Pure Mathematics 2 Quizzes: Test Your Understanding

Question 1

Find the general solution to the first-order linear differential equation:
$$rac{dy}{dx} + 2xy = x$$

Accepted Answer

$y = rac{1}{2}e^{-x^2} + C$

Answer

$y = -e^{x^2} + C$

Answer

$y = x^2 + C$

Question 2

Which of the following is NOT a complex number?

Accepted Answer

6

Answer

2 + 3i

Answer

4 - 5i

Answer

7i

Question 3

Which of the following is the equation of the line passing through the points (2, 3) and (4, 7)?

Accepted Answer

y = 3x + 1

Answer

y = x + 1

Answer

y = 2x + 1

Answer

y = 4x + 1

Question 4

Which of the following is a scalar product?

Accepted Answer

The dot product of two vectors

Answer

The cross product of two vectors

Answer

The angle between two vectors

Answer

The magnitude of a vector

Question 5

Which of the following is the solution to the equation x^2 - 5x + 6 = 0?

Accepted Answer

x = 2 or x = 3

Answer

x = 1/2 or x = 3

Answer

x = 1/4 or x = 1

Answer

x = 1/8 or x = 1/2

Question 6

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

Accepted Answer

f'(x) = 3x^2 + 4x

Answer

f'(x) = 3x^2 + 2x

Answer

f'(x) = 2x^2 + 4x

Answer

f'(x) = 2x^2 + 2x

Question 7

Which of the following is NOT a theorem in vector calculus?

Accepted Answer

The Pythagorean Theorem

Answer

Green's Theorem

Answer

The Divergence Theorem

Answer

Stokes' Theorem

Question 8

Find the determinant of the matrix {{2, 3}, {-1, 4}}.

Accepted Answer

5

Answer

10

Answer

0

Answer

8

Question 9

What is the inverse function of f(x) = 2x + 1?

Accepted Answer

g(x) = (x - 1)/2

Answer

g(x) = x/2 - 1

Answer

g(x) = 2x - 1

Answer

g(x) = 1 - 2x

Question 10

Express the complex number $2 + 3i$ in polar form, giving the modulus and argument.

Accepted Answer

$r = sqrt{13}, 	heta = arctanleft(rac{3}{2}ight)$

Answer

$r = sqrt{13}, 	heta = arctanleft(rac{2}{3}ight)$

Question 11

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

Accepted Answer

5

Answer

-5

Answer

10

Answer

-10

Question 12

Find the inverse of the matrix B = [[1, 2], [3, 5]]

Accepted Answer

[[5, -2], [-3, 1]]

Answer

[[1, -2], [-3, 5]]

Answer

[[5, 2], [3, -1]]

Answer

[[1, 2], [3, -5]]

Question 13

What is the modulus of the complex number (2 + 3i)?

Accepted Answer

√13

Answer

2

Answer

13

Answer

4

Question 14

The equation of the tangent to the curve y = x^3 - 2x^2 at the point (1, -1) is:

Accepted Answer

y = 3x - 4

Answer

y = x - 1

Answer

y = 2x + 1

Answer

y = -x + 1

Question 15

The general solution to the differential equation dy/dx = xy is:

Accepted Answer

y = Ce^(x^2/2)

Answer

y = Ce^(-x^2/2)

Answer

y = Cx

Answer

y = C/x

Question 16

Which of the following is NOT a theorem related specifically to complex numbers?

Accepted Answer

Pythagorean Theorem

Answer

De Moivre's Theorem

Answer

Euler's Formula

Answer

Triangle Inequality

Question 17

Find the discriminant of the quadratic equation ax^2 + bx + c = 0.

Accepted Answer

b^2 - 4ac

Answer

b^2 + 4ac

Answer

ac - b^2

Answer

a^2 - bc

Question 18

Expand the binomial (x + 2)^5 using the binomial theorem.

Accepted Answer

x^5 + 10x^4 + 40x^3 + 80x^2 + 80x + 32

Answer

x^5 - 10x^4 + 40x^3 - 80x^2 + 80x - 32

Answer

x^5 + 20x^4 + 80x^3 + 160x^2 + 160x + 64

Answer

x^5 - 20x^4 + 80x^3 - 160x^2 + 160x - 64

Question 19

Determine the modulus of the complex number: 2 - 3i

Accepted Answer

sqrt(13)

Answer

1

Answer

-sqrt(13)

Answer

2

Question 20

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

Accepted Answer

3e^(3x+2)

Answer

e^(3x+2)/3

Answer

e^(3x+2)

Answer

3xe^(3x+2)

Question 21

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

Accepted Answer

1, 3

Answer

2, 4

Answer

-1, -3

Answer

0, 5

Question 22

Solve the logarithmic equation: log₂(x + 3) = 4

Accepted Answer

13

Answer

17

Answer

11

Answer

15

Question 23

Find the derivative of the function f(x) = x³ - 2x² + 5.

Accepted Answer

3x² - 4x

Answer

3x³ - 2x

Answer

x² - 2x

Answer

3x - 4

Question 24

Calculate the gradient of the line passing through points (2, 3) and (-1, 5).

Accepted Answer

-1/3

Answer

3

Answer

-3

Answer

1/3

Question 25

Find the derivative of the function f(x) = x^3 + 2x^2 - 5x + 1.

Accepted Answer

3x^2 + 4x - 5

Answer

x^3 + 4x - 5

Answer

2x^3 + 4x - 5

Answer

3x^2 + 2x - 5

Question 26

A vector has components (3, 4, -2). Find its magnitude.

