Pure Mathematics 3 Quizzes: Master Key Concepts and Excel in Exams

Question 1

Which step is not involved in a two-column proof?

Accepted Answer

Drawing a picture

Answer

Stating a claim

Answer

Providing a justification

Answer

Drawing a conclusion

Question 2

Which step is **NOT** part of a mathematical proof by contradiction?

Accepted Answer

Stating that the original statement is true

Answer

Assuming the opposite of the statement to be proven

Answer

Showing that the assumption leads to a contradiction

Question 3

Which of these is NOT a trigonometric function?

Accepted Answer

Exponential

Answer

Sine

Answer

Cosine

Answer

Tangent

Question 4

For a function to have an inverse function, which condition must it satisfy?

Accepted Answer

It is one-to-one.

Answer

It is continuous.

Answer

It is differentiable.

Answer

It is increasing.

Question 5

Calculate the integral of sin(x) with respect to x.

Accepted Answer

-cos(x) + C

Answer

cos(x) + C

Answer

sin(x) + C

Answer

-sin(x) + C

Question 6

Determine the equation of a circle with a radius of 5 centered at the origin.

Accepted Answer

x^2 + y^2 = 25

Answer

x^2 + y^2 = 5

Answer

(x - 5)^2 + (y - 5)^2 = 25

Answer

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

Question 7

If a matrix A is invertible, then what is the value of its determinant?

Accepted Answer

non-zero

Answer

0

Answer

1

Answer

undefined

Question 8

Which type of function has an inverse function?

Accepted Answer

Bijective function

Answer

One-to-one function

Answer

Onto function

Answer

Surjective function

Question 9

Which technique cannot be used to prove mathematical statements?

Accepted Answer

Trial and Error

Answer

Direct Proof

Answer

Proof by Contradiction

Answer

Proof by Mathematical Induction

Question 10

Determine the derivative of f(x) = x^3 - 2x^2 + 1:

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 11

What is the sum of the first n odd integers?

Accepted Answer

n(n + 1)

Answer

n^2

Answer

n(n - 1)

Answer

n^2 + 1

Question 12

Find the value of sin(π/3):

Accepted Answer

√3/2

Answer

1/2

Answer

-1/2

Answer

-√3/2

Question 13

Which statement is mathematically correct?

Accepted Answer

All squares are rectangles.

Answer

All rectangles are squares.

Answer

Some squares are circles.

Answer

No circles are squares.

Question 14

Which of the following is a valid mathematical proof technique for proving statements?

Accepted Answer

Mathematical induction

Answer

Experimental verification

Answer

Trial and error

Answer

Guess and check

Question 15

Calculate the sum of the first 10 terms of the arithmetic sequence: 3, 7, 11, ...

Accepted Answer

110

Answer

100

Answer

120

Answer

130

Question 16

Solve the exponential equation: e^(2x) = 10

Accepted Answer

x = ln(5)

Answer

x = -ln(5)

Answer

x = 5

Answer

x = -5

Question 17

Find the derivative of the function f(x) = x³ - 2x² + 5x - 1.

Accepted Answer

f'(x) = 3x² - 4x + 5

Answer

f'(x) = x³ - 4x² + 5x

Answer

f'(x) = 3x² + 4x + 5

Answer

f'(x) = x³ + 4x² + 5x

Question 18

Which of the following physical quantities is a vector quantity?

Accepted Answer

Velocity

Answer

Temperature

Answer

Mass

Answer

Volume

Question 19

Find the range of the function f(x) = |x - 2| + 1?

Accepted Answer

[1, ∞)

Answer

[0, ∞)

Answer

(-∞, 1]

Answer

(-∞, 0)

Question 20

Given vectors a = (2, -3) and b = (-1, 5), what is the magnitude of the vector a - b?

Accepted Answer

√13

Answer

√10

Answer

√17

Answer

√20

Question 21

What is the derivative of the function f(x) = x^3 + 2x^2 - 5?

Accepted Answer

3x^2 + 4x

Answer

3x^2 - 4x

Answer

x^3 + 4x

Answer

x^3 - 4x

Question 22

Find the indefinite integral of f(x) = sin(2x) + e^x.

Accepted Answer

-1/2 cos(2x) + e^x + C

Answer

1/2 cos(2x) - e^x + C

Answer

-1/2 sin(2x) + e^x + C

Answer

1/2 sin(2x) - e^x + C

Question 23

Which of the following is a condition for the convergence of an alternating series?

