\documentclass[11pt]{article}
\usepackage[a4paper,margin=0.82in]{geometry}
\usepackage{amsmath,amssymb,mathtools}
\usepackage{enumitem}
\usepackage{xcolor}
\usepackage{booktabs,longtable,array}
\usepackage{fancyhdr}
\usepackage[colorlinks=true,linkcolor=StatBlue,urlcolor=StatBlue]{hyperref}

\definecolor{StatBlue}{HTML}{1D4ED8}
\definecolor{StatInk}{HTML}{111827}
\definecolor{StatAmber}{HTML}{B45309}

\setlength{\parindent}{0pt}
\setlength{\parskip}{5pt}
\setlist[enumerate]{itemsep=4pt,topsep=6pt}
\pagestyle{fancy}
\fancyhf{}
\lhead{\textcolor{StatBlue}{\textbf{Statstrive ExamIQ}}}
\rhead{ISI MSQE PEA 2026}
\cfoot{\thepage}

\begin{document}

\begin{center}
{\Huge \textbf{Statstrive ExamIQ}}\\[6pt]
{\Large ISI MSQE PEA 2026 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 & Sequences and Series & Ready \\
2 & Algebra & Ready \\
3 & Calculus & Ready \\
4 & Calculus & Ready \\
5 & Real Analysis & Ready \\
6 & Real Analysis & Ready \\
7 & Real Analysis & Ready \\
8 & Calculus & Ready \\
9 & Statistics & Ready \\
10 & Regression & Ready \\
11 & Hypothesis Testing & Ready \\
12 & Random Variables & Ready \\
13 & Statistics & Ready \\
14 & Descriptive Statistics & Ready \\
15 & Probability & Ready \\
16 & Consumer Theory & Ready \\
17 & General Equilibrium & Needs Review \\
18 & Public Goods & Needs Review \\
19 & Market Structure & Ready \\
20 & Game Theory & Ready \\
21 & Public Goods & Ready \\
22 & Microeconomics & Ready \\
23 & Consumer Theory & Ready \\
24 & Consumer Theory & Ready \\
25 & International Trade & Ready \\
26 & Macroeconomics & Ready \\
27 & Macroeconomics & Ready \\
28 & Growth Theory & Ready \\
29 & Growth Theory & Ready \\
30 & Intertemporal Choice & Ready \\
\bottomrule
\end{longtable}

\newpage
\tableofcontents
\newpage

\section*{Question 1}

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

\noindent\textbf{Topic:} Sequences and Series \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



Suppose $a$ and $b$ are positive integers that are relatively prime, and
\[
\frac{a}{b} \;=\; \sum_{n=1}^{2026} \frac{1}{n^2 + 15n + 56}.
\]
Then the value of $a+b$ is
\begin{enumerate}[label=(\Alph*)]
\item 9149 \item 9229 \item 8189 \item 11
\end{enumerate}

\newpage

\section*{Question 2}

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

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



If $x^2 - 3x + 1 = 0$, then the value of $\left(x^4 + \dfrac{1}{x^4}\right)$ is
\begin{enumerate}[label=(\Alph*)]
\item 43 \item 45 \item 47 \item 49
\end{enumerate}

\newpage

\section*{Question 3}

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

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



Let $f:\mathbb{R}\to\mathbb{R}$ satisfy $f(x^3+1) = x^6 + 4x^3 + 8$ for all
$x \in \mathbb{R}$. Then $\displaystyle \int_1^2 f(x)\,dx$ equals
\begin{enumerate}[label=(\Alph*)]
\item $\dfrac{29}{3}$ \item $10$ \item $11$ \item $\dfrac{31}{3}$
\end{enumerate}

\newpage

\section*{Question 4}

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

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



Determine the value of
\[
\int_0^2 |x^2 - 7x + 10|\,dx.
\]
\begin{enumerate}[label=(\Alph*)]
\item 8 \item 9 \item $\dfrac{26}{3}$ \item $-\dfrac{23}{3}$
\end{enumerate}

