Question - Multiple Mathematics Problems Involving Algebra, Geometry, and Complex Numbers

First question: Find the 4th term from the end in the expansion of \( \left( \frac{3}{x^2} - x^3 \right)^7 \).

The 4th term from the end is the same as the 4th term from the beginning, which corresponds to \( T_4 \) in the expansion:

\( T_k = \binom{n}{k-1} \cdot (a)^{n-(k-1)} \cdot (b)^{k-1} \) where \( n = 7 \), \( a = \frac{3}{x^2} \), and \( b = -x^3 \).

\( T_4 = \binom{7}{3} \cdot \left(\frac{3}{x^2}\right)^{7-3} \cdot (-x^3)^3 \)

\( T_4 = 35 \cdot \left(\frac{3}{x^2}\right)^4 \cdot (-x^3)^3 \)

\( T_4 = 35 \cdot \frac{81}{x^8} \cdot (-x^9) \)

\( T_4 = 35 \cdot 81 \cdot \frac{-x^9}{x^8} \)

\( T_4 = -2835 \cdot \frac{1}{x} \)

Second question: Find the equation of lines passing through (1,2) and making angle 30° with y-axis.

The slope of the line making a 30° angle with the y-axis is the tangent of (90°-30°), which is \( \tan(60°) \).

Slope (m) of the desired line: \( m = \tan(60°) = \sqrt{3} \)

Use the point-slope form \( y - y_1 = m(x - x_1) \) to find the equation of the line passing through (1,2).

\( y - 2 = \sqrt{3}(x - 1) \)

Equation of the line: \( y = \sqrt{3}x + (2 - \sqrt{3}) \)

Third question: Find the domain and range of the function \( f(x) = \frac{1}{\sqrt{9 - x^2}} \).

The domain is where the function is defined and the denominator is not zero.

\( 9 - x^2 > 0 \)

\( -3 < x < 3 \), so the domain is \( (-3, 3) \).

Since the function is the reciprocal of a square root, its range is all positive real numbers, \( (0, \infty) \).

Fourth question: Solve the system of equations \( Re(z) = 0, |z| = 2 \).

A complex number \( z = x + yi \) satisfies \( Re(z) = 0 \) when \( x = 0 \).

Using \( |z| = 2 \), we get \( |0 + yi| = 2 \), which means \( \sqrt{0^2 + y^2} = 2 \).

\( y = \pm 2 \), so \( z = 0 \pm 2i \).

Fifth question: Write the complex number \( 1 + 7i \) in polar form.

Let \( z = 1 + 7i \).

Magnitude \( r = |z| = \sqrt{1^2 + 7^2} = \sqrt{50} = 5\sqrt{2} \).

Angle \( \theta = \tan^{-1}\left(\frac{7}{1}\right) \), which is in the first quadrant.

Polar form: \( z = r(\cos \theta + i \sin \theta) \)

\( z = 5\sqrt{2}\left(\cos \left(\tan^{-1} 7\right) + i \sin \left(\tan^{-1} 7\right)\right) \)