Further Pure Mathematics 1 Quizzes: Master Advanced Maths Concepts

Question 1

Identify the critical point of the equation y = x^3 + 2x^2 - x - 2.

Accepted Answer

x = -1

Answer

x = 0

Answer

x = 1

Answer

x = 2

Question 2

Calculate the derivative of the function f(x) = sin(2x + 1).

Accepted Answer

2cos(2x + 1)

Answer

cos(2x + 1)

Answer

sin(2x + 1)

Answer

-cos(2x + 1)

Question 3

Determine the equation of the circle with centre (2, 3) and radius 5.

Accepted Answer

(x - 2)^2 + (y - 3)^2 = 25

Answer

(x - 2)^2 + (y - 3)^2 = 5

Answer

(x + 2)^2 + (y + 3)^2 = 25

Answer

(x + 2)^2 + (y + 3)^2 = 5

Question 4

Calculate the derivative of the function f(x) = e^x * sin(x).

Accepted Answer

e^x * sin(x) + e^x * cos(x)

Answer

e^x * sin(x) - e^x * cos(x)

Answer

e^x * cos(x) + e^x * sin(x)

Answer

e^x * cos(x) - e^x * sin(x)

Question 5

Find the amplitude of the function y = 3 cos(πx/2).

Accepted Answer

3

Answer

1.5

Answer

π

Answer

6

Question 6

Find the derivative of the function y = e^(2x + 1).

Accepted Answer

2e^(2x + 1)

Answer

e^(2x + 1)

Answer

2x * e^(2x + 1)

Answer

(2x + 1) * e^(2x + 1)

Question 7

Which of the following is NOT a valid representation of a complex number?

Accepted Answer

π

Answer

1 + 2i

Answer

√-1

Answer

2 - 3i

Question 8

Which of the following correctly expresses the formula for integration by parts?

Accepted Answer

∫udv = uv - ∫vdu

Answer

∫vdu = uv - ∫udv

Answer

∫dudv = uv - ∫vdu

Answer

∫udv = vu - ∫vdu

Question 9

Evaluate the integral ∫(3x^2 - 5x + 2) dx.

Accepted Answer

x^3 - 5x^2 / 2 + 2x + C

Answer

x^3 + 5x^2 / 2 + 2x + C

Answer

2x^3 - 5x^2 + 2x + C

Answer

2x^3 + 5x^2 + 2x + C

Question 10

Solve the equation cos(2x) = 1/2 for x in the interval [0, 2π].

Accepted Answer

π/6, π/2, 5π/6, 3π/2

Answer

π/3, 2π/3, 4π/3, 5π/3

Answer

π/4, 3π/4, 5π/4, 7π/4

Answer

π/12, 5π/12, 7π/12, 11π/12

Question 11

Which of the following statements about complex numbers is true?

Accepted Answer

The product of two complex conjugates is always a real number

Answer

The sum of two complex numbers is always a real number

Answer

The difference of two complex numbers is always a complex number

Answer

The quotient of two complex numbers is always a complex number

Question 12

Find the gradient of the normal to the curve y = x^3 - 2x at the point (1, -1).

Accepted Answer

-2

Answer

2

Answer

1

Answer

-1

Question 13

Which of the following is the general solution to the differential equation y' - 2y = e^x?

Accepted Answer

y = e^x + Ce^-2x

Answer

y = e^x + C

Answer

y = e^-2x + C

Answer

y = xe^x + C

Question 14

Find the equation of the circle with centre (-2, 1) and radius 3.

Accepted Answer

(x + 2)^2 + (y - 1)^2 = 9

Answer

(x - 2)^2 + (y + 1)^2 = 9

Answer

(x + 2)^2 + (y - 1)^2 = 3

Answer

(x - 2)^2 + (y + 1)^2 = 3

Question 15

If sin(π/6) = 1/2, what is the value of sin(π/3)?

