GATE Biotechnology Engineering Mathematics Syllabus

For $z = f(x,y)$, what is the total differential $dz$?

$dz = \dfrac{\partial f}{\partial x}\,dx + \dfrac{\partial f}{\partial y}\,dy$.

State the necessary (first-order) condition for a stationary point of $f(x,y)$.

$\dfrac{\partial f}{\partial x} = 0$ and $\dfrac{\partial f}{\partial y} = 0$ simultaneously.

State the second-derivative test for $f(x,y)$ using $D = f_{xx}f_{yy} - (f_{xy})^{2}$ at a stationary point.

If $D>0$ and $f_{xx}<0$: local maximum. If $D>0$ and $f_{xx}>0$: local minimum. If $D<0$: saddle point. If $D=0$: test inconclusive.

State Euler's theorem for a homogeneous function $f(x,y)$ of degree $n$.

$x\dfrac{\partial f}{\partial x} + y\dfrac{\partial f}{\partial y} = n f$.

What is the difference between a sequence and a series?

A sequence is an ordered list $\{a_{n}\}$ of terms; a series is the sum of those terms, $\sum_{n=1}^{\infty} a_{n}$.

When does a geometric series $\sum_{n=0}^{\infty} ar^{n}$ converge, and to what sum?

It converges when $|r| < 1$, with sum $\dfrac{a}{1 - r}$; it diverges when $|r| \geq 1$.

For what values of $p$ does the $p$-series $\sum_{n=1}^{\infty} \frac{1}{n^{p}}$ converge?

It converges if $p > 1$ and diverges if $p \leq 1$.

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

If $\lim_{n \to \infty} a_{n} \neq 0$, the series diverges. (If the limit is $0$, the test is inconclusive.)

State the ratio test (D'Alembert) for convergence of $\sum a_{n}$.

Let $L = \lim_{n \to \infty} \left|\dfrac{a_{n+1}}{a_{n}}\right|$. The series converges if $L < 1$, diverges if $L > 1$, and the test is inconclusive if $L = 1$.

State the root test (Cauchy) for convergence of $\sum a_{n}$.

Let $L = \lim_{n \to \infty} \sqrt[n]{|a_{n}|}$. Converges if $L < 1$, diverges if $L > 1$, inconclusive if $L = 1$.

What is the order and degree of a differential equation?

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

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

Integrating factor $\mu = e^{\int P\,dx}$, and the solution is $y\,\mu = \int Q\,\mu\,dx + C$.

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

$\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$.

Define a Bernoulli equation and the substitution used to solve it.

$\dfrac{dy}{dx} + P(x)y = Q(x)y^{n}$. The substitution $v = y^{1-n}$ reduces it to a linear first-order ODE.

For a higher-order linear ODE with constant coefficients, how do real distinct roots $m_{1}, m_{2}$ of the auxiliary equation appear in the complementary function?

$y_{c} = C_{1}e^{m_{1}x} + C_{2}e^{m_{2}x}$.

For a constant-coefficient linear ODE, what is the complementary function for a repeated real root $m$ (multiplicity 2)?

$y_{c} = (C_{1} + C_{2}x)e^{mx}$.

For a constant-coefficient linear ODE, what form does the complementary function take for complex roots $m = \alpha \pm i\beta$?

$y_{c} = e^{\alpha x}\left(C_{1}\cos\beta x + C_{2}\sin\beta x\right)$.

What substitution converts a Cauchy-Euler equation $x^{2}y'' + axy' + by = 0$ into a constant-coefficient ODE?

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

For the Cauchy-Euler equation, what trial solution is assumed to form its auxiliary equation directly?

$y = x^{m}$, giving an indicial (auxiliary) equation in $m$.

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

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

What is $\mathcal{L}\{1\}$, $\mathcal{L}\{t^{n}\}$, and $\mathcal{L}\{e^{at}\}$?

$\mathcal{L}\{1\} = \dfrac{1}{s}$, $\mathcal{L}\{t^{n}\} = \dfrac{n!}{s^{n+1}}$, and $\mathcal{L}\{e^{at}\} = \dfrac{1}{s-a}$.

Give $\mathcal{L}\{\sin at\}$ and $\mathcal{L}\{\cos at\}$.

