Flashcards · YMA01 · Topic 4

Unit P4: Pure Mathematics 4 flashcards

36 cards on Topic 4, subtopics P4.1 to P4.7. Pick a subtopic or study them all.

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P4.1 Proof

Proof by contradiction: the four steps: 1. Assume the opposite of the statement. 2. Make valid deductions. 3. Reach something impossible. 4. Conclude the assumption is false, so the statement is true.
Opening assumption when proving √2 is irrational: Assume √2 is rational, so √2 = p/q with p and q integers having no common factor (lowest terms). Then p² = 2q², so p is even; then q is even too, contradicting lowest terms.
Outline the proof that there are infinitely many primes: Assume there is a finite list of all the primes. Let N = (product of all of them) + 1. N leaves remainder 1 on division by every listed prime, so it has a prime factor not on the list. Contradiction.
Key fact used when proving a square root is irrational: n² is even if and only if n is even. More generally, if a prime p divides n², then p divides n.
Why is the product of a non-zero rational r and an irrational s irrational?: Assume rs = a/b is rational, with r = c/d and c ≠ 0. Then s = ad/(bc), a ratio of integers, so s would be rational. This contradicts s being irrational.

P4.2 Algebra and functions

Partial fraction form for a denominator (ax + b)(cx + d): A/(ax + b) + B/(cx + d). Three distinct linear factors give three such terms.
Partial fraction form for a denominator (ax + b)(cx + d)²: A/(ax + b) + B/(cx + d) + C/(cx + d)². A repeated factor needs a term for each power.
What must you do first if the numerator degree is at least the denominator degree?: Divide first, to get a polynomial plus a proper fraction, then split the proper fraction into partial fractions.
Express (13x + 5)/((x + 2)(3x − 1)) in partial fractions: 13x + 5 = A(3x − 1) + B(x + 2). x = −2: −21 = −7A, A = 3. x = 1/3: 28/3 = 7B/3, B = 4. Result: 3/(x + 2) + 4/(3x − 1).
Quick way to find partial fraction constants: Multiply through by the whole denominator, then substitute the root of each factor to get one constant at a time. For a repeated factor, compare coefficients (such as x²) for the last one. A constant of 0 is a valid answer.

P4.3 Coordinate geometry in the (x, y) plane

Parametric equations: x = f(t) and y = g(t), with both coordinates given in terms of a parameter t. Eliminating t gives the cartesian equation.
How do you convert trig parametric equations to cartesian form?: Write sin t and cos t (or tan t and sec t) in terms of x and y, then use sin²t + cos²t = 1 or 1 + tan²t = sec²t. If the angles are t and 2t, use a double angle formula first.
Find the cartesian equation of x = sin t, y = cos²t: cos²t = 1 − sin²t, so y = 1 − x². Because x = sin t, the curve is only the part with −1 ≤ x ≤ 1.
Find the cartesian equation of x = 1/t, y = t² + t (t > 0): t = 1/x, so y = 1/x² + 1/x, for x > 0 (since t > 0).

P4.4 Binomial expansion

The binomial series for (1 + x)ⁿ, n any rational number: (1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + n(n − 1)(n − 2)x³/3! + … Valid for |x| < 1 when n is not a positive integer. For a positive integer n it stops and is valid for all x.
How do you expand (a + bx)ⁿ for a negative or fractional n?: Take out aⁿ: (a + bx)ⁿ = aⁿ(1 + (b/a)x)ⁿ. Expand with u = (b/a)x, then multiply by aⁿ. Valid for |bx/a| < 1, that is |x| < |a/b|.
Expansion of (1 + x)⁻¹ and of (1 + x)^(1/2) up to x³: (1 + x)⁻¹ = 1 − x + x² − x³ + … (1 + x)^(1/2) = 1 + x/2 − x²/8 + x³/16 − … Both valid for |x| < 1.
Expand (2 − 3x)⁻² up to x² and state the validity: 2⁻²(1 − 3x/2)⁻² = ¼(1 + 3x + 27x²/4 + …) = 1/4 + 3x/4 + 27x²/16. Valid for |3x/2| < 1, so |x| < 2/3.
Range of validity when a rational function is expanded using partial fractions: Expand each partial fraction separately and add. The whole series is valid only where every part is valid: take the narrowest range.

