International Baccalaureate Diploma (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. State the Pythagorean trigonometric identity and the double angle formulas for sine and cosine.

   sin^2 x + cos^2 x = 1. sin 2x = 2 sin x cos x. cos 2x = cos^2 x - sin^2 x = 2cos^2 x - 1 = 1 - 2sin^2 x.

2. Define tan x in terms of sin and cos, and give its identity with secant.

   tan x = sin x / cos x; and 1 + tan^2 x = sec^2 x.

3. State the sine rule and the cosine rule for a triangle.

   Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: c^2 = a^2 + b^2 - 2ab cos C.

4. State the formula for the area of a triangle given two sides and the included angle.

   Area = (1/2) ab sin C, where a and b are two sides and C is the angle between them.

5. (HL) What is the scalar (dot) product of two vectors, and what does it indicate about the angle?

   a . b = |a||b| cos theta = a1 b1 + a2 b2 + a3 b3. If a . b = 0 (and vectors nonzero), the vectors are perpendicular.

6. (HL) What is the vector (cross) product of two vectors and what does its magnitude represent?

   a x b is a vector perpendicular to both a and b with magnitude |a||b| sin theta, which equals the area of the parallelogram they span.

7. (HL) Give the vector equation of a line in 3D.

   r = a + t*d, where a is the position vector of a point on the line, d is the direction vector, and t is a scalar parameter.

8. What is the difference between the median, mean, and mode?

   Mean: sum of values divided by count (arithmetic average). Median: middle value when ordered. Mode: most frequently occurring value.

9. How are quartiles and the interquartile range (IQR) defined?

   Q1 is the lower quartile (25th percentile), Q3 the upper quartile (75th percentile), and IQR = Q3 - Q1, a measure of spread of the middle 50%.

10. What does Pearson's correlation coefficient r indicate, including its range and meaning of sign?

    r measures the strength and direction of a linear relationship; -1 <= r <= 1. r near +1 strong positive, near -1 strong negative, near 0 little/no linear correlation.

11. What is the equation of a regression line of y on x, and what does it pass through?

    y = a x + b (least squares line minimising vertical distances); it always passes through the mean point (x-bar, y-bar).

12. State the addition rule and the formula for conditional probability.

    P(A or B) = P(A) + P(B) - P(A and B). Conditional: P(A|B) = P(A and B)/P(B).

13. What does it mean for two events to be independent, and what is the resulting multiplication rule?

    Events A and B are independent if one occurring does not affect the other: P(A and B) = P(A) * P(B), equivalently P(A|B) = P(A).

14. State the mean and variance of a binomial distribution X ~ B(n, p).

    Mean E(X) = np; Variance Var(X) = np(1 - p). P(X = r) = C(n,r) p^r (1-p)^(n-r).

15. What are the defining properties of a normal distribution and the standardisation formula?

    Symmetric, bell-shaped, defined by mean mu and standard deviation sigma; about 68/95/99.7% of data lie within 1/2/3 sigma. Standardise with z = (x - mu)/sigma.

16. For a discrete random variable, how do you find the expected value E(X)?

    E(X) = sum of x * P(X = x) over all values x. The probabilities must sum to 1.

17. State the formal limit definition of the derivative.

    f'(x) = lim as h -> 0 of [f(x + h) - f(x)] / h.

18. State the power rule, product rule, quotient rule, and chain rule for differentiation.

    Power: d/dx(x^n) = n x^(n-1). Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v - uv')/v^2. Chain: d/dx f(g(x)) = f'(g(x)) g'(x).

19. Give the derivatives of sin x, cos x, e^x, and ln x.

    d/dx(sin x) = cos x; d/dx(cos x) = -sin x; d/dx(e^x) = e^x; d/dx(ln x) = 1/x.

20. How do you classify a stationary point using the first and second derivatives?

    Stationary points occur where f'(x) = 0. If f''(x) > 0 it is a local minimum; if f''(x) < 0 a local maximum; if f''(x) = 0 test further (could be a point of inflexion).

21. State the power rule for integration and the role of the constant of integration.

    Integral of x^n dx = x^(n+1)/(n+1) + C, for n not equal to -1; C is the arbitrary constant added for indefinite integrals.

22. How do you find the area between a curve y = f(x) and the x-axis, and the volume of revolution about the x-axis?

    Area = integral from a to b of f(x) dx. Volume about x-axis = pi * integral from a to b of [f(x)]^2 dx.

23. (HL) What is the method of solving a separable differential equation dy/dx = f(x)g(y)?

    Separate variables and integrate both sides: integral of (1/g(y)) dy = integral of f(x) dx, then solve for y including the constant.

24. (HL) State the general Maclaurin series and give the expansion of e^x.

    Maclaurin: f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + ... . For e^x: e^x = 1 + x + x^2/2! + x^3/3! + ...

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Group 5: Mathematics (Analysis and Approaches) flashcards FAQ