UPSC IES/ESE (Engineering Services) General Studies and Engineering Aptitude (Common Paper) Syllabus

State the formula for integration by parts.

$$\int u\,dv = uv-\int v\,du.$$ Choose $u$ using the LIATE/ILATE priority (Logarithmic, Inverse-trig, Algebraic, Trig, Exponential).

Define the gradient of a scalar field $\phi(x,y,z)$.

$$\nabla \phi = \frac{\partial \phi}{\partial x}\hat{i}+\frac{\partial \phi}{\partial y}\hat{j}+\frac{\partial \phi}{\partial z}\hat{k}.$$ It points in the direction of maximum rate of increase of $\phi$.

Define the divergence and curl of a vector field $\vec{F}=F_1\hat{i}+F_2\hat{j}+F_3\hat{k}$.

Divergence: $\nabla\cdot\vec{F}=\dfrac{\partial F_1}{\partial x}+\dfrac{\partial F_2}{\partial y}+\dfrac{\partial F_3}{\partial z}$ (a scalar). Curl: $\nabla\times\vec{F}$ (a vector), computed as the determinant of $\begin{pmatrix}\hat{i}&\hat{j}&\hat{k}\\ \partial_x&\partial_y&\partial_z\\ F_1&F_2&F_3\end{pmatrix}$.

State the Divergence (Gauss) theorem.

$$\iint_{S} \vec{F}\cdot \hat{n}\,dS = \iiint_{V} (\nabla\cdot\vec{F})\,dV,$$ relating the outward flux through a closed surface $S$ to the volume integral of divergence over the enclosed volume $V$.

State Stokes' theorem.

$$\oint_{C} \vec{F}\cdot d\vec{r} = \iint_{S} (\nabla\times\vec{F})\cdot \hat{n}\,dS,$$ relating the line integral around a closed curve $C$ to the surface integral of the curl over any surface $S$ bounded by $C$.

What is the addition rule of probability for two events $A$ and $B$?

$$P(A\cup B)=P(A)+P(B)-P(A\cap B).$$ For mutually exclusive events, $P(A\cap B)=0$, so $P(A\cup B)=P(A)+P(B)$.

State Bayes' theorem.

$$P(A_i\mid B)=\frac{P(B\mid A_i)\,P(A_i)}{\sum_{j} P(B\mid A_j)\,P(A_j)},$$ used to update the probability of cause $A_i$ given observed evidence $B$.

Give the mean and variance of a binomial distribution with parameters $n$ and $p$.

Mean $=np$ and variance $=np(1-p)=npq$, where $q=1-p$.

State the probability mass function, mean, and variance of a Poisson distribution.

$$P(X=k)=\frac{e^{-\lambda}\lambda^{k}}{k!},\quad k=0,1,2,\dots$$ Both the mean and variance equal $\lambda$.

In the method of least squares for fitting a straight line $y=a+bx$, write the two normal equations.

$$\sum y = na + b\sum x,$$ $$\sum xy = a\sum x + b\sum x^{2}.$$ Solving these gives the best-fit intercept $a$ and slope $b$.

Write the formula for the Pearson correlation coefficient $r$.

$$r=\frac{n\sum xy-\sum x\sum y}{\sqrt{\,[n\sum x^{2}-(\sum x)^{2}]\,[n\sum y^{2}-(\sum y)^{2}]\,}}.$$ Its value lies between $-1$ and $+1$.

State the iterative formula of the Newton-Raphson method for finding a root of $f(x)=0$.

$$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}.$$ It has quadratic convergence near a simple root, provided $f'(x_n)\neq 0$.

Write the bisection method's logic and its rate of convergence.

Given $f(a)\cdot f(b)<0$, repeatedly take the midpoint $c=\frac{a+b}{2}$ and replace the endpoint whose function value has the same sign as $f(c)$. It converges linearly (error roughly halves each step).

State the Trapezoidal rule for numerical integration.

$$\int_{a}^{b} f(x)\,dx \approx \frac{h}{2}\left[ y_0+y_n+2(y_1+y_2+\cdots+y_{n-1}) \right],$$ where $h=\frac{b-a}{n}$. Error is $O(h^{2})$.

State Simpson's one-third rule for numerical integration.

$$\int_{a}^{b} f(x)\,dx \approx \frac{h}{3}\left[ (y_0+y_n)+4(y_1+y_3+\cdots)+2(y_2+y_4+\cdots) \right].$$ It requires an even number of intervals; error is $O(h^{4})$.

Write the formula for the Euler method to solve $\frac{dy}{dx}=f(x,y)$ with step size $h$.

$$y_{n+1}=y_n+h\,f(x_n,y_n).$$ It is a first-order explicit method with local truncation error $O(h^{2})$.

