https://file.aiquickdraw.com/mathbot/file/1725621246832eub5b76c.jpg

The image contains a list of mathematical problems. Here's a breakdown of the questions:

1. **If 1, \(\omega, \omega^2\) are the cube roots of unity, then show that** 
   \[
   (1 - \omega + \omega^2)^3 = (1 + \omega - \omega^2)^3 = -8
   \]

2. **Prove that** 
   \[
   \frac{\sqrt{1-\sin \theta}}{\sqrt{1+\sin \theta}} = \sec \theta + \tan \theta
   \]

3. **Simplify**
   \[
   \frac{\cos 10^\circ + \sin 10^\circ}{\cos 40^\circ \cdot \cos 50^\circ} + \frac{\cos 40^\circ + \sin 40^\circ}{\cos 10^\circ \cdot \cos 50^\circ} + \frac{\cos 50^\circ + \sin 50^\circ}{\cos 10^\circ \cdot \cos 40^\circ}
   \]

4. **Express \(1 + i\) in polar form.**

5. **Find the modulus and argument of the complex number \(1 + \sqrt{3} i\).**

6. **Find the focus, directrix, latus rectum, and axis of the parabola** 
   \[
   x^2 = 4y
   \]

7. **Find the vertex, focus, directrix, latus rectum, and axis of the parabola** 
   \[
   4x - 3y = 10
   \]

8. **Find the equation of the straight line passing through the points (0,1) and (2,3).**

9. **Find the equation of the straight line parallel to the x-axis at a distance of 3 units below it.**

10. **Find the equation of the straight line which cuts off intercepts of 2 and 4 units on the axes.**

11. **Find the equation of the straight line which has an intercept of \(\sqrt{2}\) from the negative direction of the y-axis and makes an angle of \(45^\circ\) with the x-axis.**

12. **Find the square root of the complex number \(6 - 8i\).**

Let me know which specific problem you'd like to work on first or if you'd like detailed explanations for any of the questions!



If 1, ω, ω² are the cube roots of unity, then show that (1 - ω + ω²)³ = (1 + ω - ω²)³ = -8, and related mathematical problems.

This page covers solutions to advanced mathematical problems, including cube roots of unity, complex numbers, parabolas, and trigonometric identities. Learn step-by-step methods for simplifying trigonometric expressions, finding the square root of complex numbers, and working with equations of parabolas and straight lines.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation