GATE Biotechnology Engineering Mathematics Flashcards

24 sample cards from the Engineering Mathematics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. 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 $t$.

2. 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 $m$.

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

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

4. 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 $\mathcal{L}\{e^{at}\} = \dfrac{1}{s-a}$.

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

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

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

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

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

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

8. 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.

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

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

10. Define variance and standard deviation of a population.

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

11. 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 $= \sigma^{2}$.

12. 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 $k = 0,1,\dots,n$.

13. 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 $\lambda = np$.

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

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

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

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

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

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

17. 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.

18. 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 $\mu \pm 2\sigma$).

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

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

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

21. 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).

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

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

23. 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).

24. 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.

What this deck covers

Engineering Mathematics flashcards FAQ