GATE Environmental Engineering Mathematics Foundation Flashcards

24 sample cards from the Mathematics Foundation deck

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

1. What is the integration by parts formula for indefinite integrals?

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

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

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

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

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

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

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

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

6. 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.

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

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

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

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

9. 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 $\dfrac{\partial^{2} f}{\partial x\,\partial y} = \dfrac{\partial^{2} f}{\partial y\,\partial x}$.

10. 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: $\displaystyle\int \frac{dy}{h(y)} = \int g(x)\,dx + C$.

11. 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 $y\,\mu = \int \mu\,Q\,dx + C$.

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

13. 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 $v$.

14. 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 $\alpha \pm i\beta$: $y = e^{\alpha x}(c_1 \cos\beta x + c_2 \sin\beta x)$.

15. 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.

16. 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 $t$.

17. 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 $m$.

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

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

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

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

20. 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 $Y(s) = \mathcal{L}\{y\}$.

21. 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 $y(t)$.

22. 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.

23. 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.

24. 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.

What this deck covers

Mathematics Foundation flashcards FAQ