Q1. For the set {15, 49, 0.42, 0.1, 4, 5, -π}, list all the elements that belong to (a) natural numbers, (b) whole numbers, (c) integers, (d) rational numbers, (e) irrational numbers, and (f) real numbers.

Background

Topic: Sets of Real Numbers

This question tests your understanding of the classification of numbers into different sets: natural, whole, integers, rational, irrational, and real numbers.

Key Terms:

• Natural Numbers: Counting numbers starting from 1 ()

• Whole Numbers: Natural numbers plus zero ()

• Integers: Whole numbers and their negatives ()

• Rational Numbers: Numbers that can be written as , where and are integers and

• Irrational Numbers: Numbers that cannot be written as a ratio of integers (non-repeating, non-terminating decimals, e.g., )

• Real Numbers: All rational and irrational numbers

Step-by-Step Guidance

1. Examine each element in the set and determine which categories it fits into. For example, check if 15 is a natural number, whole number, integer, rational, etc.

2. Recall that negative values (like ) are not natural or whole numbers, but may be integers, rational, or irrational.

3. Identify which numbers are decimals and whether they are terminating or repeating (rational) or non-repeating (irrational).

4. Sort each number into the appropriate sets based on their definitions.

5. Prepare to list the elements for each set, but stop before writing the final lists.

Try solving on your own before revealing the answer!

Q2. Plot the points (-2, 4) and (0, -3) and state the quadrant each point lies in, if any.

Background

Topic: Rectangular Coordinate System

This question tests your ability to plot points on the Cartesian plane and identify their locations by quadrant.

Key Terms:

• Quadrants: The four regions created by the x- and y-axes in the plane.

• Point: An ordered pair representing a location.

Step-by-Step Guidance

1. Recall the quadrant layout: Quadrant I (x > 0, y > 0), Quadrant II (x < 0, y > 0), Quadrant III (x < 0, y < 0), Quadrant IV (x > 0, y < 0).

2. Plot the point : x is negative, y is positive.

3. Plot the point : x is zero, y is negative.

4. Determine which quadrant or axis each point lies on, but do not state the final quadrant yet.

Try solving on your own before revealing the answer!

Q3. If and , find using the Pythagorean Theorem.

Background

Topic: Pythagorean Theorem

This question tests your ability to use the Pythagorean Theorem to solve for a missing side of a right triangle.

Key Formula:

Where:

• and are the legs of the triangle

• is the hypotenuse

Step-by-Step Guidance

1. Write the formula:

2. Substitute and into the equation.

3. Rearrange to solve for .

4. Prepare to take the square root to solve for , but stop before calculating the value.

Try solving on your own before revealing the answer!

Q4. Find the distance and midpoint of the line segment connecting the points (-2, 4) and (0, -3).

Background

Topic: Distance and Midpoint Formulas

This question tests your ability to use formulas to find the length and midpoint of a segment between two points in the plane.

Key Formulas:

• Distance:

• Midpoint:

Step-by-Step Guidance

1. Label the points: and .

2. Plug the coordinates into the distance formula.

3. Plug the coordinates into the midpoint formula.

4. Set up the calculations for both, but stop before computing the final values.

Try solving on your own before revealing the answer!

Q5. Write the function whose graph is , but is shifted to the left two units and up three units.

Background

Topic: Transformations of Functions

This question tests your understanding of how to apply horizontal and vertical shifts to a function.

Key Formula:

• Horizontal shift left by :

• Vertical shift up by :

Step-by-Step Guidance

1. Start with the parent function .

2. Apply the horizontal shift left by 2 units: .

3. Apply the vertical shift up by 3 units: .

4. Combine the transformations into a single function, but stop before writing the final expression.

Try solving on your own before revealing the answer!

Q6. Let . Find the difference quotient and simplify the expression.

Background

Topic: Difference Quotient

This question tests your ability to compute and simplify the difference quotient for a given function, which is foundational for calculus.

Key Formula:

Difference Quotient:

Step-by-Step Guidance

1. Write the formula for the difference quotient:

2. Compute by substituting into the function.

3. Subtract from .

4. Set up the expression to simplify, but stop before completing the simplification.

Try solving on your own before revealing the answer!