\newpage

\section*{Question 5}

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

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



Let $f:[0,1]\to\mathbb{R}$ be continuous with $\displaystyle\int_0^1 f(t)\,dt = 1$, and let
\[
P(x) = \sum_{k=1}^n a_k x^k, \qquad \sum_{k=1}^n a_k = 1.
\]
Then there exists $c \in (0,1)$ such that
\begin{enumerate}[label=(\Alph*)]
\item $f(c) = 2\sum_{k=1}^n a_k$
\item $f(c) = \int_0^1 P(t)\,dt$
\item $f(c) = P'(c)$
\item $f(c) = P(c)$
\end{enumerate}

\newpage

\section*{Question 6}

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

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



Let $f:[0,\pi/4]\to\mathbb{R}$ be continuous. Then there exists $c\in(0,\pi/4)$ such that
\begin{enumerate}[label=(\Alph*)]
\item $f(c) = \int_0^{\pi/4} f(t)\,dt$
\item $f(c) = 2\cos(2c)\,\int_0^{\pi/4} f(t)\,dt$
\item $f(c) = \sin(2c)$
\item $f(c) = \sin(2c)\,\int_0^{\pi/4} f(t)\,dt$
\end{enumerate}

\newpage

\section*{Question 7}

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

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



Suppose $f:[a,b]\to\mathbb{R}$ is continuous and
\[
\int_a^x f(t)\,dt \;=\; \int_x^b f(t)\,dt \quad \text{for all } x \in [a,b].
\]
Then $f(x)$ must be
\begin{enumerate}[label=(\Alph*)]
\item A non-zero constant
\item A linear function
\item Identically zero
\item An odd function about $\dfrac{a+b}{2}$
\end{enumerate}

\newpage

\section*{Question 8}

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

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



Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous odd function, vanishing exactly at one point and satisfying $f(1) = \tfrac{1}{2}$. If
\[
\lim_{x\to 1} \frac{F(x)}{G(x)} = \frac{1}{14},
\]
where
\[
F(x) = \int_{-1}^{x} f(t)\,dt, \qquad G(x) = \int_{-1}^{x} t\,|f(f(t))|\,dt,
\]
then the value of $f\!\left(\tfrac{1}{2}\right)$ is
\begin{enumerate}[label=(\Alph*)]
\item $\dfrac{1}{7}$ \item $7$ \item $-7$ \item $\dfrac{7}{2}$
\end{enumerate}

\newpage

\section*{Question 9}

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

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



If the correlation between $X$ and $Y$ is zero, then which of the following is always true?
\begin{enumerate}[label=(\Alph*)]
\item They are independent
\item There is no linear relationship
\item They cannot be functionally related
\item All of the above
\end{enumerate}

\newpage

\section*{Question 10}

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

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



In simple linear regression $Y = a + bX$, the least squares line based on $n$ data points always passes through
\begin{enumerate}[label=(\Alph*)]
\item $(0,0)$
\item $\left(\tfrac{1}{n}, \tfrac{1}{n}\right)$
\item $(1,1)$
\item None of the previous
\end{enumerate}

\newpage

\section*{Question 11}

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

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



Given sample size $n$, reducing the significance level $\alpha$
\begin{enumerate}[label=(\Alph*)]
\item Increases Type I error
\item Decreases Type II error
\item Increases power
\item Decreases power
\end{enumerate}

\newpage

\section*{Question 12}

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

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



If $X \sim \text{Bernoulli}(p)$, then $E(X^2)$ equals
\begin{enumerate}[label=(\Alph*)]
\item $p^2$ \item $p$ \item $p(1-p)$ \item $p + p^2$
\end{enumerate}

\newpage

\section*{Question 13}

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

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



For $X_1, \dots, X_n \sim \text{Poisson}(\lambda)$ (i.i.d.), which is unbiased for $\lambda^2$?
\begin{enumerate}[label=(\Alph*)]
\item $\bar{X}^2$
\item $\bar{X}$
\item $\bar{X}^2 - \dfrac{\bar{X}}{n}$
\item $\bar{X}^2 - \bar{X}$
\end{enumerate}

\newpage

\section*{Question 14}

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

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



A dataset consists of $9$ observations arranged in increasing order
$x_1 \le x_2 \le \cdots \le x_9$ with mean $20$ and median $18$. If $x_9$ is
replaced by a very large number, which must be true?
\begin{enumerate}[label=(\Alph*)]
\item Both the mean and the median increase
\item The mean increases but the median remains unchanged
\item The median increases but the mean remains unchanged
\item Both the mean and the median remain unchanged
\end{enumerate}

