Further Pure Mathematics 2 Quizzes - Test Your Skills

Question 1

Demonstrate that the complex function f(z) = e^z is an entire function. Provide a clear and detailed explanation of your reasoning using the Cauchy-Riemann equations.

Accepted Answer

f(z) is differentiable at every point because the Cauchy-Riemann equations are satisfied at every point in the complex plane.

Answer

f(z) is differentiable at every point because the exponential function is differentiable at every point in the complex plane.

Answer

f(z) is differentiable at every point because it is continuous at every point in the complex plane.

Question 2

Solve the differential equation dy/dx + 2y = 0, where y is the dependent variable and x is the independent variable.

Accepted Answer

y = Ce^(-2x)

Answer

y = Ce^(2x)

Answer

y = 2Ce^(-2x)

Answer

y = 2Ce^(2x)

Question 3

Find the derivative of the function f(x) = e^(2x) sin(3x)

Accepted Answer

2e^(2x) sin(3x) + 3e^(2x) cos(3x)

Answer

2e^(2x) sin(3x) - 3e^(2x) cos(3x)

Answer

-2e^(2x) sin(3x) + 3e^(2x) cos(3x)

Answer

-2e^(2x) sin(3x) - 3e^(2x) cos(3x)

Question 4

Determine the eigenvalues of the matrix C = [[2, 1], [-1, 2]]

Accepted Answer

1, 2

Answer

1, -1

Answer

-1, 2

Answer

-1, -2

Question 5

Find the complex conjugate of (3 - 4i).

Accepted Answer

(3 + 4i)

Answer

(3 - 4i)

Answer

(4 - 3i)

Answer

(-3 - 4i)

Question 6

Which of the following is a solution to the differential equation y' - 2y = 0?

Accepted Answer

y = e^(-2x)

Answer

y = e^(2x)

Answer

y = 2e^x

Answer

y = -2e^x

Question 7

Find the general solution to the differential equation y'' + 4y = 0.

Accepted Answer

y = C1*cos(2x) + C2*sin(2x)

Answer

y = C1*e^(2x) + C2*e^(-2x)

Answer

y = C1*cosh(2x) + C2*sinh(2x)

Answer

y = C1*e^(4x) + C2*e^(-4x)

Question 8

Which statement about eigenvalues is accurate?

Accepted Answer

Eigenvalues can be either real or complex numbers.

Answer

Eigenvalues are exclusively real numbers.

Answer

Eigenvalues are exclusively complex numbers.

Answer

Eigenvalues are always positive numbers.

Question 9

Find the determinant of the matrix [[2, 3], [-1, 4]]?

Accepted Answer

5

Answer

-5

Answer

8

Answer

-8

Question 10

Which differential equation is first order and linear?

Accepted Answer

y' + y = sin(x)

Answer

y'' - 2y' + y = 0

Answer

xy' + y = x^2

Answer

y' = y^2 + 1

Question 11

Solve the differential equation dy/dx + y = e^x?

Accepted Answer

y = Ce^-x + e^x

Answer

y = Ce^x + e^-x

Answer

y = e^x + C

Answer

y = Ce^-x + C

Question 12

Determine the principal argument of the complex number (-3 + 4i) in radians.

Accepted Answer

π/4

Answer

-π/4

Answer

π/2

Answer

-π/2

Question 13

Solve the first-order linear differential equation y' + 2y = e^{2x}.

Accepted Answer

y = (1/2)e^{2x} + Ce^{-2x}

Answer

y = e^{2x} + Ce^{-2x}

Answer

y = (1/2)e^{-2x} + Ce^{2x}

Answer

y = e^{-2x} + Ce^{2x}

Question 14

Find the general solution of the second-order linear homogeneous differential equation y'' - 4y' + 4y = 0.

Accepted Answer

y = C₁e^(2x) + C₂e^(-2x)

Answer

y = C₁e^(4x) + C₂e^(-4x)

Answer

y = C₁sin(2x) + C₂cos(2x)

Answer

y = C₁sin(4x) + C₂cos(4x)

Question 15

Determine the equation of the line passing through the points (1, 2) and (3, 4)

Accepted Answer

y = x + 1

Answer

y = x - 1

Answer

y = 2x

Answer

y = -2x

Question 16

Which of the following is the inverse of the matrix A = [[1, 2], [-1, 1]]?

