Q1. Simplify the expression and write the result in standard form.

Background

Topic: Complex Numbers

This question tests your ability to simplify a fraction involving complex numbers and express the result in standard form .

Key Terms and Formulas:

• Complex number: , where and are real numbers, and is the imaginary unit ().

• Standard form:

• To simplify, multiply numerator and denominator by the conjugate of the denominator.

• Conjugate of is .

Step-by-Step Guidance

1. Write the expression: .

2. Identify the conjugate of the denominator: .

3. Multiply both numerator and denominator by the conjugate: .

4. Expand the numerator and denominator using distributive property (FOIL method).

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Q2. Solve the rational inequality and write the solution set in interval notation.

Background

Topic: Rational Inequalities

This question tests your ability to solve inequalities involving rational expressions and express the solution in interval notation.

Key Terms and Formulas:

• Rational expression: Fraction with polynomials in numerator and denominator.

• Interval notation: A way to describe sets of numbers as intervals.

• Critical points: Values where numerator or denominator is zero.

Step-by-Step Guidance

1. Set numerator and denominator equal to zero to find critical points: and .

2. List critical points: and .

3. Divide the real number line into intervals based on these points: , , .

4. Test values from each interval in the original inequality to determine where it holds true.

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Q3. For the equation , complete the table of ordered pairs that are solutions of the equation.

Background

Topic: Quadratic Functions and Tables

This question tests your ability to evaluate a quadratic function for given values of and fill in the corresponding values.

Key Terms and Formulas:

• Quadratic function:

• Ordered pair:

Step-by-Step Guidance

1. Write the equation: .

2. Substitute into the equation and calculate .

3. Substitute into the equation and calculate .

4. Substitute into the equation and calculate .

Try solving on your own before revealing the answer!