Accepted Answer

√3/2

Answer

1/2

Answer

1

Answer

√2/2

Question 16

Which of the following transformations will map the graph of y = x^2 onto the graph of y = 2(x - 1)^2 + 3?

Accepted Answer

Translation 1 unit right and 3 units up

Answer

Translation 1 unit left and 3 units down

Answer

Reflection in the y-axis and translation 1 unit right and 3 units up

Answer

Reflection in the x-axis and translation 1 unit left and 3 units down

Question 17

Which of the following is the derivative of ln(x^2 + 1)?

Accepted Answer

2x / (x^2 + 1)

Answer

x / (x^2 + 1)

Answer

1 / (x^2 + 1)

Answer

2 / (x^2 + 1)

Question 18

What is the equation of the plane passing through the points (1, 0, 0), (0, 1, 0), and (0, 0, 1)?

Accepted Answer

x + y + z = 1

Answer

x + y + z = 0

Answer

x - y + z = 1

Answer

x + y - z = 1

Question 19

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

Accepted Answer

3x² - 4x

Answer

x³ - 4x + 5

Answer

3x² + 4x

Answer

x³ - 2x + 5

Question 20

Determine the domain of the function f(x) = 1/(x - 2).

Accepted Answer

All real numbers except x = 2

Answer

All real numbers

Answer

All real numbers greater than 2

Answer

All real numbers less than 2

Question 21

Identify a characteristic of a geometric sequence.

Accepted Answer

Each term is obtained by multiplying the previous term by a constant ratio.

Answer

The difference between consecutive terms is constant.

Answer

The sum of the terms is always convergent.

Answer

The sequence oscillates between two values.

Question 22

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

Accepted Answer

3x^2 + 4x - 5

Answer

3x^2 - 4x - 5

Answer

x^2 + 2x - 5

Answer

x^2 - 2x - 5

Question 23

Determine the general solution to the differential equation dy/dx = x^2 + 1.

Accepted Answer

y = x^3/3 + x + C

Answer

y = x^3/3 - x + C

Answer

y = 2x^3/3 + C

Answer

y = 2x^3/3 - C

Question 24

Solve the inequality |x - 2| < 3.

Accepted Answer

-1 < x < 5

Answer

x < -5 or x > 1

Answer

x > 5 or x < -1

Answer

-5 < x < 1

Question 25

Find the general solution of the differential equation dy/dx = 2x + 3.

Accepted Answer

y = x^2 + 3x + C

Answer

y = 2x^2 + 3x + C

Answer

y = 2x + 3 + C

Answer

y = x^2 + 3 + C

Question 26

Which of the following is NOT a subgroup of the group (Z_12, +)?

Accepted Answer

<{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}>

Answer

<{0, 6}>

Answer

<{0, 2, 4, 6, 8, 10}>

Answer

<{0, 3, 6, 9}>

Question 27

Which of the following is a vector space over the field of real numbers?

Accepted Answer

The set of all polynomials of degree less than or equal to 3

Answer

The set of all matrices of size 2x2

Answer

The set of all odd functions

Answer

The set of all prime numbers

Question 28

Determine the range of the function f(x) = 1/(x^2 - 4).

Accepted Answer

(-∞, -2) U (-2, 2) U (2, ∞)

Answer

[0, ∞)

Answer

(-2, 2)

Answer

(-∞, ∞)

Question 29

Evaluate the indefinite integral ∫(x³ + 2x²)dx.

Accepted Answer

x⁴/4 + x³/3 + C

Answer

x⁴/4 - x³/3 + C

Answer

x³/4 + x²/2 + C

Answer

x³/4 - x²/2 + C

Question 30

Expand the binomial (a + b)³.

Accepted Answer

a³ + 3a²b + 3ab² + b³

Answer

a³ + b³ + 3ab²

Answer

a³ + b³ - 3ab²

Answer

a³ + 2a²b + 2ab² + b³

Question 31

Solve the trigonometric equation sin θ = 1/2 for θ in the interval [0, 2π).

