Pure Mathematics 1: Master Algebra and Calculus Concepts

Question 1

Which of the following is NOT a transformation of the function f(x) = x^2?

Accepted Answer

Taking the inverse function

Answer

Translation 2 units to the right

Answer

Reflection across the x-axis

Answer

Dilation by a factor of 2

Question 2

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

Accepted Answer

(2x - 2) / (x - 2)^2

Answer

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

Answer

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

Answer

(x^2 + 1) / (2x - 4)

Question 3

Which of the following accurately describes the relationship between the graphs of the functions f(x) = x^2 and g(x) = x^3?

Accepted Answer

The graph of g(x) has a steeper slope than the graph of f(x) at every point.

Answer

The graph of g(x) is shifted right by 1 unit compared to the graph of f(x).

Answer

The graph of g(x) is shifted up by 1 unit compared to the graph of f(x).

Answer

The graph of g(x) is a reflection of the graph of f(x) over the x-axis.

Question 4

Find the derivative of the function $$f(x) = (sin(x) + cos(x))^2$$ using the chain rule.

Accepted Answer

$$2(sin(x) + cos(x))(cos(x) - sin(x))$$

Answer

$$2(sin(x) + cos(x))$$

Answer

$$(sin(x) + cos(x))$$

Answer

$$(cos(x) - sin(x))$$

Question 5

Identify the even function among the given options:

Accepted Answer

f(x) = |x|

Answer

f(x) = x² + 1

Answer

f(x) = x³ - x

Answer

f(x) = sin(x)

Question 6

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

Accepted Answer

1/6

Answer

1/3

Answer

1/5

Answer

1/4

Question 7

Identify the geometric sequence among the options:

Accepted Answer

1, 2, 4, 8, 16, ...

Answer

2, 4, 6, 8, ...

Answer

5, 10, 15, 20, 25, ...

Answer

3, 6, 12, 24, 48, ...

Question 8

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

Accepted Answer

x^2 + x + C

Answer

2x^2 + C

Answer

x^2 + C

Answer

2x + C

Question 9

Solve the quadratic equation x^2 - 5x + 6 = 0.

Accepted Answer

x = 2 or x = 3

Answer

x = -2 or x = -3

Answer

x = 1 or x = -6

Answer

x = -1 or x = 6

Question 10

Which of the following is a rational function?

Accepted Answer

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

Answer

f(x) = x^2 + 1

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Question 11

Which of the following is the domain of the function f(x) = sqrt(x-1)?

Accepted Answer

x ≥ 1

Answer

x ≤ 1

Answer

x > 1

Answer

All real numbers

Question 12

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 13

Find the equation of the circle with centre (2, 3) and radius 4.

Accepted Answer

(x-2)^2 + (y-3)^2 = 16

Answer

(x-2)^2 + (y-3)^2 = 8

Answer

(x+2)^2 + (y+3)^2 = 16

Answer

(x+2)^2 + (y+3)^2 = 8

Question 14

Which of the following is the equation of the normal to the curve y = x^2 at the point (2, 4)?

Accepted Answer

y = -x + 8

Answer

y = x + 4

Answer

y = -x/2 + 6

Answer

y = x/2 + 2

Question 15

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

Accepted Answer

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

Answer

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

Answer

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

Answer

f^-1(x) does not exist

Question 16

Calculate the integral of ∫(x² - 2x + 1) dx.

Accepted Answer

x³/3 - x² + x + C

Answer

x³ - 2x² + x + C

Answer

x³ - 2x² + 1 + C

Answer

x² - 2x + 1 + C

Question 17

Find the discriminant of the quadratic equation x² - 5x + 6 = 0.

Accepted Answer

25

Answer

1

Answer

5

Answer

-25

Question 18

Determine the vertex of the parabola y = x² - 4x + 3.

Accepted Answer

(2, -1)

Answer

(-2, -1)

Answer

(2, 1)

Answer

(-2, 1)

Question 19

Graph the inequality y > x² - 2x + 1.

