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\lhead{\textcolor{StatBlue}{\textbf{Statstrive ExamIQ}}}
\rhead{ISI MSQE PEA 2023}
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\begin{document}

\begin{center}
{\Huge \textbf{Statstrive ExamIQ}}\\[6pt]
{\Large ISI MSQE PEA 2023 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 & Microeconomics & Ready \\
2 & Microeconomics & Ready \\
3 & Microeconomics & Ready \\
4 & Microeconomics & Ready \\
5 & Microeconomics & Ready \\
6 & Microeconomics & Ready \\
7 & Microeconomics & Ready \\
8 & Microeconomics & Needs Review \\
9 & Macroeconomics & Ready \\
10 & Macroeconomics & Ready \\
11 & Macroeconomics & Needs Review \\
12 & Macroeconomics & Needs Review \\
13 & Macroeconomics & Ready \\
14 & Macroeconomics & Needs Review \\
15 & Econometrics / Growth & Ready \\
16 & Linear Algebra & Needs Review \\
17 & Probability & Needs Review \\
18 & Probability & Needs Review \\
19 & Probability & Needs Review \\
20 & Calculus & Needs Review \\
21 & Calculus & Needs Review \\
22 & Algebra / Combinatorics & Needs Review \\
23 & Probability & Needs Review \\
24 & Algebra & Needs Review \\
25 & Optimization / Number Theory & Needs Review \\
26 & Combinatorics & Needs Review \\
27 & Probability & Needs Review \\
28 & Calculus & Ready \\
29 & Convex Analysis & Ready \\
30 & Functional Equations & Needs Review \\
\bottomrule
\end{longtable}

\newpage
\tableofcontents
\newpage

\section*{Question 1}

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

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



A consumer's budgetary allocation for two commodities $x$ and $y$ is $m$. Her demand for $x$ is
\[
x(p_x,p_y,m)=\frac{2m}{5p_x}.
\]
Initially $m=1000$, $p_y=20$, $p_x=5$. The price of $x$ falls from $5$ to $4$. The substitution effect of the price change is:
\begin{enumerate}[label=(\Alph*)]
\item an increase in demand for $x$ from 80 to 100
\item an increase in demand for $x$ from 90 to 100
\item an increase in demand for $x$ from 80 to 90
\item an increase in demand for $x$ from 80 to 92
\end{enumerate}

\newpage

\section*{Question 2}

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

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



\begin{center}
\begin{tabular}{lcc|lcc}
\toprule
\multicolumn{3}{c|}{Year 1} & \multicolumn{3}{c}{Year 2} \\
& Qty & Price & & Qty & Price \\
\midrule
Good 1 & 100 & 100 & Good 1 & 120 & 100 \\
Good 2 & 100 & 100 & Good 2 & $x$ & 80 \\
\bottomrule
\end{tabular}
\end{center}
For which range of $x$ is the consumer's behaviour inconsistent with WARP?
\begin{enumerate}[label=(\Alph*)]
\item $75<x<80$
\item $x\ge 70$
\item $70<x<75$
\item $x\le 75$
\end{enumerate}

\newpage

\section*{Question 3}

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

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



$p=100-q$, $q=\sum_{i=1}^{23}q_i$, $c_i(q_i)=\dfrac{q_i^{2}}{2}$. Cournot--Nash equilibrium:
\begin{enumerate}[label=(\Alph*)]
\item is not well defined
\item each firm produces 3 units
\item each firm produces 4 units
\item each firm produces 5 units
\end{enumerate}

\newpage

\section*{Question 4}

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

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



$p=100-q$, $n=10$ firms, $c_i(q_i)=q_i$. Total deadweight loss is
\begin{enumerate}[label=(\Alph*)]
\item $\dfrac{9^2}{2}$ \qquad
\item $\dfrac{9^2}{2}$ \qquad
\item $\dfrac{10^2}{2}$ \qquad
\item $\dfrac{100^2}{2}$
\end{enumerate}
(The intended options correspond to $\frac{9^2}{2},\frac{9^2}{2},\frac{10^2}{2},\frac{100^2}{2}$, the first being $\frac{9^2}{2}$.)