\newpage

\section*{Question 15}

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

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



A restaurant has $60\%$ vegetarian customers (of whom $30\%$ order dessert) and
$40\%$ non-vegetarian (of whom $50\%$ order dessert). Given that a randomly
selected customer ordered dessert, what is the probability they ordered vegetarian?
\begin{enumerate}[label=(\Alph*)]
\item $0.375$ \item $0.474$ \item $0.600$ \item $0.750$
\end{enumerate}

\newpage

\section*{Question 16}

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

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



Suppose preferences are represented by $u(x,y) = x - \ln y$ for $x,y > 0$. Then the underlying preference relation must be
\begin{enumerate}[label=(\Alph*)]
\item Incomplete
\item Complete but intransitive
\item Complete and transitive but discontinuous
\item Complete, transitive and continuous but not convex
\end{enumerate}

\newpage

\section*{Question 17}

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

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

\par\noindent\textcolor{StatAmber}{\textbf{Review note:} source status is Draft.}


Suppose there are two agents $A$ and $B$ with utilities
\[
u_A = x_A + y_A - \tfrac{m}{2}(x_A - y_A)^2, \qquad u_B = x_B + y_B,
\]
where $x_i$ and $y_i$ are the amounts of goods $X$ and $Y$ consumed by agent
$i \in \{A,B\}$. The total endowments are $x$ and $y$. Initially $A$ owns the
entire endowment of $Y$ and $B$ owns the entire endowment of $X$. Then $A$ will
voluntarily transfer a positive amount of $Y$ to $B$ if
\begin{enumerate}[label=(\Alph*)]
\item $m > 0$ \item $m < 0$ \item $y > 2$ \item $my > \dfrac{1}{2}$
\end{enumerate}

\newpage

\section*{Question 18}

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

\noindent\textbf{Topic:} Public Goods \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.}


Two agents $A$ and $B$ have utilities
\[
u_A = x_A + y, \qquad u_B = 2x_B + y,
\]
where $x_A, x_B$ are amounts of a private good and $y$ is the amount of a public
good. Production satisfies $y + m(x_A + x_B) = 1$, with $m \in (0,1)$. Suppose a
social planner decides that one unit should be produced. For what value of $m$
will the resulting allocation be Pareto optimal?
\begin{enumerate}[label=(\Alph*)]
\item $\dfrac{4}{5}$ \item $\dfrac{4}{7}$ \item $\dfrac{4}{9}$ \item $\dfrac{4}{11}$
\end{enumerate}

\newpage

\section*{Question 19}

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

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



Three consumers with valuations $v_A, v_B, v_C$ each wish to buy one unit. The
monopolist has zero marginal cost and knows the valuations but must charge a
uniform price. It is given that
\[
\frac{7 v_B}{6} > v_A > v_B > \frac{8 v_C}{5} > 0.
\]
The profit-maximising price is
\begin{enumerate}[label=(\Alph*)]
\item $v_A$ \item $v_B$ \item $v_C$ \item Something in $(v_C, v_B)$
\end{enumerate}

\newpage

\section*{Question 20}

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

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



$n$ agents simultaneously choose private consumption $x_i$ and public-good
contribution $y_i$ to maximise
\[
u_i = x_i\, \Big(\sum_{j=1}^n y_j\Big),
\]
subject to $x_i + y_i = b$, $x_i, y_i \ge 0$. As $n$ increases, the total Nash
equilibrium expenditure on the public good
\begin{enumerate}[label=(\Alph*)]
\item Remains invariant
\item Monotonically increases and approaches $b$
\item Approaches $\dfrac{b}{n}$
\item Monotonically decreases and approaches $0$
\end{enumerate}

\newpage

\section*{Question 21}

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

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



$A$ and $B$ are neighbours; the snow plough cannot clear in front of $A$'s house
without clearing in front of $B$'s. Inverse demands are $P_A = 12 - q$,
$P_B = 8 - q$. Marginal cost is $16$. What is the efficient level of provision?
\begin{enumerate}[label=(\Alph*)]
\item $6$ \item $4$ \item $2$ \item $1$
\end{enumerate}

