Q1. Solve the equation: $\ln x + \ln(x-3) = 0$

Background

Topic: Exponential and Logarithmic Equations

This question tests your understanding of logarithmic properties and how to solve equations involving natural logarithms.

Key Terms and Formulas

• Natural logarithm: $\ln x$ is the logarithm base $e$.

• Logarithm property: $\ln a + \ln b = \ln(ab)$

• To solve $\ln y = 0$, recall that $y = 1$.

Step-by-Step Guidance

1. Combine the two logarithms using the property $\ln a + \ln b = \ln(ab)$ to write the equation as a single logarithm.

2. Set the argument of the logarithm equal to $1$ (since $\ln 1 = 0$).

3. Write the resulting quadratic equation and rearrange it into standard form.

4. Set up the quadratic formula or factor the equation to find possible values for $x$.

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Q2. Find the limit: $\displaystyle \lim_{x \to 0} \frac{\sin(5x)}{2x}$

Background

Topic: Limits and Trigonometric Functions

This question tests your understanding of limits involving trigonometric functions, especially the standard limit $\lim_{x \to 0} \frac{\sin x}{x} = 1$.

Key Terms and Formulas

• Standard limit: $\lim_{x \to 0} \frac{\sin x}{x} = 1$

• Substitution and scaling: $\lim_{x \to 0} \frac{\sin(ax)}{bx} = \frac{a}{b}$

Step-by-Step Guidance

1. Recognize the form $\frac{\sin(ax)}{bx}$ and recall the standard limit result.

2. Rewrite the numerator to match the denominator by factoring constants if needed.

3. Apply the standard limit property to evaluate the limit in terms of $a$ and $b$.

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Q3. Find the values of $a$ and $b$ so that the function $f(x) = \begin{cases} x^2 - a & x \leq 1 \\ \frac{3x^2 + 12x - b}{x^2 + 2x - 3} & x > 1 \end{cases}$ is continuous on $(-\infty, \infty)$.

Background

Topic: Continuity of Piecewise Functions

This question tests your ability to ensure continuity of a piecewise-defined function by matching function values at the point where the definition changes.

Key Terms and Formulas

• Continuity at a point: $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$

• For piecewise functions, set the left and right limits equal at the transition point.

Step-by-Step Guidance

1. Identify the transition point, which is $x = 1$.

2. Compute $\lim_{x \to 1^-} f(x)$ using the first piece: $x^2 - a$.

3. Compute $\lim_{x \to 1^+} f(x)$ using the second piece. Simplify the denominator and numerator at $x = 1$.

4. Set the two limits equal and solve for $a$ and $b$.

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Q4. Compute the limit: $\displaystyle \lim_{h \to 0} \frac{\ln(1 + 3h)}{7h}$

Background

Topic: Limits and Derivatives (Logarithmic Functions)

This question tests your understanding of the definition of the derivative and the use of standard limits involving logarithms.

Key Terms and Formulas

• Standard limit: $\lim_{h \to 0} \frac{\ln(1 + h)}{h} = 1$

• Chain rule for limits: $\lim_{h \to 0} \frac{\ln(1 + ah)}{bh} = \frac{a}{b}$

Step-by-Step Guidance

1. Recognize the form of the limit as similar to the standard derivative of $\ln(x)$ at $x = 1$.

2. Let $k = 3h$ to rewrite the limit in terms of $k$.

3. Express the limit in terms of $k$ and $h$, and relate it to the standard limit.

4. Apply the result to find the value in terms of $a$ and $b$.

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Q5. Using the Intermediate Value Theorem, determine on which of the intervals (0, 1), (1, 2), (2, 3), (3, 4) the equation $x^3 - 3x = 5$ has a root.

Background

Topic: Intermediate Value Theorem (IVT)

This question tests your understanding of the IVT, which states that if a function is continuous on $[a, b]$ and takes values of opposite sign at $a$ and $b$, then it must cross zero somewhere in $(a, b)$.

Key Terms and Formulas

• Intermediate Value Theorem: If $f$ is continuous on $[a, b]$ and $f(a)$ and $f(b)$ have opposite signs, then $f$ has a root in $(a, b)$.

• Check $f(a)$ and $f(b)$ for each interval.

Step-by-Step Guidance

1. Define $f(x) = x^3 - 3x - 5$.

2. Compute $f(x)$ at the endpoints of each interval: $x = 0, 1, 2, 3, 4$.

3. Check the sign of $f(x)$ at each endpoint.

4. Identify intervals where $f(x)$ changes sign between endpoints.

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