Accepted Answer

Region above the parabola y = x² - 2x + 1

Answer

Region below the parabola y = x² - 2x + 1

Answer

Line y = x² - 2x + 1

Answer

Parabola y = x² - 2x + 1

Question 20

Identify the domain restriction for the function f(x) = sqrt(2x - 5)?

Accepted Answer

x >= 5/2

Answer

x <= 0

Answer

x > 0

Answer

x != 1

Question 21

Determine the equation of the tangent to the curve y = x^2 + 3x - 4 at the point (1, 0).

Accepted Answer

y = 2x - 4

Answer

y = 5x - 4

Answer

y = 5x + 4

Answer

y = -2x + 4

Question 22

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

Accepted Answer

f^-1(x) = (x - 5)/2

Answer

f^-1(x) = 2x - 5

Answer

f^-1(x) = 5x - 2

Answer

f^-1(x) = 1/(2x + 5)

Question 23

For a quadratic function f(x) = ax² + bx + c, where a is a non-zero constant and the discriminant is positive, what is the shape of its graph?

Accepted Answer

Opens upward with a maximum point

Answer

Opens upward with a minimum point

Answer

Opens downward with a maximum point

Question 24

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

Accepted Answer

2x + 3

Answer

x^2 + 3

Answer

x - 5

Answer

2x - 3

Question 25

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

Accepted Answer

y = 2x - 1

Answer

y = -2x

Answer

y = x + 2

Answer

y = x - 2

Question 26

Find the domain and range of the function f(x) = √(x - 2).

Accepted Answer

Domain: [2, ∞); Range: [0, ∞)

Answer

Domain: (-∞, 2]; Range: [0, ∞)

Answer

Domain: [2, ∞); Range: (-∞, 0]

Answer

Domain: (-∞, 2]; Range: (-∞, 0]

Question 27

Which integral represents the area under the curve y = sin(x) between the x-coordinates 0 and π?

Accepted Answer

∫[0, π] sin(x) dx

Answer

∫[π, 0] sin(x) dx

Answer

∫[0, π] sin^2(x) dx

Answer

∫[π, 0] cos(x) dx

Question 28

Which transformation is represented by the matrix [[-1, 0], [0, -1]]?

Accepted Answer

Reflection across the y-axis

Answer

Reflection across the x-axis

Answer

Rotation by 90 degrees clockwise about the origin

Answer

Rotation by 90 degrees counterclockwise about the origin

Question 29

Which of the following series converges?

Accepted Answer

∑[n=1,∞] 1/n^2

Answer

∑[n=1,∞] 1/n

Answer

∑[n=1,∞] n

Answer

∑[n=1,∞] (-1)^n

Question 30

Calculate the integral of the expression ∫(x^2 + 2x - 1) dx.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 31

Factorize the algebraic expression x^2 - 9.

Accepted Answer

(x + 3)(x - 3)

Answer

(x + 9)(x - 9)

Answer

(x - 3)^2

Answer

(x + 3)^2

Question 32

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

Accepted Answer

y = -2x + 9

Answer

y = -x + 7

Answer

y = x + 3

Answer

y = 2x + 1

Question 33

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

Accepted Answer

All real numbers except 2

Answer

All real numbers except 0

Answer

All real numbers

Answer

All positive real numbers

Question 34

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

Accepted Answer

3x^2 + 4x - 5

Answer

3x^2 + 2x - 5

Answer

x^3 + 4x^2 - 5x

Answer

x^2 + 2x - 5

Question 35

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

Accepted Answer

All real numbers except 2

Answer

All real numbers except 0

Answer

All real numbers

Answer

Natural numbers only

Question 36

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) = 1 - 2x

Answer

f^(-1)(x) does not exist

Question 37

Calculate the limit of the function f(x) = (x^2 - 4) / (x - 2) as x approaches 2.

Accepted Answer

0

Answer

2

Answer

4

Answer

Does not exist

Question 38

Find the derivative of the logarithmic function f(x) = ln(x^2 + 1).

Accepted Answer

2x / (x^2 + 1)

Answer

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

Answer

x / (x^2 + 1)

Answer

1 / (x^2 + 1)

Question 39

Determine the indefinite integral of ∫(cos(x) + sin(x)) dx.