Accepted Answer

θ = π/6, 5π/6

Answer

θ = π/3, 2π/3

Answer

θ = π/4, 3π/4

Answer

θ = 0, π

Question 32

Which of the following is a condition for a quadratic equation ax^2 + bx + c = 0 to have complex roots?

Accepted Answer

b^2 - 4ac < 0

Answer

a ≠ 0

Answer

c ≠ 0

Answer

b = 0

Question 33

The sum of the first n positive integers is given by:

Accepted Answer

n(n+1)/2

Answer

n^2/2

Answer

n(n-1)/2

Answer

n^2

Question 34

In how many ways can a group of 8 students be seated in a row if two particular students refuse to sit next to each other?

Accepted Answer

6!

Answer

7!

Answer

5!

Answer

4!

Question 35

The complex number z = 3 - 4i is represented by the point:

Accepted Answer

(3, -4)

Answer

(-3, 4)

Answer

(4, -3)

Answer

(-4, 3)

Question 36

Find the inverse function of f(x) = 2x^3 + 5.

Accepted Answer

∛((x - 5)/2)

Answer

∛((x + 5)/2)

Answer

∛(2x^3 - 5)

Answer

(∛(x)/2) - 5

Question 37

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

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 4x - 5

Answer

x^2 + 2x - 5

Answer

3x^2 + 2x - 5

Question 38

Find the equation of the tangent to the curve y = x^3 - 2x^2 + 1 at the point (1, 0).

Accepted Answer

y = 3x - 2

Answer

y = 3x + 2

Answer

y = -3x - 2

Answer

y = -3x + 2

Question 39

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

Accepted Answer

f^-1(x) = (x - 1) / 2

Answer

f^-1(x) = -2x + 1

Answer

f^-1(x) = (x + 1) / 2

Answer

f^-1(x) = 2x - 1

Question 40

Which of the following is NOT a property of a quadratic function?

Accepted Answer

Is symmetrical about the y-axis

Answer

Has a vertex

Answer

Intercepts the x-axis at most twice

Answer

Has a maximum or minimum value

Question 41

Which of the following is the correct formula for the sum of an infinite geometric series with first term a and common ratio r, where |r| < 1?

Accepted Answer

a/(1 - r)

Answer

(1 - r)/a

Answer

a/(1 + r)

Answer

r/(1 - a)

Question 42

The derivative of (x^2 + 3x - 5) with respect to x is:

Accepted Answer

2x + 3

Answer

x^2 + 3

Answer

2x + 3x - 5

Answer

x^2 + 3x + 5

Question 43

The value of sin(π/6) is:

Accepted Answer

√3/2

Answer

1/2

Answer

1/3

Answer

√2/2

Question 44

The inverse function of f(x) = 2x + 5 is:

Accepted Answer

g(x) = (x - 5)/2

Answer

g(x) = 2x - 5

Answer

g(x) = 5 - 2x

Answer

g(x) = (x + 5)/2

Question 45

The domain of the function f(x) = ln(x^2 - 4) is:

Accepted Answer

x ∈ ℝ, x ≠ ±2

Answer

x ∈ ℝ, x > 2

Answer

x ∈ ℝ, x < -2

Answer

x ∈ ℝ, -2 < x < 2

Question 46

The general solution to the differential equation dy/dx = x^2 + 1 is:

Accepted Answer

y = (x^3)/3 + x + C

Answer

y = (x^3)/3 - x + C

Answer

y = x^2 + x + C

Answer

y = x^2 - x + C

Question 47

Expand (x + y)^4 using the binomial theorem.