\newpage

\section*{Question 22}

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

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



A lump-sum tax of $\mathrm{Rs.}\,1$ is paid by the buyer per unit of a competitive good.
The price paid by buyers rises by $80$ paise. Then
\begin{enumerate}[label=(\Alph*)]
\item Demand must be perfectly elastic
\item Demand must be perfectly inelastic
\item Demand and supply elasticities must both be positive but finite
\item Supply must be perfectly elastic
\end{enumerate}

\newpage

\section*{Question 23}

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

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



A consumer always spends one-fourth of income on $X$, and the income elasticity of demand for $X$ is $5$. Is $Y$ inferior?
\begin{enumerate}[label=(\Alph*)]
\item Yes \item No
\item $Y$ is inferior iff $p_Y \ge 2 p_X$
\item $Y$ is inferior iff $p_Y \le p_X/2$
\end{enumerate}

\newpage

\section*{Question 24}

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

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



Half the total quantity is purchased by $75$ consumers each with price elasticity
$2$, and the other half by $25$ consumers each with price elasticity $3$. What is
the aggregate price elasticity for the $100$ consumers?
\begin{enumerate}[label=(\Alph*)]
\item Cannot be determined \item $2.25$ \item $2.5$ \item $2.75$
\end{enumerate}

\newpage

\section*{Question 25}

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

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



Two countries $H$ and $F$ produce single goods with outputs $Y_H, Y_F$. Take
$H$'s good as numeraire and $p$ as the relative price of $F$'s good. Consumers
in country $i$ spend a fraction $1/4$ of their expenditure on the foreign good.
$E_i$ denotes total expenditure of country $i$ measured in its own good.
Given $Y_H = 120$, $Y_F = 100$, $E_H = 80$ and market clearing with
$Y_H + p Y_F = E_H + p E_F$, find $p$.
\begin{enumerate}[label=(\Alph*)]
\item $8$ \item $6$ \item $4$ \item $2$
\end{enumerate}

\newpage

\section*{Question 26}

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

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



An economy has one good produced one-for-one from labour. The representative
consumer has $U\!\left(C, \tfrac{M}{P}\right) = \tfrac{3}{4}\ln C + \tfrac{1}{4}\ln\!\tfrac{M}{P}$,
labour endowment $50$, money endowment $\bar M = 100$, with constant money
supply $M = \bar M$. If the price is flexible (so labour is fully employed), the
equilibrium price is
\begin{enumerate}[label=(\Alph*)]
\item $2$ \item $4$ \item $6$ \item $8$
\end{enumerate}

\newpage

\section*{Question 27}

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

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



With the same setup as Question 26 but with price fixed at $P = 10$,
the equilibrium output is
\begin{enumerate}[label=(\Alph*)]
\item $15$ \item $20$ \item $25$ \item $30$
\end{enumerate}

\newpage

\section*{Question 28}

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

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



Solow economy with $Y = K^{1/2} L^{1/2}$, no depreciation, $n = 0.02$, savings
rate $s > 0$. If the steady-state capital--labour ratio is $k^* = 9$ and the
current ratio is $k = 1$, then the growth rate of $y = Y/L$ at the current date
is
\begin{enumerate}[label=(\Alph*)]
\item $0.02$ \item $0.04$ \item $0.03$ \item Indeterminable
\end{enumerate}

\newpage

\section*{Question 29}

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

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



Two Solow economies $A, B$ share the same technology and $n = 0.02$. They have
$k_A = 3, k^*_A = 9$ and $k_B = 4, k^*_B = 16$. Then per capita output is growing
\begin{enumerate}[label=(\Alph*)]
\item Faster in $A$ than in $B$
\item Faster in $B$ than in $A$
\item At the same rate in $A$ and $B$
\item At an indeterminable relative rate
\end{enumerate}

\newpage

\section*{Question 30}

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

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



A two-period consumer maximises
\[
U(C_1, C_2) = \log C_1 + \tfrac{1}{1+\rho}\log C_2, \qquad \rho > 0,
\]
subject to $C_1 + S = w$ and $C_2 = (1+r) S$, with $w, r > 0$. If $w$ rises by
$1\%$ and $r$ falls by $1\%$, then $S$
\begin{enumerate}[label=(\Alph*)]
\item Increases by more than $1\%$
\item Increases by $1\%$
\item Decreases by $1\%$
\item Decreases by more than $1\%$
\end{enumerate}

\end{document}
