The image contains a set of mathematical questions covering various topics, including set theory, functions, calculus, combinatorics, statistics, and algebra. Here's a brief breakdown of each question and how to approach them:

Q.1 True or False with Reasons:

1. a) Collection of rich persons in India forms a set.

   • Answer: False. The term "rich" is subjective and not well-defined, so it doesn't form a set.

2. b) Given diagram: Determine if it's a function.

   • Answer: True. Each element in set A is mapped to exactly one element in set B.

3. c) Derivative problem: $\frac{d}{dx} (9 - 7x)^5 = 45(9 - 7x)^4$

   • Answer: False. Applying the chain rule correctly: $\frac{d}{dx} (9 - 7x)^5 = 5(9 - 7x)^4 \times (-7) = -35(9 - 7x)^4.$

4. d) In exclusive method, upper limit of a class is included in the same class.

   • Answer: True. In the exclusive method, the upper limit is part of the class interval.

5. e) The order of the matrix $\begin{bmatrix} 2 & 5 & 6 \\ 4 & 3 & 1 \end{bmatrix}$ is $3 \times 2$.

   • Answer: False. The matrix has 2 rows and 3 columns, so the order is $2 \times 3$.

Q.2 Combinatorics with Playing Cards:

1. a) All cards of the same suit.

   • Total ways = $\binom{4}{1} \times \binom{13}{4}$.

2. b) All cards red.

   • Total ways = $\binom{26}{4}$.

3. c) All face cards.

   • Total ways = $\binom{12}{4}$.

4. d) All cards are kings.

   • Total ways = $\binom{4}{4} = 1$.

5. e) All cards of different suits.

   • Total ways = $13^4$.

Q.3 Stem-and-Leaf Plot:

• Arrange the numbers with a stem of 5. Split numbers by the tens place as the stem and the units place as the leaf.

Q.4 Set Theory Verification:

• Verify De-Morgan's laws and the left distributive law using the given sets.

Q.5 Integration:

1. a) $\int x^2 e^{2x} \, dx$

   • Use integration by parts.

2. b) $\frac{dy}{dx}$ for $y = (4x + 5)^4 (9x + 4)^5$

   • Use the product rule.

Q.6 Matrix Proofs and Data:

1. a) Matrix identity proof involving the expression $a^2 - b^2 - c^2$.
2. b) Define primary and secondary data with examples.

Q.7 Box Plot:

• Draw a box plot for the given data, indicating quartiles, median, and outliers.

Q.8 Continuity and Histogram:

1. a) Determine $a$ and $b$ so the function is continuous at $x = 2$.
2. b) Draw a histogram and frequency polygon for the provided data.

Would you like a detailed solution for any specific question, or do you need help understanding the concepts?

Relative Questions:

1. What is the definition of a function in mathematics?
2. How do you use the chain rule in differentiation?
3. How is a stem-and-leaf plot constructed?
4. What are De-Morgan's laws in set theory?
5. What does it mean for a function to be continuous at a point?

Tip:

When working with derivatives involving products or compositions of functions, remember to apply the product and chain rules correctly to avoid common mistakes.