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The document is a mathematics test or assignment for Class XI Science, covering topics such as trigonometry, complex numbers, and angle conversions. Here's an overview of the questions:

### Section A (2 marks each, 8 marks total)
1. Convert an angle from degrees to degrees, minutes, and seconds.
2. Convert 1560 degrees into radians.
3. Find trigonometric functions of \(-210^\circ\).
4. Show that \(\left( \frac{\sqrt{3}}{2} + \frac{i}{2} \right)^3 = i\).

### Section B (3 marks each, 9 marks total)
1. Simplify \((1 + 2i)^{-3}\) and express it in the form \(a + ib\).
2. Express \(2(2 - i)(2 + i)^{-1}\) in the form \(a + ib\) and find \(a\) and \(b\).
3. If \(\csc\theta - \cot\theta = q\), find \(\tan\theta\) and \(\cot\theta\).

### Section C (4 marks each, 8 marks total)
1. Prove:
   a) \(\sin^2 \frac{\pi}{6} + \cos^2 \frac{\pi}{3} - \tan^2 \frac{\pi}{4} = -\frac{1}{2}\).
   b) \(2\sin^2 \frac{\pi}{6} + \csc^2 \frac{7\pi}{6} \cos^2 \frac{\pi}{3} = \frac{3}{2}\).
2. Simplify \((1 + 2i)(1 + 3i)(2 + i)^{-3}\) and express it in the form \(a + ib\).

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Would you like detailed solutions or assistance with specific problems?

Here are five related questions:
1. How do you convert degrees to radians?
2. What are the steps to simplify a complex number raised to a power?
3. How do you find trigonometric values for negative angles?
4. What is the method to prove trigonometric identities?
5. How do you simplify complex expressions involving multiplication and division?

**Tip**: Always remember the relation \( \pi \text{ radians} = 180^\circ \) when converting between degrees and radians!

Mathematics test for Class XI Science covering trigonometry, complex numbers, and angle conversions.

Class XI Science Mathematics test featuring problems on trigonometry, complex numbers, and angle conversion. Includes conversions from degrees to radians, simplification of complex numbers, and trigonometric identities. Perfect for Grade 11 students.

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