Pure Mathematics 3 Quizzes: Master Complex Numbers, Calculus, and Statistics

Question 1

Determine the derivative of the function f(x) = (x^2 + 2x + 1) / (x - 1) using the quotient rule.

Accepted Answer

((x^2 + 2x + 1) * (x - 1) - (x^2 + 2x + 1))/(x - 1)^2

Answer

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

Answer

1

Answer

(2x - 2) / (x - 1)

Question 2

Which mathematical expression is invalid?

Accepted Answer

2 + 3 / 0

Answer

2 + 3 * 4

Answer

(2 + 3) * 4

Answer

2 ^ (3 ^ 2)

Question 3

Determine the equation of the curve whose gradient is double its y-coordinate and passes through the point (1, 2).

Accepted Answer

y = 2x + 1

Answer

y = x^2 + 1

Answer

y = x + 2

Answer

y = 4x - 2

Question 4

Which technique is used to calculate the area under a curve?

Accepted Answer

Integration

Answer

Differentiation

Answer

Substitution

Answer

Logarithms

Question 5

Find the derivative of the function f(x) = x^2 + 2x + 1 with respect to x.

Accepted Answer

2x + 2

Answer

2

Answer

x + 2

Answer

x^2 + 2

Question 6

In hypothesis testing, what is the term for the hypothesis that assumes no significant difference?

Accepted Answer

Null hypothesis

Answer

Research hypothesis

Answer

Alternative hypothesis

Answer

Experimental hypothesis

Question 7

Which probability distribution is characterized by a bell-shaped curve with a mean of 0 and a standard deviation of 1?

Accepted Answer

Normal Distribution

Answer

Binomial Distribution

Answer

Poisson Distribution

Answer

Exponential Distribution

Question 8

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

Accepted Answer

3x^2 - 4x

Answer

3x^2 + 4x

Answer

x^2 - 4x + 5

Answer

3x^2 - 4x + 5

Question 9

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

Accepted Answer

1/6

Answer

1/3

Answer

1/4

Answer

1/5

Question 10

In a hypothesis test with a 5% significance level, if the p-value is 0.06, what is the appropriate conclusion?

Accepted Answer

Fail to reject the null hypothesis

Answer

Reject the null hypothesis

Answer

Accept the alternative hypothesis

Answer

Insufficient evidence to conclude

Question 11

Which property is **NOT** applicable to complex numbers?

Accepted Answer

Commutative property of addition

Answer

Closed under addition

Answer

Closed under multiplication

Answer

Distributive property of multiplication over addition

Question 12

Integrate the function ∫(x^2 + 1) dx.

Accepted Answer

x^3/3 + x + C

Answer

x^2/2 + x + C

Answer

3x^2 + x + C

Answer

x^3 + x^2 + C

Question 13

Identify the limit of the sequence an = (n^2 - 1) / (n^2 + 1) as n approaches infinity.

Accepted Answer

1

Answer

0

Answer

2

Answer

Infinity

Question 14

Which condition is **NOT** required for a function to be differentiable at a specific point?

Accepted Answer

Presence of an inflection point

Answer

Continuity at that point

Answer

Existence and equality of left-hand and right-hand derivatives

Answer

Definition over an open interval containing that point.

Question 15

Find the second derivative of f(x) = e^(2x) + sin(x).

Accepted Answer

4e^(2x) + cos(x)

Answer

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

Answer

e^(2x) + cos(x)

Answer

4e^(2x) - cos(x)

Question 16

Which of the following accurately represents the imaginary unit?

Accepted Answer

i

Answer

-i

Answer

1/i

Answer

-1/i

Question 17

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

Accepted Answer

3x² + 4x - 5

Answer

3x² - 4x - 5

Answer

x³ + 4x² - 5x

Answer

x³ - 4x² - 5x

Question 18

Find the antiderivative of the function f(x) = e^x.

Accepted Answer

e^x + C

Answer

e^x

Answer

x * e^x

Answer

x * e^x + C

Question 19

Which of the following statements about probability is accurate?

Accepted Answer

It can never exceed 1.

Answer

It can take on values less than 0.

Answer

It is always equal to 0 or 1.

Answer

It can be greater than 1.

Question 20

In hypothesis testing, the null hypothesis proposes that:

Accepted Answer

There is no significant difference between the groups.

Answer

The alternative hypothesis is true.

