y(L,K) \geq \bar Y$.
\begin{enumerate}[label=(\Alph*)]
\item $L^* = \bar Y$ and $K^* = \bar Y/2$
\item $L^* = \bar Y$ and $K^* = \bar Y$
\item $L^* = \bar Y/2$ and $K^* = \bar Y$
\item None of the other options is correct
\end{enumerate}

\newpage

\section*{Question 5}

\addcontentsline{toc}{section}{Question 5}

\noindent\textbf{Topic:} Market Structure \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



Inverse demand $P = 1 - q_1 - q_2$; cost $c_i(q_i) = \kappa_i q_i$, $\kappa_i \in (0,1)$.
Find firm 2's Cournot equilibrium profit.
\begin{enumerate}[label=(\Alph*)]
\item $(1 - \kappa_1 + \kappa_2)^2 / 9$
\item $(1 - \kappa_2 + \kappa_1)^2 / 9$  \quad [i.e.\ $(1 - 2\kappa_2 + \kappa_1)^2/9$ corrected below]
\item $(1 - 2\kappa_1 + \kappa_2)^2 / 9$
\item $(1 - 2\kappa_2 + \kappa_1)^2 / 9$
\end{enumerate}

\newpage

\section*{Question 6}

\addcontentsline{toc}{section}{Question 6}

\noindent\textbf{Topic:} Consumer Theory \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



Which statement is correct in a two-good world?
\begin{enumerate}[label=(\Alph*)]
\item Diminishing MU of both goods is sufficient for diminishing MRS
\item Diminishing MU of both goods is necessary for diminishing MRS
\item Diminishing MU of at least one good is necessary for diminishing MRS
\item Diminishing MU of at least one good is neither necessary nor sufficient for diminishing MRS
\end{enumerate}

\newpage

\section*{Question 7}

\addcontentsline{toc}{section}{Question 7}

\noindent\textbf{Topic:} Consumer Theory \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



A non-transitive preference relation can be represented by a utility function
\begin{enumerate}[label=(\Alph*)]
\item Always
\item Only if preferences are complete
\item Only if preferences are complete and convex
\item Never
\end{enumerate}

\newpage

\section*{Question 8}

\addcontentsline{toc}{section}{Question 8}

\noindent\textbf{Topic:} Consumer Theory \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



$U(x,y) = \min\{x,y\}$, income $M = 200$. Old prices $(p_x, p_y) = (2,2)$; new prices $(2,3)$.
Find the equivalent variation $A$ and compensating variation $B$.
\begin{enumerate}[label=(\Alph*)]
\item $A = 30, B = 70$
\item $A = 40, B = 50$
\item $A = 50, B = 75$
\item $A = 60, B = 60$
\end{enumerate}

\newpage

\section*{Question 9}

\addcontentsline{toc}{section}{Question 9}

\noindent\textbf{Topic:} Optimization \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



$U(x,y) = -\big[(10-x)^2 + (10-y)^2\big]$. All prices equal $1$, income $= 40$.
Optimal $(x,y)$?
\begin{enumerate}[label=(\Alph*)]
\item $(10,10)$
\item $(0,0)$
\item $(5,5)$
\item None of these
\end{enumerate}

\newpage

\section*{Question 10}

\addcontentsline{toc}{section}{Question 10}

\noindent\textbf{Topic:} Production Theory \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



$F(K,L) = \min\{aK, bL\}$, $a,b>0$, $a \neq b$. For fixed $\bar K$, the marginal product of labor is
\begin{enumerate}[label=(\Alph*)]
\item $0$
\item $1/a$ if $L < (a/b)\bar K$ and $0$ otherwise
\item $1/b$ if $L < (b/a)\bar K$ and $0$ if $L > (b/a)\bar K$
\item None of the above
\end{enumerate}

\newpage

\section*{Question 11}

\addcontentsline{toc}{section}{Question 11}

\noindent\textbf{Topic:} Consumer Theory \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



Let $e_i(p_0)$ be the price elasticity of demand for good $X$ of consumer $i$ at $p_0$, with
heterogeneous quantities. What is the elasticity of the aggregate demand at $p_0$?
\begin{enumerate}[label=(\Alph*)]
\item $\sum e_i(p_0)$
\item $\sum \dfrac{q_i(p_0)}{\sum_j q_j(p_0)}\, e_i(p_0)$
\item $\sum \dfrac{1}{N}\, e_i(p_0)$
\item None of these
\end{enumerate}

\newpage

\section*{Question 12}

\addcontentsline{toc}{section}{Question 12}

\noindent\textbf{Topic:} Market Structure \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