Accepted Answer

A^(-1) = [[1, -2], [1, 1]]

Answer

A^(-1) = [[1, 2], [-1, -1]]

Answer

A^(-1) = [[-1, -2], [1, 1]]

Answer

A^(-1) does not exist

Question 17

Which of the following is the correct form of the series expansion of e^x in terms of x?

Accepted Answer

e^x = 1 + x + x^2 / 2! + x^3 / 3! + ...

Answer

e^x = 1 - x + x^2 / 2! - x^3 / 3! + ...

Answer

e^x = 1 + x - x^2 / 2! + x^3 / 3! + ...

Answer

e^x = 1 - x - x^2 / 2! - x^3 / 3! + ...

Question 18

Find the magnitude and argument of the complex number z = 2 + 3i. Express the argument in radians.

Accepted Answer

Magnitude: √(13), Argument: 0.983 radians

Answer

Magnitude: 3.606, Argument: 33.69°

Answer

Magnitude: 2, Argument: 56.31°

Answer

Magnitude: 1, Argument: 33.69°

Question 19

Find the general solution to the differential equation d^2y/dx^2 - y = 0.

Accepted Answer

y = C₁e^x + C₂e^-x

Answer

y = C₁sin(x) + C₂cos(x)

Answer

y = C₁e^x + C₂x

Answer

y = C₁sin(x) + C₂x

Question 20

Find the adjoint of the matrix A = [[1, 2], [3, 4]]

Accepted Answer

[[4, -2], [-3, 1]]

Answer

[[2, -4], [-1, 3]]

Answer

[[4, -1], [-2, 3]]

Answer

[[1, -3], [-2, 4]]

Question 21

Calculate the determinant of the matrix A = [[2, 3], [4, 5]]

Accepted Answer

10

Answer

2

Answer

5

Answer

-2

Question 22

Find the general solution to the differential equation d^2y/dx^2 + 4y = 0.

Accepted Answer

y = Acos(2x) + Bsin(2x)

Answer

y = Acosh(2x) + Bsinh(2x)

Answer

y = Ax^2 + Bx

Answer

y = Ae^(2x) + Be^(-2x)

Question 23

What is the mathematical symbol for the imaginary unit?

Accepted Answer

√(-1)

Answer

-1

Answer

1

Answer

0

Question 24

Which of the following matrices is not invertible?

Accepted Answer

[[0, 1], [0, 0]]

Answer

[[1, 0], [0, 1]]

Answer

[[2, 1], [-1, 2]]

Answer

[[1, 2], [3, 4]]

Question 25

Which of the following equations represents a differential equation?

Accepted Answer

dy/dx + y = cos(x)

Answer

y = x^2 + 3

Answer

sin(x) + cos(y) = 0

Answer

x + y = 5

Question 26

Which of the following is a property of complex numbers?

Accepted Answer

They can be represented as points on a plane known as the complex plane.

Answer

They cannot be added or subtracted.

Answer

They are always real numbers.

Answer

They are always positive numbers.

Question 27

Which of the following is a characteristic of a singular matrix?

Accepted Answer

It has a determinant of zero.

Answer

It has an inverse.

Answer

It has a non-zero determinant.

Answer

It is a square matrix.

Question 28

Which of the following is a type of differential equation that can be solved using an integrating factor?

Accepted Answer

dy/dx + y = x

Answer

d^3y/dx^3 = sin(x)

Answer

d^2y/dx^2 + y = 0

Answer

dy/dx = x^2 + y^2

Question 29

What is the modulus of the complex number (3 - 4i)?

Accepted Answer

5

Answer

8

Answer

25

Answer

1

Question 30

Find the general solution of the recurrence relation (a_n = 2a_{n-1} - 5a_{n-2} + 3), given that (a_0 = 1) and (a_1 = 5).

Accepted Answer

Solution: (a_n = 2^n - 1)

Answer

Solution: (a_n = 2^n + 1)

Answer

Solution: (a_n = 5^n - 1)

Answer

Solution: (a_n = 3^n - 1)

Question 31

Determine if the following statement is true or false: Every diagonalizable matrix has distinct eigenvalues.