P4.5 Differentiation

Implicit differentiation: d/dx of f(y), and of xy: d/dx [f(y)] = f′(y) dy/dx, for example d/dx (y³) = 3y² dy/dx. d/dx (xy) = y + x dy/dx (product rule).
Find the gradient of y³ + 2xy = 12 at (1, 2): 3y² dy/dx + 2y + 2x dy/dx = 0, so dy/dx = −2y/(2x + 3y²). At (1, 2): −4/14 = −2/7.
Gradient of a parametric curve: dy/dx = (dy/dt) ÷ (dx/dt). Horizontal tangent where dy/dt = 0; vertical tangent where dx/dt = 0.
Connected rates of change: the chain rule link: dy/dt = (dy/dx) × (dx/dt). Example: A = πr², so dA/dt = 2πr × dr/dt. With r = 7 and dr/dt = 0.4 cm s⁻¹, dA/dt = 5.6π cm² s⁻¹.
Form a differential equation: "the rate of decrease of h is proportional to √h": dh/dt = −k√h, with k > 0. A decreasing quantity needs the minus sign.

P4.6 Integration

Integration by substitution: the steps: 1. Choose u and find du/dx; rewrite dx. 2. Replace every x (solve for x if needed). 3. For a definite integral, change the limits to u-values. 4. Integrate, then substitute back if indefinite.
The integration by parts formula, and how to choose u: ∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx. Choose u to simplify when differentiated: u = x for x eᵏˣ, x sin kx or x cos kx; u = ln x for xⁿ ln x.
Evaluate ∫ from 0 to 1 of x e^(3x) dx: [⅓x e^(3x) − ⅑e^(3x)] from 0 to 1 = (⅓e³ − ⅑e³) − (0 − ⅑) = (2e³ + 1)/9.
Volume of revolution about the x-axis: V = π ∫ y² dx between x-limits (rotation through 360°). Example: y = x + 1 from 0 to 2: V = π[(x + 1)³/3] = π(27/3 − 1/3) = 26π/3.
Area and volume for a parametric curve: Area = ∫ y (dx/dt) dt. Volume about the x-axis = π ∫ y² (dx/dt) dt. Convert the limits to values of t first.
Solving a separable differential equation dy/dx = f(x)g(y): Separate: ∫ 1/g(y) dy = ∫ f(x) dx, with one constant, then use given values to find it. Example: dy/dx = 3y, y = 2 at x = 0: ln y = 3x + c, so y = Ae^(3x) and A = 2, giving y = 2e^(3x). Note ln y = x + c gives y = Aeˣ, not eˣ + c.

P4.7 Vectors

Magnitude of a = a₁i + a₂j + a₃k, and the unit vector in its direction: |a| = √(a₁² + a₂² + a₃²). Unit vector â = a/|a|. Example: v = i − 2j + 2k has |v| = 3, so v̂ = (i − 2j + 2k)/3.
Vector from A to B in terms of position vectors: AB = OB − OA = b − a: "end minus start". Writing a − b gives BA.
Vector equation of a line: r = a + tb, where a is the position vector of a point on the line and b is a direction vector. Through points C and D: r = c + t(d − c).
The scalar product and the angle between two vectors: a·b = a₁b₁ + a₂b₂ + a₃b₃, and cos θ = (a·b)/(|a||b|). For non-zero vectors, a·b = 0 means perpendicular. For the angle between two lines use only the direction vectors, and give the acute angle.
Find the angle between i − 2j + 2k and 2i + 3j + 6k: Scalar product = 2 − 6 + 12 = 8. Magnitudes 3 and 7. cos θ = 8/21, θ = 67.6° (1 d.p.).
How do you decide whether two lines in 3D are parallel, intersecting or skew?: 1. Directions are multiples: parallel. 2. Otherwise use different parameters, equate components, solve two equations and test the third. 3. Third holds: they intersect. Third fails: skew.