$m$ identical firms with $C(q) = q^2 + 1$. Industry demand $D(P) = a - bP$, $a,b > 0$.
Short-run equilibrium output per firm?
\begin{enumerate}[label=(\Alph*)]
\item $0$
\item $a/(m + 2b)$
\item $a/(m^2 + b)$
\item $a/(m + b/2)$
\end{enumerate}

\newpage

\section*{Question 13}

\addcontentsline{toc}{section}{Question 13}

\noindent\textbf{Topic:} Market Structure \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



$C(q) = 3q^2 + 800$; $P = 280 - 4q$. Find the price elasticity of demand at the
profit-maximizing price.
\begin{enumerate}[label=(\Alph*)]
\item $-4.5$
\item $-3.5$
\item $-2.5$
\item $-1.5$
\end{enumerate}

\newpage

\section*{Question 14}

\addcontentsline{toc}{section}{Question 14}

\noindent\textbf{Topic:} Macroeconomics \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



Solow model with saving $s$, depreciation $\delta$, labor growth $n$, no technological progress.
Steady-state capital--output ratio?
\begin{enumerate}[label=(\Alph*)]
\item $s/(n + \delta)$
\item $n/(\delta + n)$
\item $\delta/(s + n)$
\item $1/(s + n + \delta)$
\end{enumerate}

\newpage

\section*{Question 15}

\addcontentsline{toc}{section}{Question 15}

\noindent\textbf{Topic:} Combinatorics \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



Number of arrangements of the word ``PANDEMIC'' (8 distinct letters) such that the vowels
appear together.
\begin{enumerate}[label=(\Alph*)]
\item $6 \times (3!)(5!)$
\item $5 \times (3!)(5!)$
\item $4 \times (3!)(5!)$
\item $1 \times (3!)(5!)$
\end{enumerate}

\newpage

\section*{Question 16}

\addcontentsline{toc}{section}{Question 16}

\noindent\textbf{Topic:} Algebra / Functions \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



$f(x) = x^2 - x - 1$, $g(x) = x + 1$. Let $\alpha_1 > 0$ and $\alpha_2 < 0$ be the roots of $f(x)=0$;
let $\beta_1 > 0$ and $\beta_2 < 0$ be the roots of $f(g(x))=0$. Identify which statement is
\emph{incorrect}.
\begin{enumerate}[label=(\Alph*)]
\item $\alpha_1 - \beta_1 = \alpha_2 - \beta_2 = 1$
\item $\alpha_1 + \beta_2 = \alpha_2 + \beta_1 = 0.5$  \quad [as printed]
\item $\alpha_1 + \beta_1 = -(\alpha_2 + \beta_2) = \sqrt 5$
\item $\alpha_1 + \alpha_2 = -(\beta_1 + \beta_2) = -1$
\end{enumerate}

\newpage

\section*{Question 17}

\addcontentsline{toc}{section}{Question 17}

\noindent\textbf{Topic:} Functions \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



$f(x) = \sqrt{1-x}$. Then $f(1 - f(x))$ equals
\begin{enumerate}[label=(\Alph*)]
\item $x$
\item $\sqrt{1-x}$
\item $x^2$
\item $1 - x^2$
\end{enumerate}

\newpage

\section*{Question 18}

\addcontentsline{toc}{section}{Question 18}

\noindent\textbf{Topic:} Calculus \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



$f$ is increasing, concave, $C^2$; $g$ is decreasing, convex, $C^2$. Then $G(x) = g(f(x))$ is
\begin{enumerate}[label=(\Alph*)]
\item increasing and convex
\item decreasing and convex
\item increasing and concave
\item decreasing and concave
\end{enumerate}

\newpage

\section*{Question 19}

\addcontentsline{toc}{section}{Question 19}

\noindent\textbf{Topic:} Calculus \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



$f : \mathbb R \to \mathbb R$ is $C^2$ with $f''(x) > 0$ for all $x$, $f(1)=1$, $f(2)=2$.
Then,
\begin{enumerate}[label=(\Alph*)]
\item $0 < f'(2) < 1$
\item $f'(2) > 1$
\item $f'(2) = 1$
\item $f'(2) = 0$
\end{enumerate}

\newpage

\section*{Question 20}

\addcontentsline{toc}{section}{Question 20}

\noindent\textbf{Topic:} Linear Algebra \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



$A, B$ non-singular of the same order, $C = B A B^{-1}$. For scalar $\lambda$, $\det(C + \lambda I)$ equals
\begin{enumerate}[label=(\Alph*)]
\item $\det A$
\item $\det B$
\item $\det(A + \lambda I)$
\item $\det(B + \lambda I)$
\end{enumerate}