Accepted Answer

True

Answer

False

Answer

True only if the matrix is positive definite

Answer

True only if the matrix is real

Question 32

Find the domain of the function (f(x) = rac{1}{sqrt{x-1}}).

Accepted Answer

Domain: ((1, infty))

Answer

Domain: ((-1, 1))

Answer

Domain: ((0, infty))

Answer

Domain: ((-infty, 1))

Question 33

Which of the following is a characteristic of complex numbers?

Accepted Answer

They have a real and imaginary part.

Answer

They cannot be represented on the complex plane.

Answer

They are always real numbers.

Answer

They do not have a modulus or argument.

Question 34

Identify the first-order linear differential equation:

Accepted Answer

y' + 2y = x

Answer

y'' + y = 0

Answer

y'y'' + y = 0

Answer

y''' + y' = 0

Question 35

Find the complex conjugate of (2 + 3i).

Accepted Answer

2 - 3i

Answer

4 + 6i

Answer

4 - 6i

Question 36

Which of the following is the derivative of the function f(z) = z^2 - 2z + 1?

Accepted Answer

2z - 2

Answer

z^2 - 2

Answer

z - 1

Answer

2z + 2

Question 37

Which of the following is the general solution to the homogeneous differential equation y'' + 4y = 0?

Accepted Answer

y = C1cos(2x) + C2sin(2x)

Answer

y = C1e^(2x) + C2e^(-2x)

Answer

y = C1x + C2

Question 38

Which of the following transformations is a rotation about the origin through an angle θ?

Accepted Answer

[[cos(theta), -sin(theta)], [sin(theta), cos(theta)]]

Answer

[[cos(theta), sin(theta)], [-sin(theta), cos(theta)]]

Answer

[[1, 0], [0, 1]]

Question 39

Find the value of the integral ∫(x^2 + 1)dx.

Accepted Answer

x^3/3 + x + C

Answer

x^2 - x + C

Answer

x + C

Answer

x^2 + x + C

Question 40

Find the eigenvalues of the matrix A = [[2, 1], [1, 2]]

Accepted Answer

λ = 1, 3

Answer

λ = 0, 4

Answer

λ = 2, 2

Answer

λ = -1, 3

Question 41

Given matrices A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], find the determinant of AB.

Accepted Answer

-2

Answer

2

Answer

10

Answer

-10

Question 42

Solve the differential equation dy/dx = y^2 with the initial condition y(0) = 1.

Accepted Answer

y = 1/(1-x)

Answer

y = 1 - x

Answer

y = 1/(1+x)

Answer

y = x + 1

Question 43

Find the square root of the complex number 4i.

Accepted Answer

2√2(cos(π/4) + isin(π/4))

Answer

2(cos(π/2) + isin(π/2))

Answer

2(cos(π/4) + isin(π/4))

Answer

2√2(cos(π/2) + isin(π/2))

Question 44

If A is a 2x2 matrix with eigenvalues λ1 and λ2, what is its determinant?

Accepted Answer

λ1λ2

Answer

λ1 + λ2

Answer

(λ1 - λ2)^2

Answer

λ1^2 + λ2^2

Question 45

Multiplying a complex number z = 1 + i by its conjugate produces:

Accepted Answer

2

Answer

1 - i

Answer

1 + i

Answer

0

Question 46

Find the general solution to the differential equation dy/dx = 2y.

Accepted Answer

y = Ae^(2x)

Answer

y = 2e^(2x)

Answer

y = 2e^(x)

Answer

y = Ae^(x)

Question 47

Solve the differential equation dy/dx = y/x.

Accepted Answer

y = Cx

Answer

y = Cx^2

Answer

y = x/C

Question 48

Multiplying the complex number 2 + 3i by its conjugate results in:

Accepted Answer

13

Answer

-5

Answer

10

Answer

5

Question 49

Find the characteristic equation of the matrix A = [[1, 2], [3, 4]]

Accepted Answer

λ^2 - 5λ -2 = 0

Answer

λ^2 + 5λ -2 = 0

Answer

λ^2 - 5λ +2 = 0

Answer

λ^2 + 5λ +2 = 0

Question 50

The general solution to the differential equation dy/dx + 2y = 0 is:

Accepted Answer

y = Ae^(-2x)

Answer

y = Ae^(2x)

Answer

y = 2e^(-2x)

Question 51

What is the modulus of the complex number  $z = 3 - 4i$?