Accepted Answer

The terms must approach zero as (n) approaches infinity.

Answer

The absolute value of each term must be greater than the absolute value of the next term.

Answer

The series must be divergent.

Answer

The terms must have alternating signs.

Question 24

What is the derivative of the function (f(x) = e^{2x} + sin(x))?

Accepted Answer

2e^{2x} + cos(x)

Answer

e^{2x} + cos(x)

Answer

2e^{2x} - cos(x)

Answer

e^{2x} - cos(x)

Question 25

Which of the following is a complex number?

Accepted Answer

1 + 2i

Answer

2

Answer

i^2

Answer

π

Question 26

What is the conjugate of the complex number (3 + 4i)?

Accepted Answer

(3 - 4i)

Answer

(3 + 4i)

Answer

(4 + 3i)

Answer

(4 - 3i)

Question 27

What is the domain of the function f(x) = √(x - 2)?

Accepted Answer

x ≥ 2

Answer

x ≤ 2

Answer

x > 2

Answer

All real numbers

Question 28

Which of the following is the value of sin(π/3)?

Accepted Answer

√3/2

Answer

1/2

Answer

1

Answer

√3

Question 29

Write the equation of the circle with center (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 30

Solve the differential equation y' = 2x.

Accepted Answer

y = x^2 + C

Answer

y = x^2 - C

Answer

y = 2x^2 + C

Answer

y = 2x^2 - C

Question 31

Calculate the sum of the first 10 terms of the geometric sequence 2, 4, 8, ...

Accepted Answer

1023

Answer

1022

Answer

1024

Answer

1025

Question 32

In a mathematical proof by contradiction, which step is NOT included?

Accepted Answer

Stating the final conclusion that the original statement is true

Answer

Assuming the opposite of the statement to be proven

Answer

Drawing logical conclusions from the assumption

Answer

Reaching a contradiction with the initial assumption

Question 33

Given the sequence {an} = {1, 3, 9, 27, 81, ...}, determine the term-to-term rule.

Accepted Answer

an = 3^n

Answer

an = n

Answer

an = n²

Answer

an = 3n

Question 34

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

Accepted Answer

[-1, 1]

Answer

[-1, 3]

Answer

[-2, 2]

Answer

[-3, 3]

Question 35

Solve the equation log₂(x + 1) + log₂(x - 1) = 3.

Accepted Answer

x = 4

Answer

x = 2

Answer

x = 6

Answer

x = 8

Question 36

Consider the equation y = ax² + bx + c. Which of the following is the discriminant of this equation?

Accepted Answer

b² - 4ac

Answer

b - 4ac

Answer

a² - 4bc

Answer

4a² - bc

Question 37

Calculate the volume of a sphere with radius r.

Accepted Answer

(4/3)πr³

Answer

(1/3)πr³

Answer

4πr²

Answer

πr³

Question 38

Which of the following is an invalid proof technique in mathematics?

Accepted Answer

Proof by intimidation

Answer

Direct proof

Answer

Proof by contradiction

Answer

Proof by induction

Question 39

Find the general solution to the equation 2x^2 - 5x + 2 = 0.

Accepted Answer

x = 1/2 or x = 2

Answer

x = 1 or x = 4

Question 40

If f(x) = x^3 - 2x^2 + x, find f'(x).

Accepted Answer

3x^2 - 4x + 1

Question 41

Find the area of the triangle with vertices (0, 0), (3, 4), and (-1, 2).

Accepted Answer

5 square units

Answer

10 square units

Question 42

If the sum of the first n terms of an arithmetic sequence is 2n^2 + 5n, find the nth term.

Accepted Answer

2n + 5

Answer

4n + 5

Question 43

Which of the following is NOT a requirement for a function to be continuous at a point?

Accepted Answer

The function is differentiable at the point.

Answer

The function is defined at the point.

Answer

The limit of the function as x approaches the point exists.

Answer

The graph of the function has no breaks or jumps at the point.

Question 44

Which of the following is a unique characteristic of the binomial expansion?

Accepted Answer

The exponents of the terms form an arithmetic sequence.

Answer

The coefficients of the terms are all positive.

Answer

The constant term is always 1.

Answer

The middle term has the highest coefficient.

