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I see that you've uploaded a section of a Grade 10 math homework. The questions listed cover topics in number theory, probability, geometry, and trigonometry. I'll guide you through solving them. Let’s begin with a step-by-step approach:

1. **Question 20:**
   The assertion discusses complementary probability.
   - **Assertion**: The probability of getting a bad egg in a lot of 400 is 0.035, and the number of good eggs is given as 386.
   - **Reason**: If the probability of an event is \( p \), the probability of the complementary event is \( 1 - p \).

   You can solve this by using the complementary rule of probability.

2. **Question 21:**
   You need to show that \( \frac{3 + \sqrt{7}}{5} \) is an irrational number.
   - Rational numbers can be written as the ratio of two integers. To prove that this expression is irrational, assume the contrary and show that it leads to a contradiction.

3. **Question 22:**
   - First part (geometry): Use the formula for the inradius of a triangle where \( a, b, c \) are the sides, and \( r \) is given by the formula \( r = \frac{a + b - c}{2} \).
   - Second part (circle and tangents): Use properties of tangents from an external point and angle bisector properties to prove the given equation.

4. **Question 23:**
   You need to prove \( PT \cdot QR = PR \cdot ST \). This involves using geometric properties of cyclic quadrilaterals and similar triangles.

5. **Question 24:**
   - First part (trigonometric identity): Simplify the expression \( \frac{\csc x - \sin^2 x - \cos^2 x}{\sin^2 x} = 1 \) using known trigonometric identities.
   - Second part (statements on sine and cosec): Analyze how values of sine and cosecant behave at different angles.

6. **Question 25:**
   For this question, calculate the area of the shaded region in the rhombus. The arcs form segments of circles. You'll need to calculate the area of these segments and subtract them from the area of the rhombus.

Would you like me to go deeper into any particular question or provide solutions for any of these questions? Let me know if you have specific ones in mind!

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Here are 5 related questions to expand on the above:

1. How do you prove that a number involving square roots is irrational?
2. Can you explain the relationship between probability and complementary events with more examples?
3. What are the properties of tangents drawn from a common external point to a circle?
4. How can you use the sine and cosine identities to simplify complex trigonometric expressions?
5. How do you calculate areas of circular segments or sectors in different geometric figures?

**Tip:** For trigonometric identities, remember to always start by writing down the fundamental identities, such as \( \sin^2 x + \cos^2 x = 1 \), to guide your simplifications.

Solve a series of math problems including topics such as irrational numbers, geometry, trigonometric identities, and circle properties.

This Grade 10 math homework covers a range of topics including the proof of irrational numbers, geometric properties of triangles and circles, complementary probability, and trigonometric identities. Ideal for students preparing for advanced math concepts.

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