Accepted Answer

(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4

Answer

(x + y)^4 = x^4 - 4x^3y + 6x^2y^2 - 4xy^3 + y^4

Answer

(x + y)^4 = x^3 + 4x^2y + 6xy^2 + 4y^3 + y^4

Answer

(x + y)^4 = x^3 - 4x^2y + 6xy^2 - 4y^3 + y^4

Question 48

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

Accepted Answer

y = -2x + 7

Answer

y = 2x + 1

Answer

y = -x + 4

Answer

y = x + 2

Question 49

Calculate the volume of the solid generated by rotating the region bounded by y = x^2 and y = 2x about the y-axis.

Accepted Answer

V = π∫[0,1](2x - x^2)^2 dx

Answer

V = π∫[0,1](2x - x^4)^2 dx

Answer

V = 2π∫[0,1](2x - x^2)^2 dx

Answer

V = 2π∫[0,1](2x - x^4)^2 dx

Question 50

Find the sum of the first 10 terms of the geometric sequence 2, 6, 18, ...

Accepted Answer

2046

Answer

1023

Answer

511.5

Answer

255.75

Question 51

What is the sum of the first $n$ positive integers?

Accepted Answer

$rac{n(n+1)}{2}$

Answer

$rac{(n+1)(n+2)}{2}$

Answer

$rac{n(n-1)}{2}$

Answer

$rac{n(n+2)}{2}$

Question 52

What is the volume of a sphere with radius $r$?

Accepted Answer

$rac{4}{3}πr^3$ cubic units

Answer

$4πr^3$ cubic units

Answer

$rac{3}{4}πr^3$ cubic units

Answer

$3πr^3$ cubic units

Question 53

Find the complex conjugate of z = 3 + 4i.

Accepted Answer

3 - 4i

Answer

3 + 4i

Answer

-3 + 4i

Answer

-3 - 4i

Question 54

Evaluate sin(π/3) + cos(π/4).

Accepted Answer

2

Answer

-1

Answer

0

Answer

1

Question 55

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 56

Find the derivative of the function f(x) = x^3 - 2x^2 + 5x - 7.

Accepted Answer

3x^2 - 4x + 5

Answer

x^3 + 4x^2 - 5x - 7

Answer

3x^3 - 2x

Answer

x^3 - 4x^2 + 5x

Question 57

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

Accepted Answer

y = 2x - 5

Answer

y = x - 3

Answer

y = x + 1

Answer

y = -2x + 3

Question 58

Solve the equation log2(x + 3) = 4.

Accepted Answer

15

Answer

23

Answer

7

Answer

31

Question 59

Identify the graph of the function y = |x - 2|.

Accepted Answer

V-shaped graph with vertex at (2, 0)

Answer

Parabola opening upwards

Answer

Circle

Answer

Straight line

Question 60

Determine the probability of rolling a 6 on a fair six-sided die.

Accepted Answer

1/6

Answer

1

Answer

1/2

Answer

1/3

Question 61

Calculate the integral of the function f(x) = x^3 + 2x^2 - 5x + 1 with respect to x.

Accepted Answer

(x^4)/4 + (2x^3)/3 - (5x^2)/2 + x + C, where C is an arbitrary constant.

Answer

(x^4)/4 + (2x^3)/3 - (5x^2)/2 - x + C

Answer

(x^4)/4 + (2x^3)/3 + (5x^2)/2 + x + C

Answer

(x^4)/4 + (2x^3)/3 + (5x^2)/2 - x + C

Question 62

Determine the equation of the plane that passes through the points P(1, 2, 3), Q(2, 1, 4), and R(3, 3, 2).

Accepted Answer

x + y - z = 4

Answer

x - y + z = 4

Answer

x - y - z = 4

Answer

x + y + z = 4

Question 63

Solve the differential equation dy/dx = xy + x^2 for y.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 64

Find the general solution of the trigonometric equation sin(2x) = 1/2.

Accepted Answer

x = (π/6) + nπ or x = (5π/6) + nπ, where n is any integer.

Answer

x = (π/6) + (nπ/2) or x = (5π/6) + (nπ/2), where n is any integer.

Answer

x = (π/3) + nπ or x = (2π/3) + nπ, where n is any integer.