$\mathcal{L}\{\sin at\} = \dfrac{a}{s^{2}+a^{2}}$ and $\mathcal{L}\{\cos at\} = \dfrac{s}{s^{2}+a^{2}}$.

State the Laplace transform of a derivative $f'(t)$.

$\mathcal{L}\{f'(t)\} = sF(s) - f(0)$.

State the first shifting (s-shifting) theorem for Laplace transforms.

If $\mathcal{L}\{f(t)\} = F(s)$, then $\mathcal{L}\{e^{at}f(t)\} = F(s-a)$.

Define mean, median, and mode of a data set.

Mean is the arithmetic average $\bar{x} = \frac{1}{n}\sum x_{i}$; median is the middle value when data are ordered; mode is the most frequently occurring value.

Give the empirical relationship between mean, median, and mode for a moderately skewed distribution.

$\text{Mode} \approx 3\,\text{Median} - 2\,\text{Mean}$.

Define variance and standard deviation of a population.

Variance $\sigma^{2} = \dfrac{1}{n}\sum (x_{i} - \bar{x})^{2}$; standard deviation $\sigma = \sqrt{\sigma^{2}}$.

Give the mean and variance of the Poisson, Binomial, and Normal distributions.

Poisson: mean $= \lambda$, variance $= \lambda$. Binomial: mean $= np$, variance $= npq$. Normal $N(\mu,\sigma^{2})$: mean $= \mu$, variance $= \sigma^{2}$.

State the Binomial probability mass function for $k$ successes in $n$ trials.

$P(X=k) = \binom{n}{k} p^{k} q^{n-k}$, where $q = 1-p$ and $k = 0,1,\dots,n$.

State the Poisson probability mass function and when it approximates the binomial.

$P(X=k) = \dfrac{e^{-\lambda}\lambda^{k}}{k!}$. It approximates the binomial when $n$ is large and $p$ is small, with $\lambda = np$.

What is the formula for the Pearson correlation coefficient $r$?

$r = \dfrac{\text{cov}(x,y)}{\sigma_{x}\sigma_{y}} = \dfrac{\sum (x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum (x_{i}-\bar{x})^{2}\sum (y_{i}-\bar{y})^{2}}}$.

What is the iterative formula of the Newton-Raphson method for solving $f(x)=0$?

$x_{n+1} = x_{n} - \dfrac{f(x_{n})}{f'(x_{n})}$.

State the composite trapezoidal rule for $\int_{a}^{b} f(x)\,dx$ with step size $h$.

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

State Simpson's $\frac{1}{3}$ rule for numerical integration with step size $h$.

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

What is the standard normal variable $z$, and what fraction of data lies within $\pm 1\sigma$ of the mean?

$z = \dfrac{x - \mu}{\sigma}$; about $68\%$ of data lie within $\mu \pm \sigma$ (and $\approx 95\%$ within $\mu \pm 2\sigma$).

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

$y_{n+1} = y_{n} + h\,f(x_{n}, y_{n})$.

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

$y_{n+1} = y_{n} + \dfrac{1}{6}(k_{1} + 2k_{2} + 2k_{3} + k_{4})$, where $k_{1}=hf(x_{n},y_{n})$, $k_{2}=hf(x_{n}+\tfrac{h}{2}, y_{n}+\tfrac{k_{1}}{2})$, $k_{3}=hf(x_{n}+\tfrac{h}{2}, y_{n}+\tfrac{k_{2}}{2})$, $k_{4}=hf(x_{n}+h, y_{n}+k_{3})$.

What is a random variable, and how do discrete and continuous random variables differ?

A random variable assigns a numerical value to each outcome of an experiment. A discrete RV takes countable values (described by a PMF); a continuous RV takes values over an interval (described by a PDF).

For a continuous random variable with PDF $f(x)$, give the expected value (mean).

$E[X] = \mu = \int_{-\infty}^{\infty} x\,f(x)\,dx$.

What does the slope $b$ of the regression line $y = a + bx$ represent, and how is it computed?

$b = \dfrac{\sum(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum(x_{i}-\bar{x})^{2}}$; it is the change in $y$ per unit change in $x$ that minimizes the sum of squared residuals (least squares).

Compare the bisection method and the Newton-Raphson method for root finding.

Bisection is always convergent (bracketing, needs sign change) but slow with linear convergence; Newton-Raphson is faster (quadratic convergence) but needs the derivative and a good initial guess, and may diverge.