GATE Environmental Engineering Mathematics Foundation Syllabus

What is a Maclaurin series, and write the expansion of $e^{x}$.

A Maclaurin series is a Taylor series about $a=0$. $$e^{x} = \sum_{n=0}^{\infty} \frac{x^{n}}{n!} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots$$

Write the Maclaurin series for $\sin x$ and $\cos x$.

$$\sin x = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n+1}}{(2n+1)!}, \qquad \cos x = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!}$$

State the Ratio Test for convergence of the series $\sum a_n$.

Let $L = \lim_{n\to\infty} \left|\dfrac{a_{n+1}}{a_n}\right|$. If $L < 1$ the series converges absolutely; if $L > 1$ it diverges; if $L = 1$ the test is inconclusive.

State the $p$-series test: for which values of $p$ does $\sum_{n=1}^{\infty} \dfrac{1}{n^{p}}$ converge?

The $p$-series converges if $p > 1$ and diverges if $p \leq 1$ (the harmonic series $p=1$ diverges).

State the $n$-th term (divergence) test for a series $\sum a_n$.

If $\lim_{n\to\infty} a_n \neq 0$ (or does not exist), the series diverges. If the limit is $0$, the test is inconclusive.

State the Root Test for the series $\sum a_n$.

Let $L = \lim_{n\to\infty} \sqrt[n]{|a_n|}$. If $L < 1$ converges absolutely; if $L > 1$ diverges; if $L = 1$ inconclusive.

State the Leibniz (alternating series) test for $\sum (-1)^{n} b_n$ with $b_n > 0$.

The alternating series converges if $b_n$ is monotonically decreasing and $\lim_{n\to\infty} b_n = 0$.

State the Fundamental Theorem of Calculus (evaluation part) for a definite integral.

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

What is the integration by parts formula for indefinite integrals?

$\displaystyle\int u\,dv = uv - \int v\,du$

Give the general power rule for indefinite integration of $x^{n}$ (with $n \neq -1$).

$\displaystyle\int x^{n}\,dx = \frac{x^{n+1}}{n+1} + C$, and for $n=-1$, $\displaystyle\int \frac{1}{x}\,dx = \ln|x| + C$.

Write the formula for the area between two curves $y=f(x)$ and $y=g(x)$ (with $f \geq g$) over $[a,b]$.

$$A = \int_{a}^{b} \big[f(x) - g(x)\big]\,dx$$

Write the disk method formula for the volume of a solid of revolution of $y=f(x)$ about the $x$-axis from $a$ to $b$.

$$V = \pi \int_{a}^{b} \big[f(x)\big]^{2}\,dx$$

Write the shell method formula for the volume of revolution about the $y$-axis of $y=f(x)$ from $a$ to $b$.

$$V = 2\pi \int_{a}^{b} x\,f(x)\,dx$$

What is the geometric meaning of a partial derivative $\dfrac{\partial f}{\partial x}$ for $f(x,y)$?

It is the rate of change of $f$ with respect to $x$ while holding $y$ constant — the slope of the surface in the $x$-direction.

Write the total differential $df$ of a function $f(x,y)$.

$$df = \frac{\partial f}{\partial x}\,dx + \frac{\partial f}{\partial y}\,dy$$

State the chain rule for the total derivative of $z=f(x,y)$ where $x=x(t)$ and $y=y(t)$.

$$\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$$

State the condition (equality of mixed partials) for a function $f(x,y)$ to satisfy Clairaut's theorem.

If the mixed partials are continuous, then $\dfrac{\partial^{2} f}{\partial x\,\partial y} = \dfrac{\partial^{2} f}{\partial y\,\partial x}$.

What is the general form and solution method for a separable first-order ODE?

Form $\dfrac{dy}{dx} = g(x)h(y)$. Separate variables and integrate: $\displaystyle\int \frac{dy}{h(y)} = \int g(x)\,dx + C$.

Write the general form of a linear first-order ODE and its integrating factor.

Form: $\dfrac{dy}{dx} + P(x)y = Q(x)$. Integrating factor: $\mu(x) = e^{\int P(x)\,dx}$, giving solution $y\,\mu = \int \mu\,Q\,dx + C$.

What is the condition for the first-order ODE $M(x,y)\,dx + N(x,y)\,dy = 0$ to be exact?

It is exact if $\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$.

What substitution converts a Bernoulli equation $\dfrac{dy}{dx} + P(x)y = Q(x)y^{n}$ into a linear ODE?

The substitution $v = y^{1-n}$ transforms it into a linear first-order ODE in $v$.

For a homogeneous higher-order linear ODE with constant coefficients, what are the solution forms for distinct real, repeated real, and complex conjugate roots of the auxiliary equation?

Distinct real roots $m_1, m_2$: $y = c_1 e^{m_1 x} + c_2 e^{m_2 x}$. Repeated root $m$: $y = (c_1 + c_2 x)e^{m x}$. Complex roots $\alpha \pm i\beta$: $y = e^{\alpha x}(c_1 \cos\beta x + c_2 \sin\beta x)$.

How is the general solution of a non-homogeneous linear ODE structured?

$y = y_c + y_p$, where $y_c$ (complementary function) solves the homogeneous equation and $y_p$ is any particular solution of the non-homogeneous equation.

What substitution reduces a Cauchy–Euler equation $a x^{2} y'' + b x y' + c y = 0$ to a constant-coefficient ODE?

The substitution $x = e^{t}$ (i.e., $t = \ln x$) converts it into a linear ODE with constant coefficients in $t$.

For the Cauchy–Euler equation $a x^{2} y'' + b x y' + c y = 0$, what trial solution is assumed and what equation results?

Assume $y = x^{m}$, giving the indicial (auxiliary) equation $a\,m(m-1) + b\,m + c = 0$, solved for $m$.

Define the Laplace transform $\mathcal{L}\{f(t)\}$ of a function $f(t)$.

$$\mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t)\,dt$$

State the Laplace transforms of $1$, $t^{n}$, $e^{at}$, $\sin(at)$, and $\cos(at)$.

$\mathcal{L}\{1\} = \frac{1}{s}$; $\mathcal{L}\{t^{n}\} = \frac{n!}{s^{n+1}}$; $\mathcal{L}\{e^{at}\} = \frac{1}{s-a}$; $\mathcal{L}\{\sin at\} = \frac{a}{s^{2}+a^{2}}$; $\mathcal{L}\{\cos at\} = \frac{s}{s^{2}+a^{2}}$.

State the Laplace transform of the first and second derivatives of $y(t)$.

$\mathcal{L}\{y'\} = sY(s) - y(0)$ and $\mathcal{L}\{y''\} = s^{2}Y(s) - s\,y(0) - y'(0)$, where $Y(s) = \mathcal{L}\{y\}$.

How is the Laplace transform used to solve a linear ODE with initial conditions?

Take the Laplace transform of both sides (turning derivatives into algebraic terms using initial conditions), solve algebraically for $Y(s)$, then apply the inverse Laplace transform to obtain $y(t)$.

Define the mean, median, and mode in descriptive statistics.

Mean: arithmetic average $\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$. Median: the middle value when data are ordered. Mode: the most frequently occurring value.

Write the formula for the population variance $\sigma^{2}$ and standard deviation $\sigma$.

$$\sigma^{2} = \frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^{2}, \qquad \sigma = \sqrt{\sigma^{2}}$$ where $\mu$ is the population mean.

What is the coefficient of variation, and what does it measure?

$\text{CV} = \dfrac{\sigma}{\bar{x}} \times 100\%$. It measures relative dispersion (standard deviation relative to the mean), allowing comparison of variability between data sets with different units or scales.