Question 45

Find the limit of the sequence {1/n} as n approaches infinity.

Accepted Answer

0

Answer

1

Answer

∞

Answer

-∞

Question 46

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

Accepted Answer

cos(x)

Answer

-cos(x)

Answer

sin(x)

Answer

-sin(x)

Question 47

Solve the equation log(x) = 2.

Accepted Answer

10

Answer

0

Answer

1

Answer

100

Question 48

Which of the following is a valid mathematical proof technique that involves assuming the opposite of what is to be proven and deducing a contradiction?

Accepted Answer

Proof by contradiction

Answer

Proof by induction

Answer

Proof by analogy

Answer

Proof by exhaustion

Question 49

Which of the following is the equation of a circle with radius 3 and center at the point (2, 5)?

Accepted Answer

(x - 2)^2 + (y - 5)^2 = 9

Answer

x^2 + y^2 = 9

Answer

x^2 + y^2 - 4x + 10y = 9

Answer

(x + 2)^2 + (y + 5)^2 = 9

Question 50

Which of the following is not a type of function? Select the option that represents a constant value rather than a variable relationship.

Accepted Answer

Constant function

Answer

Linear function

Answer

Quadratic function

Answer

Polynomial function

Question 51

What is the integral of the function f(x) = sin(x) with respect to x? The integral is a function that represents the area under the curve of the original function.

Accepted Answer

-cos(x) + C

Answer

cos(x) + C

Answer

sin(x) + C

Answer

-sin(x) + C

Question 52

Which of the following is not a trigonometric identity? Select the option that is not a mathematical equation involving trigonometric functions that is valid for all values of the variables.

Accepted Answer

sec(x) = tan(x) + cot(x)

Answer

sin^2(x) + cos^2(x) = 1

Answer

tan(x) = sin(x) / cos(x)

Answer

cot(x) = cos(x) / sin(x)

Question 53

What is the equation of the plane that passes through the points (1, 2, 3), (2, 3, 4), and (3, 4, 5)? The equation of a plane can be determined using the coordinates of three non-collinear points.

Accepted Answer

x + y + z = 11

Answer

x + y + z = 10

Answer

x + y + z = 12

Answer

x + y + z = 13

Question 54

Which of the following is the derivative of the function f(x) = ln(x)?

Accepted Answer

1/x

Answer

x

Answer

ln(x)

Answer

e^x

Question 55

Which of the following is the equation of the line that passes through the points (1, 2) and (3, 6)?

Accepted Answer

y = 3x - 1

Answer

y = x + 1

Answer

y = 2x

Answer

y = x - 2

Question 56

Which of the following is a necessary condition for the convergence of an infinite series?

Accepted Answer

The terms of the series must approach zero.

Answer

The terms of the series must be positive.

Answer

The terms of the series must be decreasing.

Answer

The sum of the series must be finite.

Question 57

Determine the gradient of the line perpendicular to the line defined by the equation y = 2x + 3.

Accepted Answer

-1/2

Answer

-2

Answer

2

Answer

1/2

Question 58

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

Accepted Answer

3x^2 + 4x

Answer

x^2 + 4x

Answer

x^3 + 4x

Answer

3x^3 + 4x

Question 59

Evaluate the integral of the function f(x) = 2x + 1 with respect to x from 0 to 2.

Accepted Answer

6

Answer

2

Answer

12

Answer

10

Question 60

State the domain of the function f(x) = √(x - 2).

Accepted Answer

[2, ∞)

Answer

[-2, ∞)

Answer

(2, ∞)

Question 61

Which of the following is a necessary condition for a function to be one-to-one?

Accepted Answer

The function is injective.

Answer

The function is continuous.

Answer

The function is strictly increasing.

Answer

The function is strictly decreasing.

Question 62

Which of the following is the inverse function of f(x) = log2(x)?

Accepted Answer

2^x

Answer

e^x

Answer

log2(x)

Answer

x^2

Question 63

Which of the following is the equation of the line that has a gradient of 2 and passes through the point (1, 3)?

Accepted Answer

y = 2x + 1

Answer

y = x + 2

Answer

y = -x + 4

Answer

y = -2x + 5

Question 64

What is the gradient of the curve y = x^3 - 2x at the point (1, -1)?

Accepted Answer

1

Answer

-1

Answer

-2

Answer

2