Q1. Simplify the complex numbers:

Background

Topic: Complex Numbers

This question tests your understanding of how to simplify expressions involving imaginary numbers and perform arithmetic with complex numbers.

Key Terms and Formulas:

• Imaginary unit: , where

• Complex number: , where and are real numbers

Step-by-Step Guidance

1. For part (a), recognize that can be rewritten using the imaginary unit .

2. Recall that for any positive real number .

3. For part (b), subtract the real and imaginary parts separately: .

4. Combine like terms: subtract $20 for the real part, and from for the imaginary part.

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Q2. Solve the following quadratic equations using any method:

Background

Topic: Quadratic Equations

This question tests your ability to solve quadratic equations using factoring, the square root property, completing the square, or the quadratic formula.

Key Terms and Formulas:

• Quadratic Formula:

• Factoring: Expressing the quadratic as a product of two binomials

• Square Root Property: If , then

Step-by-Step Guidance

1. For each equation, identify , , and (the coefficients in ).

2. Decide which method is most appropriate (factoring, quadratic formula, etc.).

3. If using the quadratic formula, substitute the values of , , and into the formula.

4. Simplify under the square root (the discriminant) and set up the next steps to solve for .

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Q3. Quadratic Applications

Background

Topic: Applications of Quadratic Functions

This section tests your ability to apply quadratic equations to real-world scenarios, such as projectile motion and the Pythagorean theorem.

Key Terms and Formulas:

• Projectile motion: (height as a function of time)

• Pythagorean theorem: (for right triangles)

Step-by-Step Guidance

1. For the ball problem, set to find when the ball hits the ground.

2. To find the time to maximum height, use the vertex formula for .

3. Plug this value back into to find the maximum height.

4. For the ladder problem, use the Pythagorean theorem to solve for the height up the wall.

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Q4. Solve the radical and rational equations:

Background

Topic: Radical and Rational Equations

This question tests your ability to solve equations involving square roots and rational expressions.

Key Terms and Formulas:

• Radical equation: An equation with a variable inside a root

• Rational equation: An equation involving fractions with polynomials in the numerator and denominator

Step-by-Step Guidance

1. For the radical equation, isolate the square root on one side.

2. Square both sides to eliminate the square root, then solve the resulting quadratic equation.

3. For the rational equation, find a common denominator to combine terms.

4. Solve the resulting equation for , being careful to check for extraneous solutions.

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Q5. State (or describe) the transformations and then graph the functions:

Background

Topic: Graph Transformations

This question tests your understanding of how changes to a function's equation affect its graph, including shifts, reflections, and stretches/compressions.

Key Terms and Formulas:

• Vertical shift: shifts up/down

• Horizontal shift: shifts left/right

• Reflection: reflects over the x-axis

• Vertical stretch/compression: stretches if , compresses if

Step-by-Step Guidance

1. Identify the parent function (e.g., , , , ).

2. Describe each transformation in the order: horizontal shift, reflection, vertical stretch/compression, vertical shift.

3. Sketch the graph using the transformations.

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Q6. Write the equation of a square root function with a reflection over the x-axis, a horizontal shift of 2 units to the right, and a vertical shift 5 units down.

Background

Topic: Function Transformations

This question tests your ability to write equations for transformed functions.

Key Terms and Formulas:

• General form:

• Reflection over x-axis:

• Horizontal shift right:

• Vertical shift down:

Step-by-Step Guidance

1. Start with the parent function .

2. Apply the reflection by multiplying by .

3. Shift the function right by replacing with .

4. Shift the function down by subtracting $5$.

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Q7. For the following quadratic functions, find the vertex, axis of symmetry, x-intercepts, y-intercept, graph, domain, and range:

Background

Topic: Quadratic Functions and Their Graphs

This question tests your ability to analyze and graph quadratic functions, and to find key features such as vertex, intercepts, and domain/range.

Key Terms and Formulas:

• Vertex: ,

• Axis of symmetry:

• X-intercepts: Set and solve for

• Y-intercept:

• Domain: All real numbers for quadratics

• Range: Depends on whether parabola opens up or down

Step-by-Step Guidance

1. Identify , , and in the quadratic function.

2. Find the vertex using and .

3. Find the axis of symmetry using the vertex's -value.

4. Find the x-intercepts by solving .

5. Find the y-intercept by evaluating .

6. State the domain and range based on the direction the parabola opens.

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Q8. Use the remainder theorem to determine if is a zero of the polynomial:

Background

Topic: Remainder Theorem

This question tests your ability to use the remainder theorem to check if a given value is a zero of a polynomial.

Key Terms and Formulas:

• Remainder theorem: The remainder when is divided by is .

• If , then is a zero of .

Step-by-Step Guidance

1. Substitute into the polynomial .

2. Calculate step by step.

3. If , then is a zero; otherwise, it is not.

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Q9. Use the factor theorem to determine if is a factor of the polynomial function:

Background

Topic: Factor Theorem

This question tests your ability to use the factor theorem to check if a binomial is a factor of a polynomial.

Key Terms and Formulas:

• Factor theorem: is a factor of if and only if .

Step-by-Step Guidance

1. Substitute into the polynomial .

2. Calculate step by step.

3. If , then is a factor; otherwise, it is not.

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Q10. Find all the zeros and multiplicities of the function:

Background

Topic: Zeros and Multiplicities of Polynomials

This question tests your ability to identify the zeros of a polynomial and their multiplicities from factored form.

Key Terms and Formulas:

• Zero: Value of where

• Multiplicity: The exponent on the factor corresponding to the zero

Step-by-Step Guidance

1. Set each factor equal to zero and solve for .

2. Identify the multiplicity by looking at the exponent of each factor.

3. List each zero and its multiplicity.

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Q11. Graph the following polynomial functions using the parts listed below:

Background

Topic: Polynomial Graphs

This question tests your ability to analyze and graph polynomial functions, including finding zeros, intercepts, degree, leading coefficient, and end behavior.

Key Terms and Formulas:

• Degree: Highest power of

• Leading coefficient: Coefficient of the highest power term

• End behavior: Determined by degree and leading coefficient

Step-by-Step Guidance

1. List the zeros and their multiplicities by setting each factor to zero.

2. Find the y-intercept by evaluating .

3. Determine the degree and leading coefficient.

4. Describe the end behavior based on the degree and leading coefficient.

5. Sketch the graph using the intercepts and multiplicities.

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Q12. Graph the following rational functions using the parts listed below:

Background

Topic: Rational Functions

This question tests your ability to analyze and graph rational functions, including finding asymptotes and intercepts.

Key Terms and Formulas:

• Vertical asymptote: Set denominator equal to zero and solve for

• Horizontal asymptote: Compare degrees of numerator and denominator

• X-intercept: Set numerator equal to zero and solve for

• Y-intercept: Evaluate

Step-by-Step Guidance

1. Find the vertical asymptote(s) by setting the denominator equal to zero.

2. Find the horizontal asymptote by comparing the degrees of the numerator and denominator.

3. Find the x-intercept by setting the numerator equal to zero.

4. Find the y-intercept by evaluating the function at .

5. Sketch the graph using the asymptotes and intercepts.

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