\newpage

\section*{Question 21}

\addcontentsline{toc}{section}{Question 21}

\noindent\textbf{Topic:} Calculus \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



$\displaystyle \lim_{x \to e} \frac{\ln x - 1}{x - e}$.
\begin{enumerate}[label=(\Alph*)]
\item $0$
\item $e$
\item $1/e$
\item None of these
\end{enumerate}

\newpage

\section*{Question 22}

\addcontentsline{toc}{section}{Question 22}

\noindent\textbf{Topic:} Calculus \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



$f : [0,\infty) \to \mathbb R$, $f(0) = 0$, $f''(x) > 0$ for $x > 0$. Then $f(x)/x$ is
\begin{enumerate}[label=(\Alph*)]
\item increasing in $(0,\infty)$
\item decreasing in $(0,\infty)$
\item increasing in $(0,1]$ and decreasing in $(1,\infty)$
\item decreasing in $(0,1]$ and increasing in $(1,\infty)$
\end{enumerate}

\newpage

\section*{Question 23}

\addcontentsline{toc}{section}{Question 23}

\noindent\textbf{Topic:} Combinatorics \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



$f : A \to B$ with $|A|=5$, $|B|=2$. How many onto functions?
\begin{enumerate}[label=(\Alph*)]
\item $5^2 - 1$
\item $5^2 - 2$
\item $2^5 - 1$
\item $2^5 - 2$
\end{enumerate}

\newpage

\section*{Question 24}

\addcontentsline{toc}{section}{Question 24}

\noindent\textbf{Topic:} Functions \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



$f : \mathbb R \to \mathbb R$, $f(x) = \dfrac{x}{1+x^2}$. Then
\begin{enumerate}[label=(\Alph*)]
\item $-1 \leq f(x) \leq 1$
\item $-1 \leq f(x) \leq 1/2$
\item $-1/2 \leq f(x) \leq 1$
\item $-1/2 \leq f(x) \leq 1/2$
\end{enumerate}

\newpage

\section*{Question 25}

\addcontentsline{toc}{section}{Question 25}

\noindent\textbf{Topic:} Linear Algebra \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



$A$ is $3 \times 3$ with rank $3$; $B$ is $3 \times 4$ with rank $3$. Then $\operatorname{rank}(AB)$ is
\begin{enumerate}[label=(\Alph*)]
\item $3$
\item $4$
\item $6$
\item $7$
\end{enumerate}

\newpage

\section*{Question 26}

\addcontentsline{toc}{section}{Question 26}

\noindent\textbf{Topic:} Linear Algebra \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Needs Review

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} answer option could not be parsed confidently; source status is Needs Review.}


Let $A$ be a specified $6 \times 8$ (or square, source unclear) matrix [the entries of $A$ were
not legibly transcribed in the source]. Which one is an eigenvalue of $A$?
\begin{enumerate}[label=(\Alph*)]
\item $2$
\item $1$
\item $3$
\item $5$
\end{enumerate}

\newpage

\section*{Question 27}

\addcontentsline{toc}{section}{Question 27}

\noindent\textbf{Topic:} Linear Algebra \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



$A$ is a $5\times 5$ non-null singular matrix. Then $Ax = 0$ has
\begin{enumerate}[label=(\Alph*)]
\item only the trivial solution
\item exactly $5$ solutions
\item no solution
\item infinitely many solutions
\end{enumerate}

\newpage

\section*{Question 28}

\addcontentsline{toc}{section}{Question 28}

\noindent\textbf{Topic:} Probability \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



A family has two children. Find the probability that both are boys given that at least one
is a boy.
\begin{enumerate}[label=(\Alph*)]
\item $1/2$
\item $2/3$
\item $1/3$
\item $1/4$
\end{enumerate}

\newpage

\section*{Question 29}

\addcontentsline{toc}{section}{Question 29}

\noindent\textbf{Topic:} Probability \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



Box 1 contains $\{1$ black, $1$ white$\}$; Box 2 contains $\{1$ black, $2$ white$\}$. Select a
box at random and then a ball at random. Probability that the drawn ball is black?
\begin{enumerate}[label=(\Alph*)]
\item $5/12$
\item $2/5$
\item $1/6$
\item $5/11$
\end{enumerate}

\newpage

\section*{Question 30}

\addcontentsline{toc}{section}{Question 30}

\noindent\textbf{Topic:} Functions \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



$f : \mathbb R \to \mathbb R$, $f(x) = (x^2 + 1)^{2022}$.
\begin{enumerate}[label=(\Alph*)]
\item one-one but not onto
\item onto but not one-one
\item both one-one and onto
\item neither one-one nor onto
\end{enumerate}

\end{document}