Write the fourth-order Runge-Kutta (RK4) update formula for $\frac{dy}{dx}=f(x,y)$.

$$y_{n+1}=y_n+\frac{1}{6}(k_1+2k_2+2k_3+k_4),$$ with $k_1=h f(x_n,y_n)$, $k_2=h f(x_n+\tfrac{h}{2},y_n+\tfrac{k_1}{2})$, $k_3=h f(x_n+\tfrac{h}{2},y_n+\tfrac{k_2}{2})$, $k_4=h f(x_n+h,y_n+k_3)$.

What is the regula falsi (false position) iterative formula for a root of $f(x)=0$?

$$x=\frac{a\,f(b)-b\,f(a)}{f(b)-f(a)},$$ where $f(a)$ and $f(b)$ have opposite signs; the bracket is updated like the bisection method but using this interpolated point.

What was the National Infrastructure Pipeline (NIP) and its headline investment target?

The NIP (launched 2019) is a coordinated plan for infrastructure projects across sectors (energy, roads, railways, urban) with a projected investment of around ₹111 lakh crore, aimed at improving project preparation and attracting investment.

What is the PM Gati Shakti National Master Plan?

A 2021 digital platform for integrated, multi-modal infrastructure planning that brings together ministries (roads, railways, ports, etc.) on a single GIS-based map to break silos and ensure coordinated, time-bound project execution.

What is the significance of the India–Middle East–Europe Economic Corridor (IMEC) announced at the 2023 G20 Summit?

IMEC is a multi-modal connectivity initiative (rail and shipping links plus energy and digital cables) connecting India to Europe via the Gulf, announced during India's G20 presidency to boost trade and engineering/logistics cooperation.

What is a caselet in analytical reasoning, and how should it be approached?

A caselet is a short paragraph of data/conditions from which multiple questions are derived. Approach: extract all given facts into a table/diagram, identify fixed and variable elements, then apply the constraints systematically to deduce answers.

In a linear arrangement puzzle, six people sit in a row; what is the key first step to solve such arrangements?

Identify and place the most restrictive (definite) clues first — e.g., people fixed at the ends or with exact positions — then use relative clues ('immediate left/right', 'between') to fill remaining seats, keeping multiple possibilities until eliminated.

In quantitative aptitude, state the formula relating speed, distance, and time, and the average speed for equal distances at speeds $u$ and $v$.

$\text{Distance}=\text{Speed}\times\text{Time}$. For two equal distances covered at speeds $u$ and $v$, the average speed is the harmonic mean $$\frac{2uv}{u+v}.$$

What is the compound interest formula for principal $P$, annual rate $r\%$, compounded annually for $n$ years?

$$A=P\left(1+\frac{r}{100}\right)^{n},$$ and the compound interest is $CI=A-P$.

In non-verbal reasoning, what is a 'mirror image' versus a 'water image'?

A mirror image is the lateral (left–right) reflection of a figure as seen in a vertical mirror. A water image is the vertical (top–bottom) reflection, as seen reflected in water below the figure.

What is the National Solar Mission's role within India's renewable energy targets?

The Jawaharlal Nehru National Solar Mission (part of the National Action Plan on Climate Change) promotes grid-connected and off-grid solar power; it underpins India's larger goal of achieving 500 GW of non-fossil installed capacity by 2030.

Name a major recent science/technology achievement of ISRO involving a soft landing near the Moon's south pole.

Chandrayaan-3 (2023) achieved a soft landing near the lunar south pole, making India the first country to land in that region and the fourth nation overall to soft-land on the Moon. Its lander was named Vikram and rover Pragyan.

In decision-making problems, what is the difference between a 'necessary' condition and a 'sufficient' condition for taking an action?

A necessary condition must be met for the action to be valid (its absence rules the action out), while a sufficient condition, if met, alone justifies the action. A condition can be necessary, sufficient, both, or neither.

State the Rank–Nullity theorem for a linear transformation / matrix $A$ with $n$ columns.

$$\operatorname{rank}(A)+\operatorname{nullity}(A)=n,$$ where rank is the dimension of the column space and nullity is the dimension of the null space (solution space of $Ax=0$).

State the chain rule for differentiating a composite function $y=f(g(x))$.

$$\frac{dy}{dx}=f'(g(x))\cdot g'(x)=\frac{dy}{du}\cdot\frac{du}{dx},\quad \text{where } u=g(x).$$

What is the geometric meaning of a definite integral $\int_a^b f(x)\,dx$ when $f(x)\geq 0$?

It represents the area bounded by the curve $y=f(x)$, the $x$-axis, and the vertical lines $x=a$ and $x=b$. If $f(x)$ is negative over part of the interval, that area is counted as negative.