Accepted Answer

5

Answer

7

Answer

3

Answer

4

Question 52

Which of the following is a solution to the differential equation $\frac{dy}{dx} = 2x + y$?

Accepted Answer

$y = -2(x + 1) + Ce^x$

Answer

$y = x^2 + C$

Answer

$y = 2x + C$

Answer

$y = Ce^{2x}$

Question 53

Find the determinant of the matrix  $A = \begin{pmatrix} 2 & -1 \ 4 & 3 \end{pmatrix}$

Accepted Answer

10

Answer

-2

Answer

14

Answer

5

Question 54

Given the complex numbers $z_1 = 2 + i$ and $z_2 = 1 - 2i$, what is $z_1 - z_2$?

Accepted Answer

1 + 3i

Answer

3 - i

Answer

3 + 3i

Answer

1 - i

Question 55

If  $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$  and  $B = \begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix}$, what is $AB$?

Accepted Answer

$ \begin{pmatrix} -2 & 1 \ -4 & 3 \end{pmatrix}$

Answer

Not defined

Answer

$ \begin{pmatrix} 3 & 4 \ 1 & 2 \end{pmatrix}$

Answer

$ \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$

Question 56

Convert the complex number $z = 2	ext{cis}(60°)$ to rectangular form.

Accepted Answer

1 + √3i

Answer

√3 + i

Answer

2 - 2√3i

Answer

2 + 2√3i

Question 57

Solve the differential equation $\frac{dy}{dx} = 2y$ given that $y(0) = 1$.

Accepted Answer

$y = e^{2x}$

Answer

$y = Ce^{2x}$

Answer

$y = 1 + 2x$

Answer

$y = 2e^x$

Question 58

Find the inverse of the matrix $A = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}$.

Accepted Answer

$ \begin{pmatrix} 1 & -1 \ 0 & 1 \end{pmatrix}$

Answer

$ \begin{pmatrix} 0 & 1 \ 1 & -1 \end{pmatrix}$

Answer

$ \begin{pmatrix} 1 & 0 \ -1 & 1 \end{pmatrix}$

Answer

The inverse does not exist

Question 59

Which of the following is the correct way to write the complex number (2 + 3i) in polar form?

Accepted Answer

(2√10, π/6)

Answer

(2√10, 5π/6)

Answer

(√10, π/3)

Answer

(√10, 2π/3)

Question 60

Find the sum of the eigenvalues of the matrix A = [[1, 2], [-2, 1]]

Accepted Answer

3

Answer

0

Answer

1

Answer

-1

Question 61

Which of the following is the Taylor series expansion of f(x) = cos(x) about x = 0?

Accepted Answer

1 - x^2/2! + x^4/4! - ...

Answer

x + x^3/3! + x^5/5! + ...

Answer

x - x^3/3! + x^5/5! - ...

Answer

1 + x^2/2! + x^4/4! + ...

Question 62

Find the general solution to the differential equation y'' + y = sin(2x)

Accepted Answer

y = c1cos(x) + c2sin(x) + (x - 1/2)sin(2x)

Answer

y = c1cos(x) + c2sin(x) - (x - 1/2)sin(2x)

Answer

y = c1cos(2x) + c2sin(2x) + (x - 1/2)sin(x)

Answer

y = c1cos(2x) + c2sin(2x) - (x - 1/2)sin(x)

Question 63

Which of the following is the inverse of the matrix A = [[1, 2], [0, 1]]?

Accepted Answer

[[1, -2], [0, 1]]

Answer

[[1, 2], [0, -1]]

Answer

[[2, -1], [0, 1]]

Answer

[[2, 1], [0, 1]]

Question 64

Which of the following is the correct representation of the complex number (3 + 4i) in polar form?

Accepted Answer

3(cos 53.1° + i sin 53.1°)

Answer

4(cos 36.9° + i sin 36.9°)

Answer

4(cos 53.1° + i sin 53.1°)

Answer

3(cos 36.9° + i sin 36.9°)