Accepted Answer

√29

Answer

7

Answer

5

Answer

9

Question 27

Find the inverse of the function f(x) = x^2 - 1, given that x ≥ 0.

Accepted Answer

f^-1(x) = √(x + 1)

Answer

f^-1(x) = √(x - 1)

Answer

f^-1(x) = -√(x - 1)

Answer

f^-1(x) = -√(x + 1)

Question 28

Prove that the equation x^4 + 2x^2 + 1 = 0 has no real solutions.

Accepted Answer

Use the fact that x^2 ≥ 0 for any real x.

Answer

Use the principle of mathematical induction.

Answer

Use the quadratic formula.

Answer

Use the binomial theorem.

Question 29

Find the derivative of log₂(x) with respect to x.

Accepted Answer

1 / (x * ln(2))

Answer

ln(2) / x

Answer

1 / ln(2)

Answer

x * ln(2)

Question 30

Determine the equation of the normal to the curve y = x³ + 2x at the point (1, 3).

Accepted Answer

y = -x + 4

Answer

y = x + 2

Answer

y = 2x + 1

Answer

y = -x + 6

Question 31

Calculate the modulus of the complex number z = (3 + 4i)(-2 + 5i).

Accepted Answer

53

Answer

125

Answer

10

Answer

25

Question 32

Find the determinant of the matrix A = [2 -1; 3 4].

Accepted Answer

11

Answer

-11

Answer

7

Answer

1

Question 33

What is the modulus of the complex number 3 + 4i?

Accepted Answer

5

Answer

√7

Answer

1

Answer

7

Question 34

Given matrix A = [[2, 1], [3, 4]], what is the determinant of A?

Accepted Answer

5

Answer

2

Answer

-5

Answer

11

Question 35

What is the derivative of the function f(x) = x³ + 2x² - 5x?

Accepted Answer

3x² + 4x - 5

Answer

3x² + 4x

Answer

x³ + 4x - 5

Answer

x² + 2x - 5

Question 36

What is the equation of the tangent line to the curve y = x² at the point (2, 4)?

Accepted Answer

y = 4x - 4

Answer

y = 4x + 4

Answer

y = x² + 4

Answer

y = 2x

Question 37

If a = (2, 1) and b = (3, -2), what is the dot product of a and b?

Accepted Answer

4

Answer

-1

Answer

7

Answer

5

Question 38

What are the solutions to the equation |x - 3| = 5?

Accepted Answer

x = 8 or x = -2

Answer

x = 3 or x = 8

Answer

x = -2

Answer

x = 8

Question 39

The sum of the first n terms of an arithmetic series is given by Sₙ = 2n² + 3n. What is the common difference?

Accepted Answer

4

Answer

6

Answer

2

Answer

3

Question 40

What is the inverse function of f(x) = 2x + 1?

Accepted Answer

f⁻¹(x) = (x - 1)/2

Answer

f⁻¹(x) = -2x - 1

Answer

f⁻¹(x) = 2x - 1

Answer

f⁻¹(x) = (x + 1)/2

Question 41

Find the indefinite integral of the function f(x) = 2x + 1.

Accepted Answer

x² + x + C

Answer

2x² + x + C

Answer

2x + C

Answer

x² + C

Question 42

What is the equation of the line passing through the points (1, 2) and (3, 6)?

Accepted Answer

y = 2x

Answer

y = 3x - 1

Answer

y = x + 2

Answer

y = x + 1

Question 43

What is the complex conjugate of z = 3 + 4i?

Accepted Answer

3 - 4i

Answer

4 + 3i

Answer

-4 + 3i

Answer

-3 - 4i

Question 44

Find the determinant of the matrix A = [2 1; 3 4].

Accepted Answer

5

Answer

8

Answer

1

Answer

-5

Question 45

Solve for x: 2x^2 - 5x + 2 = 0

Accepted Answer

x = 1/2 or x = 2

Answer

x = 1/4 or x = 4

Answer

x = 2 or x = -1

Answer

x = 1 or x = -2

Question 46

Find the derivative of f(x) = x^3 + 2x^2 - 5.

Accepted Answer

f'(x) = 3x^2 + 4x

Answer

f'(x) = 3x^2 - 4x

Answer

f'(x) = x^2 + 4x

Question 47

Determine the direction of the vector v = <2, -3>

Accepted Answer

South-west

Answer

South-east

Answer

North-east

Answer

North-west

Question 48

Which of the following is NOT a type of conic section?

Accepted Answer

Parabola

Answer

Circle

Answer

Hyperbola

Answer

Ellipse

Question 49

Find the inverse of the matrix B = [1 2; -1 3].

Accepted Answer

B^-1 = [3 -2; 1 1]

Answer

B^-1 = [3 -2; -1 -1]

Answer

B^-1 = [3 2; -1 1]

Answer

B^-1 = [-3 2; 1 -1]

Question 50

Evaluate the integral ∫(x^2 + 3x - 2) dx

Accepted Answer

x^3/3 + (3/2)x^2 - 2x + C

Answer

x^3/3 - (3/2)x^2 + 2x - C

Answer

x^3/3 + (3/2)x^2 - 2x - C

Answer

x^3/3 - (3/2)x^2 + 2x + C

Question 51

Which of the following is the equation of the plane that contains the points (1, 0, 1), (2, 1, 0), and (0, 2, 1)?

Accepted Answer

x - 2y + 3z = 1

Answer

x + 2y - 3z = 1

Answer

2x - y + 3z = 1

Answer

x + y + z = 1

Question 52

Find the value of sin(π/3).

Accepted Answer

√3/2

Answer

1

Answer

√2/2

Answer

1/2