International Baccalaureate Diploma Programme (IB) Group 5: Mathematics (Analysis and Approaches) Flashcards

24 sample cards from the Group 5: Mathematics (Analysis and Approaches) deck

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

1. For vectors $\vec{a}$ and $\vec{b}$, give the scalar (dot) product in component and geometric form (HL).

   $\vec{a}\cdot\vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 = |\vec{a}||\vec{b}|\cos\theta$

2. How do you test whether two non-zero vectors are perpendicular, and how is the magnitude of $\vec{a}$ found (HL)?

   They are perpendicular iff $\vec{a}\cdot\vec{b} = 0$; magnitude $|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$.

3. Give the vector equation of a line through point with position vector $\vec{a}$ in direction $\vec{b}$ (HL).

   $\vec{r} = \vec{a} + \lambda\vec{b}$, where $\lambda \in \mathbb{R}$

4. State the formula for the vector (cross) product magnitude and its geometric meaning (HL).

   $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$, equal to the area of the parallelogram formed by $\vec{a}$ and $\vec{b}$; the result is perpendicular to both.

5. Distinguish between the mean, median and mode as measures of central tendency.

   Mean: arithmetic average of all values. Median: the middle value when data are ordered. Mode: the most frequently occurring value.

6. Define the interquartile range (IQR) and state a common rule for outliers.

   $\text{IQR} = Q_3 - Q_1$. An outlier is any value below $Q_1 - 1.5\,\text{IQR}$ or above $Q_3 + 1.5\,\text{IQR}$.

7. What does the Pearson product-moment correlation coefficient $r$ measure, and what is its range?

   It measures the strength and direction of a linear relationship between two variables; $-1 \leq r \leq 1$, where $\pm 1$ is perfect linear correlation and $0$ is none.

8. State the addition rule and the definition of mutually exclusive events.

   $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. Events are mutually exclusive when $P(A \cap B) = 0$, giving $P(A \cup B) = P(A) + P(B)$.

9. Define conditional probability and state the condition for two events to be independent.

   $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$. $A$ and $B$ are independent iff $P(A \cap B) = P(A)P(B)$, equivalently $P(A\mid B) = P(A)$.

10. For a binomial distribution $X \sim B(n,p)$, give $P(X=r)$, the mean and the variance.

    $P(X=r) = \binom{n}{r}p^r(1-p)^{n-r}$; mean $E(X)=np$; variance $\operatorname{Var}(X)=np(1-p)$.

11. State the key properties of the normal distribution and the empirical 68-95-99.7 rule.

    Symmetric, bell-shaped about the mean $\mu$, with mean = median = mode. About $68\%$ of data lie within $1\sigma$, $95\%$ within $2\sigma$, and $99.7\%$ within $3\sigma$ of the mean.

12. How is a normal variable $X \sim N(\mu, \sigma^2)$ standardised to the $z$-score?

    $z = \frac{x - \mu}{\sigma}$, giving the standard normal distribution $Z \sim N(0, 1)$.

13. State the limit definition of the derivative $f'(x)$.

    $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

14. State the product rule, quotient rule and chain rule for differentiation.

    Product: $(uv)' = u'v + uv'$. Quotient: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$. Chain: $\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$.

15. Give the derivatives of $\sin x$, $\cos x$, $\mathrm{e}^x$ and $\ln x$.

    $\frac{d}{dx}\sin x = \cos x$; $\frac{d}{dx}\cos x = -\sin x$; $\frac{d}{dx}\mathrm{e}^x = \mathrm{e}^x$; $\frac{d}{dx}\ln x = \frac{1}{x}$

16. How do you classify a stationary point using the second derivative test?

    At a stationary point ($f'(x)=0$): if $f''(x) > 0$ it is a local minimum; if $f''(x) < 0$ it is a local maximum; if $f''(x) = 0$ the test is inconclusive (check further).

17. State the fundamental theorem of calculus linking integration and differentiation.

    If $F'(x) = f(x)$, then $\int_a^b f(x)\,dx = F(b) - F(a)$.

18. Give the power rule for integration and the integral of $\frac{1}{x}$.

    $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$); $\int \frac{1}{x}\,dx = \ln|x| + C$

19. State the formula for the volume of revolution about the $x$-axis between $x=a$ and $x=b$.

    $V = \pi \int_a^b [f(x)]^2\,dx$

20. What is the method of separation of variables for a differential equation $\frac{dy}{dx} = g(x)h(y)$ (HL)?

    Rearrange to $\frac{1}{h(y)}\,dy = g(x)\,dx$, then integrate both sides: $\int \frac{1}{h(y)}\,dy = \int g(x)\,dx$.

21. State the general Maclaurin series of a function $f(x)$ (HL).

    $f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$

22. Give the Maclaurin series for $\mathrm{e}^x$, $\sin x$ and $\cos x$ (HL).

    $\mathrm{e}^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}$; $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$; $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$

23. In mathematical modelling, what is the role of validating a model against data?

    Validation compares the model's predictions with observed/real data to assess accuracy; large discrepancies indicate the model's assumptions must be refined or the model adjusted.

24. What is the purpose of the Internal Assessment (IA) mathematical exploration in IB Maths AA?

    It is an individual written exploration (about 12-20 pages) of an area of mathematics of the student's choice, assessed on criteria including presentation, mathematical communication, personal engagement, reflection and use of mathematics.

What this deck covers

Group 5: Mathematics (Analysis and Approaches) flashcards FAQ