UPSC ESE Electrical Engineering General Studies and Engineering Aptitude Syllabus

Define the order and degree of a differential equation.

Order is the highest derivative present; degree is the power of that highest-order derivative after the equation is made free of radicals/fractions in derivatives.

What is the general solution form of a first-order linear ODE $\frac{dy}{dx} + P(x)y = Q(x)$?

Using integrating factor $\mu = e^{\int P\,dx}$: $$y\,\mu = \int Q\,\mu\,dx + C$$

What is the condition for an equation $M\,dx + N\,dy = 0$ to be exact?

$$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$

State the Cauchy-Riemann equations for a complex function $f(z)=u(x,y)+iv(x,y)$ to be analytic.

$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$

State Cauchy's integral formula for an analytic function $f$ inside a closed contour $C$.

$$f(a) = \frac{1}{2\pi i} \oint_{C} \frac{f(z)}{z - a}\,dz$$

What is the residue theorem used for in complex analysis?

It evaluates a closed-contour integral as $\oint_C f(z)\,dz = 2\pi i \sum \text{Res}$, the sum of residues of $f$ at the poles enclosed by $C$.

Define a harmonic function in the context of complex variables.

A real function $u(x,y)$ is harmonic if it satisfies Laplace's equation $\dfrac{\partial^{2}u}{\partial x^{2}} + \dfrac{\partial^{2}u}{\partial y^{2}} = 0$; real and imaginary parts of an analytic function are harmonic.

State 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, $P(A\cap B)=0$.

State Bayes' theorem.

$$P(A_i \mid B) = \frac{P(B \mid A_i)\,P(A_i)}{\sum_{j} P(B \mid A_j)\,P(A_j)}$$

What are the mean and variance of a binomial distribution with $n$ trials and success probability $p$?

Mean $= np$; Variance $= np(1-p) = npq$, where $q = 1-p$.

Give the probability density function of the normal distribution.

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}$$ with mean $\mu$ and standard deviation $\sigma$.

What is the mean of a Poisson distribution and its relation to variance?

For a Poisson distribution with parameter $\lambda$, both the mean and variance equal $\lambda$.

Define standard deviation in terms of variance.

Standard deviation $\sigma = \sqrt{\text{Variance}}$, i.e., $\sigma = \sqrt{\dfrac{1}{N}\sum_{i=1}^{N}(x_i - \bar{x})^{2}}$ for a population.

State the iteration formula of the Newton-Raphson method for solving $f(x)=0$.

$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$

What is the order of convergence of the Newton-Raphson method versus the bisection method?

Newton-Raphson converges quadratically (order 2); the bisection method converges linearly (order 1) but is always guaranteed to converge if the root is bracketed.

State the trapezoidal rule for numerically approximating $\int_a^b f(x)\,dx$ with step $h$.

$$\int_a^b f(x)\,dx \approx \frac{h}{2}\left[y_0 + y_n + 2(y_1 + y_2 + \cdots + y_{n-1})\right]$$

State Simpson's 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.

List the common types of pencils used in engineering drawing by hardness grade.

H-grade (hard, light lines: 2H, 3H for construction/dimension lines), HB (medium, lettering), and B-grade (soft, dark lines: B, 2B for outlines and sketching).

What are the main drawing instruments and their primary uses?

T-square (horizontal lines), set squares/45° & 30°-60° (angled lines), compass (circles/arcs), divider (transferring distances), protractor (angles), and scales (measuring/proportioning).

What are the four standard line types and their meaning in engineering drawing?

Continuous thick (visible/object edges), continuous thin (dimension, extension, leader lines), short dashes (hidden edges), and long chain/dash-dot thin (center lines and axes).

State the recommended single-stroke lettering proportions in engineering drawing.

Single-stroke (Gothic) lettering is standard; common height-to-width ratio is about 7:5 (or 5:3), with uniform stroke thickness and consistent spacing for legibility.

What are the two main systems of placing dimensions in engineering drawing?

Aligned system (dimensions read parallel to the dimension line, rotated with it) and unidirectional system (all dimensions read horizontally from the bottom of the sheet).

Define 'leader line' and 'extension line' in dimensioning.

A leader line is a thin line (often inclined) directing a note or dimension to a feature; an extension (projection) line extends from the object outward to indicate the limits of a dimension without touching the object outline.

How do you geometrically bisect a given line segment using compass and straightedge?

With the compass set to more than half the length, draw arcs from each endpoint above and below the line; the straight line joining the two arc intersections bisects the segment perpendicularly.

Describe the geometric construction to divide a line into $n$ equal parts.

Draw an auxiliary line at an angle from one endpoint, step off $n$ equal divisions on it with the compass, join the last point to the other endpoint, then draw lines parallel to that final line through each division to intersect the original line.

In orthographic projection, what is the projection of a point that lies in the first quadrant?

Its front view (elevation) lies above the XY reference line and its top view (plan) lies below the XY line, since the point is above the horizontal plane and in front of the vertical plane.

What is the difference between first-angle and third-angle projection?

In first-angle (used in India/ISO), the object is between observer and plane, so the top view is placed below the front view. In third-angle (US), the plane is between observer and object, so the top view is placed above the front view.

How does a line inclined to both reference planes appear in its projections?

Such a line appears foreshortened (shorter than true length) in both front and top views; its true length and true inclinations are found by rotation/auxiliary-plane methods.

What is the difference between quality control (QC) and quality assurance (QA)?

Quality control is product-oriented: detecting defects through inspection and testing of the output. Quality assurance is process-oriented: preventing defects by establishing and following proper processes and standards.

Define Total Quality Management (TQM) and name its core principle.

TQM is an organization-wide, continuous improvement approach involving all employees to maximize customer satisfaction. Its core principle is continuous improvement (Kaizen) with a customer-focused, fact-based, total-participation culture.

Name the seven basic tools of quality control.

Cause-and-effect (Ishikawa/fishbone) diagram, check sheet, control chart, histogram, Pareto chart, scatter diagram, and flowchart (stratification is sometimes listed in place of flowchart).