Pure Maths

Question 1

What are the solutions of a homogeneous SODE when the auxiliary equation has:

• Two distinct roots
• Repeated roots
• Complex roots ?

Answer 1

y = Ae^(mx)+Be^(nx) if roots are m and n
y = (Ax+b)e^(mx) if repeated root is m
y = e^(αx)(Acos β+Bcos β) where α±βi

Question 2

What is the solution of a homogeneous SODE when the auxiliary equation has repeated roots?

Answer 2

y = (Ax+b)e^(mx)

Question 3

What is the solution to:

y’ + ky = 0 ?

Answer 3

y = Ae^(-kx)

Question 4

What is the formula of volume of revolution?

Answer 4

π∫f(x)^2 dx

Question 5

How do you calculate the inverse of a 3x3 matrix?

Answer 5

M = 1 2 3
0 1 4
5 6 0
Calculate the determinant:

det(M) = 1(0-24) - 2(0-20) +3(0-5) = 1

Transpose the matrix:
M^T = 1 0 5
           2 1 6
           3 4 0
Calculate the determinant of each of the 2x2 minor matrices:

Det = -24 -18 5

      - 20 -16 4
      - 5    -4  1 

Create the matrix of cofactors:

Adj(M) = -24 -18 5 + - +

          - 20 -16 4  x  - + -
          - 5    -4  1      + - +

Adj(M) = -24 18 5
20 -15 -4
-5 4 1

Multiply the adjugate matrix by 1/det(M):

M^-1 = -24 18 5
20 -15 -4
-5 4 1

Question 6

Roots of cubics

Answer 6

ax^3 + bx^2 + cx + d
a(x-p)(x-q)(x-r)
product of the roots = -d/a
sum of product pairs = c/a
sum of the roots = -b/a

Question 7

For higher polynomials

Answer 7

Sum of the roots = -b/a

product of the roots = z/a for even degree polynomials whilst -z/a for odd degree polynomials

Question 8

What is z^n ± 1/(z^n) equal to?

Answer 8

2 cos(nϴ) or (2cosϴ)^n
2isin(nϴ) or (2isinϴ)^n

Question 9

General proof by induction?

Answer 9

Show it is true for n = 1

Assume true for n = k

Show it is true for n = k+1

State:
If it is true for some n=k it is also true for n=k+1. It is also true for n=1, therefore I have proved by mathematical induction that it is true for all the natural numbers n.

Question 10

Prove that 4^n + 6n -1 is divisible by 9 for all positive integers.

Answer 10

for n = 1
4^1 + 6(1) -1 = 9 therefore true for n = 1

Assume true for n = k
4^k + 6k -1 = 9N for some integer N

4^k+1 + 6(k+1) -1 = [4(4^k+6k-1)-24k+4] + 6(k+1) -1
= 49N - 18k + 4 +6k + 6 -1
= 94N - 18k + 9 = 9(4N-2k+1) therefore multiple of 9

If result true for n = 1 and n= k , then true for n = k+1
Therefore, statement is true for all positive integers

Question 11

Find the distance between:
R = (2+λ)i + (-3-3λ)j +(0+2λ)k and
r = (4-2μ)i + (2+6μ)j + (1-4μ)k

Answer 11

Choose a point on r:
μ = 0 ⟹ p = (4, 2, 1)

Let q be the point on R closest to P
q = (2+λ, -3-3λ, 2λ)
PQ = (-2+λ, -5-3λ, 2λ-1)

PQ is perpendicular to R = (1, -3, 2), so
Do dot product of PQ and R and equate it to 0

λ = -11/14

Substitute λ into PQ

PQ = 1/14(-39, -37, -36)

Shortest distance = |PQ| =
√ (299/14)

Question 12

What is the loci arg(z-a) = θ?

Answer 12

Half life starting from z = a, at an angle θ measured anticlockwise from the positive x-axis.

Question 13

What is the loci |z-a| = |z-b|?

Answer 13

Perpendicular bisector between points z = a and z = b

Question 14

What are the partial fractions of:
qx+r
_________
(ax+b)(cx+d)

Answer 14

A B
____ + _____
ax+b cx + d

Question 15

What are the partial fractions of:
px^2+qx+r
______________
(ax+b)(cx+d)(ex+f)

Answer 15

A B C
____ + _____ + _____
ax+b cx + d ex + f

Question 16

What is the loci |z-a| = b

Answer 16

Circle with centre a and radius b

Question 17

What are the partial fractions of:
px^2+qx+r
___________
(ax+b)(cx+d)^2

Answer 17

A B C
____ + _____ + _______
ax+b cx + d (cx+d)^2