Accepted Answer

sin(x) + cos(x) + C

Answer

sin(x) - cos(x) + C

Answer

cos(x) - sin(x) + C

Answer

cos(x) + sin(x) - C

Question 40

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

Accepted Answer

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

Answer

f'(x) = 3x + 4

Answer

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

Answer

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

Question 41

Which of the following is the graph of the function h(x) = |x - 2|?

Accepted Answer

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

Answer

A U-shaped graph with a vertex at (0, 2).

Answer

A straight line with a slope of 1.

Answer

A parabola with a vertex at (2, 0).

Question 42

Simplify the expression (2x + 3)(x - 1).

Accepted Answer

2x^2 - x + 6

Answer

x^2 - x + 6

Answer

2x^2 + x - 6

Answer

x^2 + x + 6

Question 43

Find the domain of the function f(x) = 1 / (x - 3).

Accepted Answer

All real numbers except x = 3.

Answer

All real numbers.

Answer

All real numbers except x = 0.

Question 44

Determine the range of the function g(x) = -x^2 + 4.

Accepted Answer

All real numbers less than or equal to 4.

Answer

All real numbers.

Answer

All real numbers less than 4.

Answer

All real numbers greater than or equal to 4.

Question 45

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

Accepted Answer

3x² + 4x - 5

Answer

3x³ + 2x² - 5

Answer

x³ + 4x² - 5x

Answer

x² + 4x - 5

Question 46

Identify the graph of the function f(x) = |x - 2|.

Accepted Answer

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

Answer

Line with slope 1 passing through (2, 0)

Answer

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

Answer

Parabola opening upwards with vertex at (2, 0)

Question 47

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

Accepted Answer

(x - 3)/2

Answer

(x + 3)/2

Answer

2x - 3

Question 48

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

Accepted Answer

[2, ∞)

Answer

(-∞, 2]

Answer

(-∞, 2)

Question 49

Find the derivative of the function f(x) = sin(2x).

Accepted Answer

2cos(2x)

Answer

4cos(x)

Answer

sin(4x)

Answer

-2cos(2x)

Question 50

Evaluate the indefinite integral ∫(cos(x) + sin(x)) dx.

Accepted Answer

sin(x) - cos(x) + C

Answer

-sin(x) - cos(x) + C

Answer

cos(x) + sin(x) + C

Question 51

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

Accepted Answer

3x^2 - 4x

Answer

x^2 + 4x

Answer

3x - 4

Answer

x^2 - 4x

Question 52

Determine the equation of the circle with its center at (2, -1) and a radius of 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

Question 53

Solve the equation log2 (x + 2) = 3.

Accepted Answer

x = 7

Answer

x = 5

Answer

x = 9

Question 54

Simplify the expression: (x^2 - 4)/(x^2 + 2x + 4).

Accepted Answer

(x - 2)/(x + 2)

Answer

(x + 2)/(x - 2)

Answer

x^2 - 4

Question 55

What is the derivative of the function  *f(x) = x³ + 2x² - 5x + 1*?

Accepted Answer

3x² + 4x - 5

Answer

x² + 4x - 5

Answer

x³ + 4x² - 5x + 1

Answer

3x² - 4x - 5

Question 56

What is the integral of  *∫(x² + 3x - 2) dx*?

Accepted Answer

(x³/3) + (3x²/2) - 2x + C

Answer

(x³) + (3x²/2) - 2x + C

Answer

(x²/2) + (3x/2) - 2x + C

Answer

(x³/3) + (3x/2) - 2x + C

Question 57

What is the equation of the line that passes through the points (2, 3) and (-1, 5)?

Accepted Answer

y = -2x + 7

Answer

y = 2x + 7

Answer

y = 2x - 7

Answer

y = -2x + 5

Question 58

What is the domain of the function *f(x) = 1/(x - 2)*?

Accepted Answer

All real numbers except x = 2

Answer

All real numbers

Answer

x > 2

Answer

x < 2