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!