CP1

Question 1

State the scalar product of perpendicular vectors

Answer 1

0

Question 2

Give the equation used to find the angle between 3D vectors

Answer 2

cosθ = a.b / |a||b|

Question 3

Give the equation for parametric integration dx

Answer 3

∫ y(x) dx = ∫ y(t) * dx/dt dt
Change limits from x to t

Question 4

Give the equation for the inverse of 2x2 matrix

Answer 4

1/detM *
(d -b)
(-c a)

Question 5

Give the equation for the volume of a curve rotated 360° round the x-axis

Answer 5

π ∫ y^2 dx

Question 6

Give the equation for the volume of a curve rotated 360° round the y-axis

Answer 6

π ∫ x^2 dy

Question 7

State how to find the Maclaurin series for a differentiable function

Answer 7

f(0) + xf’(0) + 1/2! x^2f’‘(0) + … + 1/n! * x^n ^ f^n(0)

Question 8

z^n + z^-n = ?

Answer 8

2 cos nθ

Question 9

z^n - z^-n = ?

Answer 9

2i sin nθ

Question 10

State De Moivre’s Theorem

Answer 10

(cosθ + isinθ)^n = cosnθ + isinnθ

Question 11

Describe how to show that cos^nθ = acosnθ + …

Answer 11

Write cos^nθ as
(2 cos θ)^n
Substitute in
z^n + z^-n and expand

Question 12

Give the equation for the volume of a parametric curve rotated 360° round the y-axis

Answer 12

π ∫ y^2 * dx/dt dt
Limits appropriate t-values

Question 13

Give the equation for the volume of a parametric curve rotated 360° round the x-axis

Answer 13

π ∫ x^2 * dy/dt dt
Limits appropriate t-values

Question 14

Define x in terms of r and θ

Answer 14

x = r cosθ

Question 15

Define y in terms of r and θ

Answer 15

y = r sinθ

Question 16

Define r^2 in terms of x and y

Answer 16

r^2 = x^2 + y^2

Question 17

Define tanθ in terms of y and x

Answer 17

tanθ = y / x

Question 18

When is the tangent of a polar curve horizontal?

Answer 18

dy/dθ = 0

Question 19

When is the tangent of a polar curve vertical?

Answer 19

dx/dθ = 0

Question 20

Define sinhx in terms of e and x

Answer 20

sinhx = (e^x - e^-x)/2

Question 21

Define coshx in terms of e and x

Answer 21

coshx = (e^x + e^-x)/2

Question 22

Give the formula for the shortest distance between a point and two points on a line

Answer 22

p - point
v,w - points on line
|(p-v) x (p-w)| / |v-w|

Question 23

Give the formula for the shortest distance between 2 skew lines

Answer 23

Lines
a + tb, p+tq
|(a-p).(bxq)/|bxq||

Question 24

Give the equation for the inverse of a 3 x 3 matrix

Answer 24

1/detM [bxc cxa axb]

Question 25

State the equation for the **displacement of a particle**, x, with maximum displacement a and acceleration -w^2x **following SHM**

Answer 25

x = a sin (wt + k)

Question 26

Give the equation for the **mean value** of a function

Answer 26

1/(b-a) * ∫a to b f(x) dx

Question 27

Give the equation for the **integrating factor** of a first-order differential equation.

Answer 27

For the equation **y' + P(x)y = Q(x)**, the integrating factor is **e^∫P(x)dx**.

Question 28

State the **complimentary functions** for each solution set of the auxiliary quadratic

Answer 28

m = a, b y(x) = Ae^ax + Be^bx m = a y(x) = Ae^ax + Bxe^ax m = a +- bi y(x) = Ae^ax cosbx + Be^ax sinbx

Question 29

Give the equation for the **full solution** to a **second order** differential equation

Answer 29

Complimentary function + particular integral

Question 30

State what **(w*)** is if w is an nth root of unity

Answer 30

w = e^2**k**iπ/n w* = e^2**(n-k)**iπ/n