\newpage

\section*{Question 5}

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

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



$p=100-q$; large number of identical firms with
\[
c(q_i)=\begin{cases}10+2q_i, & q_i>0,\\ 0, & q_i=0.\end{cases}
\]
The competitive equilibrium price is:
\begin{enumerate}[label=(\Alph*)]
\item not well defined \quad (B) $2$ \quad (C) $10$ \quad (D) $2.1$
\end{enumerate}

\newpage

\section*{Question 6}

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

\noindent\textbf{Topic:} Microeconomics \quad \textbf{Difficulty:} Hard \quad \textbf{Status:} Ready



$p=100-q$. Two firms with cost
\[
c_i(q_i)=\begin{cases}0, & q_i\le 10\\ \infty, & q_i>10\end{cases}
\]
i.e.\ each firm has capacity $10$. Demands: undercut wins the market, ties split. Bertrand--Nash equilibrium:
\begin{enumerate}[label=(\Alph*)]
\item $(0,0)$ \quad (B) $(20,20)$ \quad (C) $(80,80)$ \quad (D) $(90,90)$
\end{enumerate}

\newpage

\section*{Question 7}

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

\noindent\textbf{Topic:} Microeconomics \quad \textbf{Difficulty:} Hard \quad \textbf{Status:} Ready



$500$ consumers uniform on $[0,1]$. Two hospital locations $a<b$. Travel cost = distance. Each individual values service at $4$. Fixed cost per hospital = $300$; marginal cost of treatment = $2$. Optimal locations?
\begin{enumerate}[label=(\Alph*)]
\item no hospital \quad (B) both at $1/2$ \quad (C) $1/3,2/3$ \quad (D) $1/4,3/4$
\end{enumerate}

\newpage

\section*{Question 8}

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

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

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} source TeX does not provide a full four-option MCQ block.}


Unit mass of consumers, valuation $\theta\sim F$ on $[\underline\theta,\overline\theta]$. Each buys $1$ unit iff $\theta\ge p$. Inverse demand $p(q)$. Find $p'(q)$.

\newpage

\section*{Question 9}

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

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



Demand-determined output. Workers receive share $\lambda Y$, capitalists $(1-\lambda)Y$. Saving rates $s_w$ (workers) and $s_c$ (capitalists), with $s_w<s_c$. Investment $\bar I$ autonomous. If both $s_c$ and $s_w$ rise, in the new equilibrium:
\begin{enumerate}[label=(\Alph*)]
\item savings $\uparrow$, income $\downarrow$
\item savings $\downarrow$, income $\uparrow$
\item savings $\uparrow$, income unchanged
\item savings unchanged, income $\downarrow$
\end{enumerate}

\newpage

\section*{Question 10}

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

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



With saving rates unchanged but $\lambda$ rising (more income to workers), the new equilibrium has:
\begin{enumerate}[label=(\Alph*)]
\item savings $\uparrow$, income $\downarrow$
\item savings $\downarrow$, income $\uparrow$
\item savings $\uparrow$, income unchanged
\item savings unchanged, income $\downarrow$
\end{enumerate}

\newpage

\section*{Question 11}

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

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

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} source TeX does not provide a full four-option MCQ block.}


$Y=\bar K^\alpha L^{1-\alpha}$, $0<\alpha<1$. Perfectly competitive. Find $L^d$ as a function of real wage $W/P$.

\newpage

\section*{Question 12}

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

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

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} source TeX does not provide a full four-option MCQ block.}


Household has endowment $\bar L$, supplies $L^s$, enjoys $\bar L-L^s$ leisure. $u=C^\beta+(\bar L-L^s)^\beta$, $0<\beta<1$. Budget: $PC=WL^s$. Find $L^s$ as a function of $W/P$.

\newpage

\section*{Question 13}

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

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



Given the labour demand and supply derived above, the aggregate supply curve is:
\begin{enumerate}[label=(\Alph*)]
\item upward sloping \item downward sloping \item vertical \item horizontal
\end{enumerate}

\newpage

\section*{Question 14}

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

\noindent\textbf{Topic:} Macroeconomics \quad \textbf{Difficulty:} Hard \quad \textbf{Status:} Needs Review

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} source TeX does not provide a full four-option MCQ block.}


Goods market: $S(Y,r)=I(r)$ with $0<S_Y<1,\;S_r>0,\;I_r<0$. Money: $M/P=L(Y,r)$ with $L_Y>0,\;L_r<0$. Find $dY/dP$.

\newpage

\section*{Question 15}

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

\noindent\textbf{Topic:} Econometrics / Growth \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



$g_i=\alpha+\beta_0\log y_{i,0}+\beta_1\log n_i+\beta_2\log s_i+\gamma X_i+\varepsilon_i$. The Solow model predicts the sign of $\beta_0$:
\begin{enumerate}[label=(\Alph*)]
\item negative \item positive \item zero \item inconclusive
\end{enumerate}

\newpage

\section*{Question 16}

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

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

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} source TeX does not provide a full four-option MCQ block.}


\[
A=\begin{pmatrix}1&0&1\\ 0&1&1\\ 1&1&0\end{pmatrix}.
\]
Rank of $A$?

\newpage

\section*{Question 17}

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

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

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} source TeX does not provide a full four-option MCQ block.}


Bowl A: 2 red. Bowl B: 2 white. Bowl C: 1 white, 1 red. A bowl is selected at random and a coin drawn at random. Given the drawn coin is white, what is the probability that the other coin in the bowl is red?

