UPSC ESE Civil Engineering General Studies and Engineering Aptitude Syllabus

State the product rule and the quotient rule of differentiation.

Product rule: $(uv)' = u'v + uv'$. Quotient rule: $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^{2}}$.

What is the Fundamental Theorem of Calculus?

If $F$ is an antiderivative of a continuous function $f$ on $[a,b]$, then $$\int_{a}^{b} f(x)\,dx = F(b) - F(a).$$

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

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

Give the Maclaurin series expansion of $e^{x}$.

$$e^{x} = \sum_{n=0}^{\infty} \frac{x^{n}}{n!} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots$$

What condition identifies a point $x=c$ as a local maximum using the second derivative test?

If $f'(c) = 0$ and $f''(c) < 0$, then $f$ has a local maximum at $x=c$ (if $f''(c) > 0$ it is a local minimum).

Define the order and degree of a differential equation.

The order is the highest derivative present; the degree is the power of the highest-order derivative after the equation is made polynomial (free of radicals/fractions) in derivatives.

Solve the general method for a first-order linear ODE $\dfrac{dy}{dx} + P(x)y = Q(x)$.

Multiply by the integrating factor $\mu = e^{\int P\,dx}$; the solution is $$y\,e^{\int P\,dx} = \int Q\,e^{\int P\,dx}\,dx + C.$$

What is the general solution of the homogeneous ODE $y'' - 5y' + 6y = 0$?

The auxiliary equation $m^{2} - 5m + 6 = 0$ gives $m = 2, 3$, so $$y = C_{1}e^{2x} + C_{2}e^{3x}.$$

Define a differential equation with separable variables and how it is solved.

An equation that can be written as $\dfrac{dy}{dx} = g(x)h(y)$; it is solved by separating variables: $\displaystyle \int \frac{dy}{h(y)} = \int g(x)\,dx + C$.

State the form of the auxiliary roots that give an oscillatory (sinusoidal) solution for a second-order linear ODE.

Complex conjugate roots $m = \alpha \pm i\beta$, giving $y = e^{\alpha x}\left(C_{1}\cos\beta x + C_{2}\sin\beta x\right)$.

State the addition rule of probability for two events $A$ and $B$.

$$P(A \cup B) = P(A) + P(B) - P(A \cap B).$$

Define conditional probability $P(A\mid B)$ and Bayes' theorem.

$P(A\mid B) = \dfrac{P(A \cap B)}{P(B)}$ for $P(B) > 0$. Bayes' theorem: $$P(A\mid B) = \frac{P(B\mid A)\,P(A)}{P(B)}.$$

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

Mean $\mu = np$ and variance $\sigma^{2} = np(1-p) = npq$, where $q = 1-p$.

What is the relation between mean, median and mode for a moderately skewed distribution?

The empirical relation: $$\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}.$$

Define standard deviation and its relation to variance.

Standard deviation $\sigma$ is the positive square root of the variance, $\sigma = \sqrt{\sigma^{2}}$; it measures the spread of data about the mean in the same units as the data.

State the Poisson distribution probability mass function.

$$P(X = k) = \frac{e^{-\lambda}\lambda^{k}}{k!}, \quad k = 0,1,2,\dots$$ where $\lambda$ is the mean number of events; mean = variance = $\lambda$.

In numerical methods, state the iteration formula of the Newton-Raphson method for solving $f(x)=0$.

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

State the trapezoidal rule for numerical integration over $[a,b]$ with step size $h$.

$$\int_{a}^{b} f(x)\,dx \approx \frac{h}{2}\left[(y_{0}+y_{n}) + 2(y_{1}+y_{2}+\cdots+y_{n-1})\right].$$

State Simpson's $\tfrac{1}{3}$ rule for numerical integration.

$$\int_{a}^{b} f(x)\,dx \approx \frac{h}{3}\left[(y_{0}+y_{n}) + 4(y_{1}+y_{3}+\cdots) + 2(y_{2}+y_{4}+\cdots)\right],$$ requiring an even number of intervals.

How does the bisection method locate a root, and what type of convergence does it have?

It repeatedly halves an interval $[a,b]$ where $f(a)f(b) < 0$, choosing the half that brackets the root. It always converges (for a continuous sign change) but only linearly.

What is the basis of the Gauss-Seidel method, and a sufficient condition for its convergence?

It is an iterative method for solving linear systems that uses the most recently updated values within each iteration. A sufficient condition for convergence is that the coefficient matrix be diagonally dominant.

In logical reasoning, what is a syllogism, and what are its three components?

A form of deductive argument consisting of a major premise, a minor premise, and a conclusion drawn from them, e.g., 'All A are B; all B are C; therefore all A are C.'

Distinguish between deductive and inductive reasoning.

Deductive reasoning moves from general premises to a specific, logically certain conclusion; inductive reasoning moves from specific observations to a probable general conclusion.

In a coding-decoding problem, if 'CAT' is coded as 'DBU', what is the coding rule?

Each letter is shifted forward by one position in the alphabet ($+1$): $C\to D$, $A\to B$, $T\to U$.

What is the next term in the series $2, 6, 12, 20, 30, \dots$ and the underlying rule?

$42$. The terms follow $n(n+1)$ for $n=1,2,3,\dots$, so differences increase by 2 each time ($4,6,8,10,12$).

In analytical ability, if A is the brother of B, B is the sister of C, and C is the father of D, how is A related to D?

A is the uncle (paternal) of D, since A and C are siblings and C is D's father.

A clock shows 3:15. What is the angle between the hour and minute hands?

$7.5^{\circ}$. The minute hand is at $90^{\circ}$; the hour hand is at $3\times30 + 15\times0.5 = 97.5^{\circ}$; the difference is $97.5 - 90 = 7.5^{\circ}$.

In data interpretation, how do you compute percentage change between an old value and a new value?

$$\text{Percentage change} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100\%.$$

In a pie chart, how is a category's value converted into the central angle of its sector?

$$\text{Angle} = \frac{\text{Category value}}{\text{Total value}} \times 360^{\circ}.$$

Define ethics and distinguish it from law.

Ethics is a system of moral principles governing what is right and wrong in human conduct based on values; law is a set of formal rules enforced by the state. An act can be legal yet unethical, and ethical yet illegal.

In professional engineering ethics, what is the engineer's paramount obligation according to most codes of conduct?

To hold paramount the safety, health and welfare of the public in the performance of professional duties, above personal or client interests.

Define 'conflict of interest' in a professional ethics context.

A situation in which a professional's personal interests or external loyalties could improperly influence, or appear to influence, their professional judgment and duty to a client or the public.

What is 'whistle-blowing' in professional ethics, and when is it considered justified?

The act of an employee disclosing illegal, unethical or harmful practices within an organization. It is generally justified when there is serious threat to public safety, the concern is reported through proper internal channels first, and there is documented evidence.