Engineering and Science Admissions Test (ESAT) Mathematics 2 (Advanced Mathematics Module) Flashcards

24 sample cards from the Mathematics 2 (Advanced Mathematics Module) deck

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

1. How do you solve a separable differential equation $\frac{dy}{dx}=f(x)g(y)$?

   Separate variables and integrate both sides: $$\int\frac{1}{g(y)}\,dy=\int f(x)\,dx$$ then include a constant and apply any initial condition.

2. What is the general solution of $\frac{dy}{dx}=ky$, and what does it model?

   $y=Ae^{kx}$. It models exponential growth ($k>0$) or decay ($k<0$).

3. What does a 'general solution' versus a 'particular solution' of a differential equation mean?

   The general solution contains arbitrary constant(s); a particular solution fixes those constants using given boundary or initial conditions.

4. State the Pythagorean trigonometric identities.

   $\sin^{2}\theta+\cos^{2}\theta=1$, $\quad 1+\tan^{2}\theta=\sec^{2}\theta$, $\quad 1+\cot^{2}\theta=\csc^{2}\theta$.

5. State the compound angle formulae for $\sin(A\pm B)$ and $\cos(A\pm B)$.

   $\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$; $\quad\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$.

6. State the double angle formulae for $\sin 2\theta$ and the three forms of $\cos 2\theta$.

   $\sin 2\theta=2\sin\theta\cos\theta$; $\quad\cos 2\theta=\cos^{2}\theta-\sin^{2}\theta=2\cos^{2}\theta-1=1-2\sin^{2}\theta$.

7. How is $a\sin\theta+b\cos\theta$ written in the form $R\sin(\theta+\alpha)$?

   $R=\sqrt{a^{2}+b^{2}}$ and $\tan\alpha=\frac{b}{a}$ (with $R>0$, $\alpha$ acute).

8. How do you convert parametric equations $x=f(t)$, $y=g(t)$ to Cartesian form?

   Eliminate the parameter $t$ — solve one equation for $t$ (or use an identity) and substitute into the other to get a relation between $x$ and $y$.

9. Give the parametric equations of a circle of radius $r$ centred at the origin.

   $x=r\cos t$, $\quad y=r\sin t$, for $0\leq t<2\pi$ (since $x^{2}+y^{2}=r^{2}$).

10. State the formula for the $n$th term of an arithmetic sequence and the sum of the first $n$ terms.

    $u_{n}=a+(n-1)d$; $\quad S_{n}=\frac{n}{2}\big(2a+(n-1)d\big)=\frac{n}{2}(a+l)$.

11. State the $n$th term and sum to $n$ terms of a geometric sequence.

    $u_{n}=ar^{n-1}$; $\quad S_{n}=\frac{a(1-r^{n})}{1-r}$ for $r\neq 1$.

12. When does an infinite geometric series converge, and to what sum?

    It converges when $|r|<1$, to $$S_{\infty}=\frac{a}{1-r}$$

13. What does an iterative formula $x_{n+1}=g(x_{n})$ converge to, if it converges?

    A root of $x=g(x)$ (a fixed point), which is a solution of the equation rearranged into the form $x=g(x)$.

14. How do you find the magnitude of the vector $\vec{a}=\begin{pmatrix}x\\y\\z\end{pmatrix}$?

    $$|\vec{a}|=\sqrt{x^{2}+y^{2}+z^{2}}$$

15. State the scalar (dot) product formula and the condition for perpendicular vectors.

    $\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}$. Vectors are perpendicular iff $\vec{a}\cdot\vec{b}=0$.

16. How is the angle between two vectors found from their dot product?

    $$\cos\theta=\frac{\vec{a}\cdot\vec{b}}{|\vec{a}|\,|\vec{b}|}$$

17. State the vector equation of a straight line through point $\vec{a}$ with direction $\vec{d}$.

    $$\vec{r}=\vec{a}+\lambda\vec{d},\quad \lambda\in\mathbb{R}$$

18. How does the graph of $y=f(x-a)+b$ relate to $y=f(x)$?

    It is translated by the vector $\begin{pmatrix}a\\b\end{pmatrix}$: $a$ units in the positive $x$-direction and $b$ units in the positive $y$-direction.

19. Describe the transformations $y=f(ax)$ and $y=af(x)$.

    $y=f(ax)$ is a horizontal stretch with scale factor $\frac{1}{a}$; $y=af(x)$ is a vertical stretch with scale factor $a$.

20. What transformation maps $y=f(x)$ to $y=|f(x)|$ versus $y=f(|x|)$?

    $y=|f(x)|$ reflects any part below the $x$-axis up above it. $y=f(|x|)$ keeps $x\geq 0$ and reflects it in the $y$-axis.

21. What is the condition for a function to have an inverse, and how do their graphs relate?

    It must be one-to-one (injective) on its domain. The graph of $y=f^{-1}(x)$ is the reflection of $y=f(x)$ in the line $y=x$.

22. State the Newton-Raphson iterative formula for solving $f(x)=0$.

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

23. How does the change-of-sign method locate a root of $f(x)=0$?

    If $f$ is continuous and $f(a)$ and $f(b)$ have opposite signs, then there is at least one root in the interval $(a,b)$.

24. State the trapezium rule for estimating $\int_{a}^{b}y\,dx$ with $n$ strips of width $h$.

    $$\int_{a}^{b}y\,dx\approx\frac{h}{2}\Big(y_{0}+y_{n}+2(y_{1}+y_{2}+\cdots+y_{n-1})\Big),\quad h=\frac{b-a}{n}$$

What this deck covers

Mathematics 2 (Advanced Mathematics Module) flashcards FAQ