Q1. If and , where and are in the interval , evaluate .

Background

Topic: Trigonometric Identities and Angle Addition Formulas

This question tests your ability to use trigonometric identities, specifically the sine addition formula, and to determine the values of trigonometric functions given information about angles in specific quadrants.

Key Terms and Formulas

• Recall the signs of trigonometric functions in different quadrants.

Step-by-Step Guidance

1. Start by drawing reference triangles for and in the correct quadrants (since ).

2. For , use to find and . Remember that is negative in quadrants II and IV, but since , $A$ is in quadrant III or IV.

3. For , use to find and . is negative in quadrants II and III, but so $B$ is in quadrant III.

4. Express , , , and in terms of their reference triangles, making sure to use the correct signs based on the quadrant.

5. Substitute these values into the formula and simplify as much as possible.

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Q2. Find all values of in the interval that satisfy the equation: .

Background

Topic: Solving Trigonometric Equations

This question tests your ability to manipulate and solve trigonometric equations within a given interval.

Key Terms and Formulas

• Standard values and zeros of , , and on

Step-by-Step Guidance

1. Rewrite as to simplify the equation.

2. Group like terms and factor the equation if possible.

3. Set each factor equal to zero and solve for in .

4. Check for extraneous solutions, especially where or may be undefined.

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Q3. Find a formula for the inverse of .

Background

Topic: Inverse Functions

This question tests your ability to find the inverse of a rational function by solving for in terms of and then interchanging variables.

Key Terms and Formulas

• Inverse function: If , then

• To find the inverse, solve for in terms of , then swap $x$ and $y$

Step-by-Step Guidance

1. Let .

2. Multiply both sides by to clear the denominator.

3. Expand and collect all terms involving on one side and constants on the other.

4. Solve for in terms of .

5. Swap and to write .

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Q4. Prove the identity:

Background

Topic: Trigonometric Identities

This question tests your ability to manipulate and prove trigonometric identities using algebraic and trigonometric properties.

Key Terms and Formulas

• Pythagorean identities:

Step-by-Step Guidance

1. Start with the left side: .

2. Multiply numerator and denominator by to rationalize the denominator.

3. Simplify the denominator using the Pythagorean identity.

4. Express the result in terms of .

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Q5. Solve for :

Background

Topic: Exponential Equations

This question tests your ability to solve equations involving exponential functions by substitution and factoring.

Key Terms and Formulas

• Let to reduce the equation to a quadratic in .

• Quadratic formula:

• Properties of logarithms and exponentials.

Step-by-Step Guidance

1. Let , so the equation becomes .

2. Factor or use the quadratic formula to solve for .

3. Set equal to each solution for and solve for using logarithms.

4. Check for extraneous solutions (e.g., must be positive).

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Q6. Find the limit if it exists (Do not use L'Hospital's Rule):

Background

Topic: Limits and Continuity

This question tests your understanding of evaluating limits using algebraic manipulation, factoring, rationalization, and the Squeeze Theorem.

Key Terms and Formulas

• Limit laws

• Squeeze Theorem

• Algebraic simplification techniques

Step-by-Step Guidance

1. Identify the form of the limit (e.g., , , etc.).

2. Try to factor, rationalize, or otherwise simplify the expression.

3. If the limit is indeterminate, consider using the Squeeze Theorem or other limit properties.

4. Substitute the value if possible after simplification.

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Q7. Find the horizontal asymptote(s) of the curve .

Background

Topic: Asymptotes of Rational and Exponential Functions

This question tests your ability to find horizontal asymptotes by analyzing the end behavior of the function as and .

Key Terms and Formulas

• Horizontal asymptote: and

• Dominant terms for large

Step-by-Step Guidance

1. Analyze the behavior as : Which terms dominate in numerator and denominator?

2. Analyze the behavior as .

3. Compute the limits to determine the horizontal asymptotes.

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Q8. Use the Intermediate Value Theorem to show that the equation has at least one real root.

Background

Topic: Intermediate Value Theorem (IVT)

This question tests your understanding of the IVT and how to apply it to show the existence of a root for a continuous function.

Key Terms and Formulas

• IVT: If is continuous on and and have opposite signs, then there exists such that .

Step-by-Step Guidance

1. Define .

2. Choose two values and in (or another interval) and compute and .

3. Show that and have opposite signs.

4. State that is continuous on .

5. Conclude by the IVT that there is at least one root in .

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Q9. Use the definition of derivative to find the derivative of (No marks will be given for computing the derivative by any other method).

Background

Topic: Definition of the Derivative

This question tests your ability to use the limit definition of the derivative to compute for a given function.

Key Terms and Formulas

• Definition:

Step-by-Step Guidance

1. Write and .

2. Compute and simplify the numerator.

3. Divide by and simplify further.

4. Take the limit as to find .

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Q10. Find (You do not need to simplify your answer):

Background

Topic: Product Rule, Chain Rule, and Derivatives of Logarithmic, Exponential, and Trigonometric Functions

This question tests your ability to differentiate composite and product functions using the appropriate rules.

Key Terms and Formulas

• Product Rule:

• Chain Rule:

• Derivatives of , , , ,

Step-by-Step Guidance

1. For each part, identify which rules (product, chain, etc.) are needed.

2. Apply the product rule where appropriate.

3. Apply the chain rule for composite functions.

4. Write out the derivative expressions, but do not simplify.

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