Tricky Topics (Pure/Mech)

Question 1

Finding Invariant Points

Answer 1

Points that don’t move under a given transformation.

(a b) (x) - (x)
(c d) (y) - (y)

Finds line of invariant points

Question 2

Finding Invariant Lines

Answer 2

Finds points on a line that, when transformed, move to another point on the line.

(a b) (x) - (X)
(c d) (mx + c) - (mX + c)

Simultaneous Equations
(sub in top equation to remove big X)

Question 3

Proof by Induction - Final Statement

Answer 3

• Since true for n = 1
• And true for n = k + 1
• WHEN ASSUMED true for n = k
• Thus true for all n ∈ ℤ*

Question 4

Acceleration = 0

Answer 4

• No resultant force -> Resolve to = 0
• Constant Velocity (could be 0) (N’s 1st Law)
  → Remember vice versa
• Use GCSE Speed = D/T

Question 5

Acceleration = Constant

Answer 5

SUVAT

Question 6

Acceleration = Variable

Answer 6

Differentiate given expression
(usually dx/dt)

Question 7

Vector Projection of a onto b
(Vectors in Further Mechanics)

Answer 7

(a.b / |b|^2) x b

Question 8

Forgotten Equation: Further Mech
Vectors, Impulse, e

Answer 8

-e(u . I) = v . I

Question 9

Elasticity Question Techniques

Answer 9

Work Energy Principle
- Asking for energy
- Distance
- Rest

Resolve Forces
- Acceleration

Question 10

Trig Identities (De Moivre’s)

Answer 10

e.g. cos^5(θ) = …
(z + 1/z)^5 = (2cosθ)^5 = 32cos^5(θ)
(z + 1/z)^5 = BINOMIAL EXPANSION

e.g. cos5θ = …
(cosθ + isinθ)^5 = cos5θ + isin5θ
REAL part of BINOMIAL EXPANSION

Question 11

Oblique Collisions - N’s Law of Restitution

Answer 11

v(sinβ) = eu(sinα)
v(cosβ) = u(cosα)

tanβ = e(tanα)

Question 12

Proof by Induction - Mathematical (Σ)

Answer 12

e.g. nΣ (2r-1) = n^2

Basis, Assumption…

(k+1)Σ = (k+1)^2 → AIM
(k+1)Σ = kΣ + (2(k+1) -1)
→ Sub in (k+1) to formula

Question 13

Proof by Induction - Divisibility Results

Answer 13

e.g. Divisible by 4

f(k+1) - f(k) = 4(x)
f(k+1) -f(k) = af(k) +4(x)
f(k+1) = (a+1)f(k) + 4(x)

Question 14

Proof by Induction - Matrices

Answer 14

(M)^k+1 = (M)^k x (M)^1

Question 15

Vector ACUTE ANGLE Formulae

Answer 15

cos/sinθ = | (a .b) / |a||b||

Two Lines = cos
(Direction vectors)

Line & Plane = sin
(Directions vector & Normal)

Two Planes = cos
(Normals)

Question 16

Sums of Series (Complex No. - Yr2)

Answer 16

+/-1e^aiθ +/- 1 → xe^-1/2iθ

Question 17

Types of Integration

Answer 17

• Inspection
• Parts
• Substitution
• (Partial Fractions)

Question 18

∫ ln(x) dx

Answer 18

x(lnx) - x [+ c]

Question 19

Differentiating & Integrating: a^x

Answer 19

∫dx = 1/lna (a^x) [+ c]
d/dx = a^x(lnx)

Question 20

Improper Integrals

Answer 20

∫ f(x) dx = Improper if…
- One/both of the limits is infinite
- If f(x) is undefined at some point
between the limits

Question 21

∫ f(x) dx between ∞ and a

Answer 21

lim t → ∞
∫ f(x) dx between t and a

Either 1/∞ tends towards 0, which is then ignored in the final answer, or the entire graph tends towards ∞.