Answer

The research hypothesis is true.

Answer

The sample is not representative of the population.

Question 21

Determine the equation of the line that passes through the points (2, 3) and (5, 7).

Accepted Answer

y = 2x + 1

Answer

y = x + 1

Answer

y = x + 3

Answer

y = 2x - 1

Question 22

Find the complex conjugate of 3 + 4i.

Accepted Answer

3 - 4i

Answer

3 + 4i

Answer

4 + 3i

Answer

4 - 3i

Question 23

What is the probability of rolling a six on a standard six-sided die?

Accepted Answer

1/6

Answer

1/3

Answer

1/4

Answer

1/5

Question 24

Which characteristic is unique to a binomial distribution?

Accepted Answer

Discrete values

Answer

Continuous values

Answer

Symmetric shape

Answer

Bell-shaped curve

Question 25

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

Accepted Answer

6x^2 - 10x + 3

Answer

x^2 - 10x + 3

Answer

6x - 10

Answer

x - 10

Question 26

Which of the following statistical measures represents the average value of a dataset?

Accepted Answer

Mean

Answer

Median

Answer

Mode

Answer

Standard deviation

Question 27

Which of the following types of correlation indicates that as one variable increases, the other variable decreases?

Accepted Answer

Negative correlation

Answer

Positive correlation

Answer

No correlation

Answer

Perfect correlation

Question 28

Find the derivative of the function f(x) = sin(x)cos(x) using the product rule.

Accepted Answer

2cos(x)sin(x)

Answer

cos(2x)

Answer

sin(2x)

Answer

2sin(x)cos(x)

Question 29

Find the modulus of the complex number (3 + 4i) / (2 - i).

Accepted Answer

5/√17

Answer

√17/5

Answer

5/√5

Answer

√5/5

Question 30

Find the next term in the sequence: 2, 5, 10, 17, ...

Accepted Answer

26

Answer

27

Answer

28

Answer

29

Question 31

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

Accepted Answer

1 / 6

Answer

1 / 3

Answer

1 / 4

Answer

1 / 5

Question 32

Find the modulus of the complex number z = 3 + 4i.

Accepted Answer

5

Answer

1

Answer

10

Answer

15

Question 33

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

Accepted Answer

3x^2 - 4x + 5

Answer

x^3 - 4x^2 + 5x

Answer

3x^2 - 2x + 5

Answer

x^3 - 2x^2 + 5

Question 34

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

Accepted Answer

(x^3)/3 + x + C

Answer

x^2 + x + C

Answer

(x^2 + 1)/2 + C

Answer

x^3 + C

Question 35

Determine the indefinite integral of the function f(x) = sin(x) cos(x).

Accepted Answer

(1/2) sin(2x) + C

Answer

(1/2) cos(2x) + C

Answer

-cos(2x) + C

Answer

sin(2x) + C

Question 36

Evaluate the integral (int(x^2 + 1) dx).

Accepted Answer

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

Answer

x^2 + x + C

Answer

((x^3 + 1) / 3)

Answer

((x^2 / 2) + 1)

Question 37

What is the probability of obtaining a 6 when rolling a standard six-sided die?

Accepted Answer

1 / 6

Answer

1 / 2

Answer

1 / 4

Answer

1 / 3

Question 38

State the null hypothesis for a two-sample t-test.

Accepted Answer

There is no significant difference between the means of the two populations.

Answer

The variances of the two populations are equal.

Answer

There is a significant difference between the means of the two populations.

Question 39

Which of the following is NOT a characteristic of a normal distribution?

Accepted Answer

The mean is less than the median

Answer

It is bell-shaped

Answer

The standard deviation measures the spread of the distribution

Answer

It is symmetric around the mean

Question 40

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

Accepted Answer

3 - 4i

Answer

-3 + 4i

Answer

4 - 3i

Answer

4 + 3i

Question 41

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

Accepted Answer

(cos(x))

Answer

(cos(2x))

Answer

(sin(x))

Answer

-(sin(x))

Question 42

Evaluate the integral (int(e^x + x) dx).

Accepted Answer

((e^x + x^2 / 2) + C)

Answer

((e^x + x) + C)

Answer

((e^x - x) + C)

Answer

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

Question 43

Find the modulus of the complex number (2 + 3i).

Accepted Answer

√13

Answer

2

Answer

3

Answer

5

Question 44

Which of the following probability distributions has a mean of 0 and standard deviation of 1?

Accepted Answer

Standard Normal Distribution

Answer

Uniform Distribution

Answer

Poisson Distribution

Answer

Binomial Distribution

Question 45

Which of the following differential equations represents exponential growth?

Accepted Answer

dy/dx = ky

Answer

dy/dx = kx

Answer

d^2y/dx^2 = ky

Answer

d^2y/dx^2 = kx

Question 46

What is the modulus of the complex number *z* = *a* + *bi*?

Accepted Answer

√(*a*² + *b*²)

Answer

√(*b*² - *a*)

Answer

√(*a* - *b*²)

Question 47

What is the value of *i*³?

Accepted Answer

-*i*

Answer

-1

Answer

1

Answer

*i*

Question 48

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

Accepted Answer

3*x*² - 4*x* + 5

Answer

*x*³ - 4*x* + 5

Answer

3*x*³ - 4*x* - 5

Answer

3*x*² + 4*x* - 5

Question 49

What is the indefinite integral of (*x*² + 1) with respect to *x*?

Accepted Answer

(*x*³/3) + *x* + *C*

Answer

(2*x* + 1)*dx*

Answer

(*x*²) + *x* + *C*

Answer

(*x*² - 1)*dx*

Question 50

A bag contains 5 red balls, 3 blue balls, and 2 green balls. If two balls are drawn randomly without replacement, what is the probability that both balls are blue?

Accepted Answer

3/20

Answer

1/10

Answer

3/10

Answer

1/20

Question 51

A normal distribution has a mean of 50 and a standard deviation of 10. What is the probability of a randomly selected value falling between 40 and 60?

Accepted Answer

0.6827

Answer

0.75

Answer

0.5

Answer

0.9545

Question 52

What is the primary purpose of a confidence interval?

Accepted Answer

To estimate the true population parameter with a specified level of confidence.

Answer

To determine whether a hypothesis is true or false.

Answer

To compare the means of two groups.

Question 53

Find the modulus of the complex number **z = 3 + 4i**.

Accepted Answer

5

Answer

1

Answer

7

Answer

**√7**

Question 54

Apply the **power rule of differentiation** to determine the derivative of the function **f(x) = 3x^2 + 2x - 1**.

Accepted Answer

6x + 2

Answer

6x^2 + 2x

Answer

3x + 2

Question 55

Evaluate the definite integral of **f(x) = x^2** from **x = 1 to x = 3**.

Accepted Answer

8

Answer

1

Answer

26/3

Answer

9

Question 56

A fair coin is flipped four times. What is the probability of getting exactly two heads?

Accepted Answer

3/8

Answer

1/2

Answer

1/16

Answer

1/4

Question 57

A bag contains **5 red balls** and **3 blue balls**. Two balls are drawn at random **without replacement**. What is the probability that both balls are **red**?

Accepted Answer

5/14

Answer

1/2

Answer

3/8

Answer

5/8

Question 58

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

Accepted Answer

y = x - 2

Answer

y = 2x - 3

Answer

y = -x

Answer

y = 3x - 4

Question 59

Given the complex number **z = 2 - i**, calculate the result of multiplying **z** by its conjugate.

Accepted Answer

5

Answer

3 - 2i

Answer

2 + i

Answer

4 + i

Question 60

Determine the area enclosed by the curve **y = x^2**, the **x-axis**, and the lines **x = 1** and **x = 3**.

Accepted Answer

26/3

Answer

9

Answer

8

Answer

1

Question 61

A hypothesis test is conducted to determine if the mean height of students in a school differs from **1.75 meters**. The null hypothesis is rejected. What conclusion can be drawn?

Accepted Answer

There is sufficient evidence to suggest that the mean height is different from 1.75 meters.

Answer

There is no evidence to suggest that the mean height is different from 1.75 meters.

Answer

The mean height is definitely 1.75 meters.

Question 62

Given the equation of a circle, **(x - 2)^2 + (y + 1)^2 = 9**, identify the radius of the circle.

Accepted Answer

3

Answer

2

Answer

1

Answer

9

Question 63

Which of the following is the modulus of the complex number (5 + 3i)?

Accepted Answer

√(34)

Answer

3

Answer

5

Answer

√(58)

Question 64

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

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 2x - 5

Answer

3x^3 + 4x^2 - 5

Answer

3x^3 + 2x^2 - 5