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### SECTION C (Short Answer Questions, 2 marks each)

**Q26.** Find the range of the function \( f(x) = \frac{1}{x-2} \).

**Q27.** If \( A = (4, -1), B = (-3, 2), C = (x, y) \) and \( B \) lies in the second quadrant, find the values of \( x \) and \( y \) such that \( A, B, C \) are collinear.

**Q28.** Let \( A, B, C \) be three sets such that \( A = \{ 1, 2, 3, 4 \}, B = \{ 1, 3, 5 \}, C = \{ 2, 4, 6 \} \). Show that \( A - (B \cup C) = (A - B) \cap (A - C) \).

**Q29.** Find the sum of the lengths of the semi-major and semi-minor axes of the ellipse:  
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]

**Q30.** By using a binomial theorem, show that \( 7^n - 3^n \) is divisible by 4 for all positive integers \( n \).

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### SECTION D (Long Answer Questions, 4 marks each)

**Q31.** Find the value of \( \lim_{x \to 0} \frac{(a^x - b^x)}{x} \) where \( a, b > 0 \).

**Q32.** Solve the following differential equation:  
\[ \frac{dy}{dx} + \frac{3x^2}{y} = 0 \]

**Q33.** Find the equation of the circle passing through the vertices of a triangle whose vertices are represented by the complex numbers \( z_1 = 1 + i \), \( z_2 = -1 - i \), and \( z_3 = 1 - i \).

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Would you like to go over the detailed solutions for any of these questions? If yes, please specify which one.

Here are some follow-up questions that can expand your understanding:

1. How do you determine the range of a function like \( f(x) = \frac{1}{x-2} \)?
2. What conditions must be met for three points to be collinear in coordinate geometry?
3. What is the significance of the binomial theorem in proofs involving divisibility?
4. How is the limit involving exponential functions calculated when \( x \to 0 \)?
5. How do you approach solving first-order differential equations like \( \frac{dy}{dx} + \frac{3x^2}{y} = 0 \)?

**Tip:** When solving limits involving exponentials, using logarithmic expansions can often simplify the problem, especially for small values of \( x \).

Find the range of the function f(x) = 1 / (x - 2).

Explore how to find the range of the function f(x) = 1 / (x - 2). This detailed solution covers essential mathematical concepts for rational functions and is suitable for Grades 10-12 students.

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