\newpage

\section*{Question 18}

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

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

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} source TeX does not provide a full four-option MCQ block.}


Choose $n\in\{1,2,\dots,6\}$ uniformly, then choose $m\in\{1,\dots,n\}$ uniformly. Probability that $m=5$?

\newpage

\section*{Question 19}

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

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

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} source TeX does not provide a full four-option MCQ block.}


$F(1.4)=0.92,\;F(0.14)=0.555,\;F(-0.2)=0.42,\;F(-1.6)=0.055$. Diameter $\sim N(\mu,\sigma^2)$. $P(\text{diam}<1.8)=0.08$ and $P(\text{diam}>2.4)=0.055$. Find $\mu$.

\newpage

\section*{Question 20}

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

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

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} source TeX does not provide a full four-option MCQ block.}


$f$ differentiable, $f'$ strictly increasing. $f(1/2)=1/2,\;f(1)=1$. Compare $f'(1/2),\;1,\;f'(1)$.

\newpage

\section*{Question 21}

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

\noindent\textbf{Topic:} Calculus \quad \textbf{Difficulty:} Hard \quad \textbf{Status:} Needs Review

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} source TeX does not provide a full four-option MCQ block.}


$f(x)=\lfloor x\rfloor$. Evaluate $\displaystyle\int_0^{\sqrt 5}f(x^2)\,dx$.

\newpage

\section*{Question 22}

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

\noindent\textbf{Topic:} Algebra / Combinatorics \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Needs Review

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} source TeX does not provide a full four-option MCQ block.}


Find the constant term in $\bigl(x+\tfrac{1}{x^2}\bigr)^{19}$.

\newpage

\section*{Question 23}

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

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

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} source TeX does not provide a full four-option MCQ block.}


Arjun and Gukesh each toss 3 fair coins. $p_1=P(\text{Arjun's heads}>\text{Gukesh's heads})$. Find $p_1$.

\newpage

\section*{Question 24}

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

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

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} source TeX does not provide a full four-option MCQ block.}


Find the number of real solutions of $x|x|+1=3|x|$.

\newpage

\section*{Question 25}

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

\noindent\textbf{Topic:} Optimization / Number Theory \quad \textbf{Difficulty:} Hard \quad \textbf{Status:} Needs Review

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} source TeX does not provide a full four-option MCQ block.}


Positive integers $k_1,\dots,k_n$ (not necessarily distinct) with
\[
k_1+\cdots+k_n=5n-4,\qquad \tfrac{1}{k_1}+\cdots+\tfrac{1}{k_n}=1.
\]
Maximum $n$?

\newpage

\section*{Question 26}

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

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

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} source TeX does not provide a full four-option MCQ block.}


Monkey at $(0,0)$ in $\mathbb R^2$ jumps a distance of $5$ to an integer point. Number of possible end positions?

\newpage

\section*{Question 27}

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

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

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} source TeX does not provide a full four-option MCQ block.}


Strip of $n+2$ squares; squares $1$ and $n+2$ are black; squares $2,\dots,n+1$ are white. Girl picks a white square uniformly, then picks one of its two neighbours uniformly. Probability the neighbour is white?

\newpage

\section*{Question 28}

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

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



$f(x)=\max(|x|,x^2)$ for all $x\in\mathbb R$. Which is true?
\begin{enumerate}[label=(\Alph*)]
\item $f$ is increasing \item $f$ not continuous \item $f$ continuous but not differentiable \item $f$ decreasing
\end{enumerate}

\newpage

\section*{Question 29}

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

\noindent\textbf{Topic:} Convex Analysis \quad \textbf{Difficulty:} Hard \quad \textbf{Status:} Ready



With $f$ as in Q28, $D=\{(x,y)\in\mathbb R^2:y\ge f(x)\}$ (epigraph of $f$). Which is true?
\begin{enumerate}[label=(\Alph*)]
\item $\mathbb R^2\setminus D$ is convex \item $\mathbb R^2_+\setminus D$ is convex \item $D$ is not convex \item None of the above
\end{enumerate}

\newpage

\section*{Question 30}

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

\noindent\textbf{Topic:} Functional Equations \quad \textbf{Difficulty:} Hard \quad \textbf{Status:} Needs Review

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} source TeX does not provide a full four-option MCQ block.}


$f:[-1,1]\to\mathbb R$ with
\[
f(x)=f\!\left(\frac{2-x^2}{2}\right)\cdot\frac{x^2}{2-x^2}\quad\text{... [as printed]}
\]
(Standard interpretation: $f(x)=f\!\Bigl(\dfrac{2x^2}{2-x^2}\Bigr)$ or similar.) Find $f(-1)$.

\end{document}
