6.) Let f(x) = 3x² + 2x – 4 and g(x) = x³ – 4x.

a) Find the domain and range of f(x) and g(x).
The domain of f(x) is all real numbers because the function is defined for all real values of x. The range of f(x) is also all real numbers, because the function is a polynomial and can take on any real value. Similarly, the domain of g(x) is all real numbers and the range of g(x) is all real numbers.

b) Find the inverse of f(x) and g(x).
To find the inverse of f(x) we need to solve f(x) = y for x.
3x² + 2x – 4 = y
x² + 2/3 x – 4/3 = y
x = √(y + 4/3) – ⅓

To find the inverse of g(x) we need to solve g(x) = y for x.
x³ – 4x = y
x = (y + 4x)^(1/3)

c) Find the x-coordinate of the point of intersection between f(x) and g(x).
To find the x-coordinate of the point of intersection between f(x) and g(x) we need to solve the equation f(x) = g(x)
3x² + 2x – 4 = x³ – 4x

We can use the polynomial division or the rational root theorem to find the roots of the equation.
3x² + 2x – 4 = x³ – 4x
x³ – 3x² – 2x + 4 = 0

By Factorization or synthetic division we can find the roots of the equation which are x = 1, x = -2, x = 2

7.) The following diagram shows a circle of centre O, and a radius of 15 cm. The arc ACB subtends an angle of 2 radians at the centre O.

a) Find the length of arc ACB.
ACB = 2 × OA= 30cm

b) Find the area of the shaded region.
AOB = 2π – 2
Area = ½(r)²𝜃
=½(15)²(2π – 2)
=482cm²

8.)

a) The population of village A increases by 8% every year. If the population today is 1000 people find
FV= 1000(1+(8/100))ⁿ = 1000(1.08)ⁿ

i) the population after 5 years;
n=5, hence 1469

ii) the population 5 years ago;
n=-5, hence 681

iii) the number of full years after the population will exceed 2000.
Solve for n : 1000(1.08)ⁿ =2000, n=9.0064 so n=10

b) The population of village B decreases by 8% every year. If the population today is 1000 people find
FV= 1000(1-(8/100))ⁿ = 1000(0.92)ⁿ

i) the population after 5 years;
n=5, hence 659

ii) the population 5 years ago;
n=-5, 1517

iii) the number of full years after the population will fall under 500
Solve for n : 1000(0.92)ⁿ = 500, n=8.31 so n=9

9.) Consider the geometric sequence 10, 5, 2.5, 1.25, …

a) Write down the first term u1and the common ratio r.
u1= 10, r= 0.5

b) Find the 10th term of the sequence.
0.0195

c) Find the sum of the first 10 terms.
19.98

d) Express the general term un in terms of n.
10*0.5^(n-1)

e) Hence find the value of n given that un.03125
n= 6

10.) Let A be the matrix
[1 2 3]
[4 5 6]
[7 8 9]

a) Find the characteristic polynomial of A
To find the characteristic polynomial of A, we find the determinant of the matrix A – λI, where λ is a scalar and I is the identity matrix.
Characteristic polynomial of A = det(A – λI) = det([1-λ 2 3] [4 5-λ 6] [7 8 9-λ]) = (1-λ)((5-λ)(9-λ) – 86) – 24*7 = λ³ – 6λ² + 11λ – 6

b) Find the eigenvalues of A
To find the eigenvalues of A, we find the roots of the characteristic polynomial.
Eigenvalues of A = roots of λ³ – 6λ² + 11λ – 6 = 0 = λ³ – 6λ² + 11λ – 6 = (λ-3)(λ-2)(λ-1) = λ = 3, 2, 1

c) Find the eigenvectors of A corresponding to the eigenvalues found in (b)
To find the eigenvectors of A corresponding to the eigenvalues found in (b), we solve the system of equations (A – λI)v = 0 for each eigenvalue λ.
For λ = 3, we have (A – 3I)v = 0, which gives us the equation (1-3)x + 2y + 3z = 0, 4x + (5-3)y + 6z = 0, 7x + 8y + (9-3)z = 0
This system can be written as [1 -2 3][x] = [0] , [4 -1 6][y] = [0] , [7 8 -6][z] = [0]
Eigenvectors of A corresponding to λ=3 are [1 -2 3]^T
For λ = 2, we have (A – 2I)v = 0, which gives us the equation (1-2)x + 2y + 3z = 0, 4x + (5-2)y + 6z = 0, 7x + 8y + (9-2)z = 0
This system can be written as [ -1 2 3][x] = [0] , [4 3 6][y] = [0] , [7 8 7][z]
Eigenvectors of A corresponding to λ=2 are [ -1 2 3]^T
For λ = 1, we have (A – I)v = 0, which gives us the equation (1-1)x + 2y + 3z = 0, 4x + (5-1)y + 6z = 0, 7x + 8y + (9-1)z = 0
This system can be written as [0 2 3][x] = [0] , [4 4 6][y] = [0] , [7 8 8][z]
Eigenvectors of A corresponding to λ=1 are [0 2 3]^T